Research Article Lattice-Valued Topological Systems as a...

Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 506275 33 pageshttpdxdoiorg1011552013506275

Research ArticleLattice-Valued Topological Systems as a Framework forLattice-Valued Formal Concept Analysis

Sergey A Solovyov12

1 Department of Mathematics and Statistics Faculty of Science Masaryk University Kotlarska 2 611 37 Brno Czech Republic2 Institute of Mathematics and Computer Science University of Latvia Raina bulvaris 29 Riga 1459 Latvia

Correspondence should be addressed to Sergey A Solovyov solovjovsmathmunicz

Received 26 August 2012 Accepted 9 November 2012

Academic Editor Alfred Peris

Copyright copy 2013 Sergey A Solovyov This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recently Denniston Melton and Rodabaugh presented a new categorical outlook on a certain lattice-valued extension of FormalConcept Analysis (FCA) of Ganter and Wille their outlook was based on the notion of lattice-valued interchange system anda category of Galois connections This paper extends the approach of Denniston et al clarifying the relationships between Chuspaces of Pratt many-valued formal contexts of FCA lattice-valued interchange systems and Galois connections

1 Introduction

This paper considers a particular application of the theoryof variety-based topological systems introduced in [1] as ageneralization of topological systems of Vickers [2] whichin turn provide a common framework for both topologi-cal spaces and their underlying algebraic structures frames(also called locales) thereby allowing researchers to switchfreely between the spatial and localic viewpoints Vickersrsquoconcept whichwasmotivated by the theory of crisp topologyhas recently drawn the attention of fuzzy topologists whohave incorporated the notion in their topic of study Thefirst attempt in this respect was done by Denniston andRodabaugh [3] who considered various functorial relation-ships between topological systems of Vickers and lattice-valued topological spaces of Rodabaugh [4] Soon afterwardsthe study of Guido [5 6] followed he used topologicalsystems in order to construct a functor from the categoryof 119871-topological spaces to the category of crisp topologicalspaces thereby providing a common framework for variousapproaches to the hypergraph functor of the fuzzy commu-nity (see eg [7])

It soon appeared though that the translation of themost important tools of the theory of topological systems(eg system spatialization and localification procedures) intothe language of lattice-valued topology required a suitable

fuzzification of the former concept which was accomplishedsuccessfully by Denniston et al in [8] The authors broughttheir theory into maturity in [9] where they considered itsapplications to both lattice-valued variable-basis topology ofRodabaugh [4] and (119871119872)-fuzzy topology of Kubiak andSostak [10] Later on they introduced a particular instanceof their concept called interchange system [11] which wasmotivated by certain aspects of program semantics (the so-called predicate transformers) initiated by Dijkstra [12] Itshould be noted however that an interchange system is aparticular instance of the well-known concept of Chu spacein the sense of Pratt [13] (the original notion goes back to Barr[14])Moreover the interchange systemmorphisms of [11] areprecisely the Chu space morphisms (called Chu transforms)of [13] (In [15] Denniston et al use the terms transformersystem and transformer morphism) The underlying moti-vations of interchange systems and Chu spaces are quitedifferent which is reflected well enough in both their namesand their respective theories

Parallel to the previously mentioned developments weintroduced the concept of variety-based topological system[1] which not only extended the setting of Denniston et albut also included the case of state property systems of Aerts[16ndash18] (introduced as the basic mathematical structure inthe Geneva-Brussels approach to foundations of physics)considered in [19] in full detail and we brought the functor of

2 Journal of Mathematics

Guido to a new level [20] (eg made it variable-basis in thesense of Rodabaugh [4] as well as extended the machinery ofHohle [7] to construct its right adjoint functor) Moreoverin [21] we provided a thorough categorical extension of thesystem spatialization procedure to the variety-based settingthe respective localification procedure being elaborated in[22] Additionally certain aspects of the new theory bearinglinks to bitopological spaces of Kelly [23] as well as tononcommutative topology of Mulvey and Pelletier [24 25]were treated extensively in [26 27] respectively

Seeing the fruitfulness of their newly introduced notionsboth Denniston et al and the author of this paper indepen-dently turned their attention to possible applications of thearising system framework to the areas beyond topology Inparticular in [28] we started the theory of lattice-valuedsoft universal algebra whereas Denniston et al presentedin [29 30] a challenging categorical outlook on a certainlattice-valued extension of Formal Concept Analysis (FCA)of Ganter and Wille [31] Similar to the case of lattice-valuedtopological systems their research motivated us to take adeeper look into the proposed topic of lattice-valued FCAthereby streamlining the motivating approach of Dennistonet al and also extending some of their results

Modifying slightly the definition of the category of Galoisconnections of McDill et al [32] Denniston et al [29]consider its relationships to the already mentioned categoryof interchange systems the objects of both categories beingessentially formal contexts of FCA The main result is acategorical embedding of each category into its counterpartthe ultimate conclusion though amounting to the categoriesin question being rather different that is the two viewpointson FCA are not categorically isomorphic The rest of thepaper provides a lattice-valued fixed-basis extension of thecrisp achievements based on the lattice-valued FCA ofBelohlavek [33] Moreover towards the end of the paperwith the help of the constructed embeddings the authorscompare the categories of (lattice-valued) formal contextsand (lattice-valued) topological systems inside the categoryof (lattice-valued) interchange systems making a significant(and rather striking) metamathematical conclusion on theirbeing disjoint

The just mentioned approach of Denniston et al canbe extended taking into consideration the following factsFirstly it never gives much attention to the links betweeninterchange systems and Chu spaces thereby making nouse of the well-developed machinery of the latter Secondlywhile pursuing their approach to many-valued FCA theauthors never mention the already existing many-valuedformal contexts of Ganter and Wille [31] which (beingsimilar to Chu spaces) essentially extend their own Thirdlybeing categorically oriented the paper introduces two typesof formal context morphisms in terms of Chu transformsand Galois connections respectively but misses the well-established context morphisms of FCA which are differentfrom those of the paper Lastly (andmost important) turningto fuzzification of their notions the authors concentrateon the fixed-basis lattice-valued approach notwithstandingthat the more promising variable-basis setting (initiated byone of them) has already gained its popularity in the fuzzy

community currently being a kind of de facto standard forfuzzy structures

The main purposes of this paper are to extend the resultsof Denniston et al and to clarify the relationships betweenChu spaces many-valued formal contexts of FCA lattice-valued interchange systems and Galois connections Inparticular we introduce three categories (each of themhavinga twofold modification arising from its respective variable-basis) whose objects are lattice-valued versions of formalcontexts of FCA but whose morphisms are quite differenteach instance reflecting the approach of Pratt Ganter andWille andDenniston et al respectively Following the resultsof [29] we embed (nonfully) each of the categories into itstwo counterparts showing however that the constructedembeddings fail to provide a categorical isomorphism (evenan adjoint situation) between any two of the presentedthree approaches As a consequence we obtain three dif-ferent categorical outlooks on FCA which are done in thevariable-basis lattice-valued framework of [4] Moreoverthe respective underlying lattices of the fuzzified formalcontexts are quantales [34 35] instead of complete residuatedlattices of Denniston et al that is the heavy assumption oncommutativity of the quantale multiplication is dropped Inparticular we show lattice-valued generalizations of formalconcept preconcept and protoconcept of [36] which arebased on an arbitrary (namely non-commutative) quantale(actually an algebra having a quantale as its reduct)

Similar to Denniston et al we develop the requiredmachinery of quantale-valued Galois connections whichextends that of Erne et al [37] and which is motivatedby the commutative approach of fuzzy Galois connectionsof Belohlavek [38] For example we provide yet anothergeneralization to the quantale setting of the well-knownfact that every order-reversing Galois connection betweenpowersets arises as a pair of Birkhoff operators with respectto a binary relation [39] The generalization though appearsto be somewhat truncated in the sense that while lattice-valued binary relations still give rise to order-reversingGaloisconnections the converseway is not always available ([38 40]do restore the full result even in the lattice-valued case butrely on the more demanding notion of fuzzy Galois con-nection) Moreover we define a category of order-preservingGalois connections as a certain subcategory of the categoryof Chu spaces in which (unlike McDill et al and Dennistonet al) Galois connections play the role of morphisms ratherthan objects thereby providing particular instances of Chutransforms We fail to provide a similar machinery for theorder-reversing case

At the end of the paper we will notice that the approachto FCA through order-reversing Galois connections takenup by Denniston et al does not suit exactly the underlyingideas of FCA of Ganter and Wille which are formulatedin terms of the incidence relations of formal contexts Themain sticking point here is the previously mentioned factthat the important crisp case one-to-one correspondencebetween order-reversing Galois connections on powersetsand pairs of Birkhoff operators breaks down in the lattice-valued case (quantales) As a consequence the incidencerelation viewpoint provides one with a Galois connection

Journal of Mathematics 3

whereas the latter alone lacks the strength of bringing arespective incidence relation in play (cf Problems 1 and 2)

The paper is based on both category theory and universalalgebra relyingmore on the formerThenecessary categoricalbackground can be found in [41ndash43] For algebraic notionsthe reader is referred to [34 35 44 45] Although we triedto make the paper as much self-contained as possible itis expected from the reader to be acquainted with basicconcepts of category theory that is those of category andfunctor

2 Algebraic Preliminaries

For convenience of the reader we begin with those algebraicpreliminaries which are crucial for the understanding of theresults of the paper Experienced researchers may skip thissection consulting it for the notations of the author only

21 Varieties of Algebras An important foundational aspectof the paper is the abstract algebraic structure (called algebrafor short) which is a set equippedwith a family of operationssatisfying certain identities The theory of universal algebracalls a class of finitary algebras (induced by a set of finitaryoperations) which is closed under the formation of homo-morphic images subalgebras and direct products a varietyIn this paper we extend the notion of varieties to includeinfinitary algebraic theories Our motivation includes ideasof [45ndash47]

Definition 1 LetΩ = (119899120582)120582isinΛ be a (possibly proper or empty)class of cardinal numbers AnΩ-algebra is a pair (119860 (120596119860

120582)120582isinΛ)

which comprises a set 119860 and a family of maps 119860119899120582120596119860

120582

997888997888rarr

119860 (119899120582-ary primitive operations on 119860) An Ω-homomorphism(1198601 (120596

1198601

120582)120582isinΛ)

120593

997888rarr (1198602 (1205961198602

120582)120582isinΛ) is a map 1198601

120593

997888rarr 1198602 suchthat the diagram

1198601 1198602

1198601198991205821 119860

1198991205822

1205961198601120582

1205961198602120582

120593

120593

119899120582

(1)

commutes for every 120582 isin Λ Alg(Ω) is the construct of Ω-algebras andΩ-homomorphisms

Every concrete category of this paper has an underlyingfunctor | minus | to the respective ground category (eg thecategory Set of sets andmaps) the latter mentioned explicitlyin each case

Definition 2 Let M (resp E) be the class of Ω-homomorphisms with injective (resp surjective) underlyingmaps A variety of Ω-algebras are a full subcategory ofAlg(Ω) closed under the formation of products M-subobjects (subalgebras) and E-quotients (homomorphicimages) The objects (resp morphisms) of a variety arecalled algebras (resp homomorphisms)

Definition 3 Given a varietyA a reduct ofA is a pair (minusB)where B is a variety such that ΩB sube ΩA and A

minus

997888997888rarr B is aconcrete functorThe pair (A minus) is called then an extensionof B

The following definitions recall several well-known vari-eties [34 35 48 49] thereby providing the background forthe subsequent developments of this paper

Definition 4 CSLat (Ξ) is the variety of Ξ-semilattices thatis partially ordered sets (posets) which have arbitrary Ξ isin

⋀⋁ (meets and joins resp)

Given aΞ-semilattice119860 we have both⋀ and⋁ (howeveronly one of them is respected by Ξ-semilattice homomor-phisms) and denote its largest (resp smallest) element⊤119860 (resp perp119860) Given a ⋁-semilattice 119860 we denote ≀119860≀

its respective ⋀-semilattice and vice versa The standardresult of [50 Sections 0ndash3] says that every ⋁-semilatticehomomorphism 1198601

120593

997888rarr 1198602 has the upper adjoint map

|1198602|120593⊢

997888997888rarr |1198601| which is characterized uniquely by thecondition 120593(1198861) ⩽ 1198862 iff 1198861 ⩽ 120593

⊢(1198862) for every 1198861 isin

1198601 and every 1198862 isin 1198602 which for order-preserving mapsonly is equivalent to the properties 120593 ∘ 120593

⊢⩽ 11198602

and11198601

⩽ 120593⊢∘ 120593 (given an algebra 119860 1119860 denotes the identity

map on 119860) The explicit formula for the map is given by120593⊢(1198862) = ⋁1198861 isin 1198601 | 120593(1198861) ⩽ 1198862 whereas the map

itself is easily shown to be ⋀-preserving The dual descrip-tion of the lower adjoint map (based on a ⋀-semilattice

homomorphism 1198601

120593

997888rarr 1198602 and denoted by |1198602|120593⊣

997888997888rarr

|1198601|) is left to the reader (see also Section 23 on Galoisconnections)

Definition 5 Quant is the variety of quantales that is triples(119876⋁ otimes) where

(1) (119876⋁) is a⋁-semilattice(2) (119876 otimes) is a semigroup that is 1199021 otimes (1199022 otimes 1199023) = (1199021 otimes

1199022) otimes 1199023 for every 1199021 1199022 1199023 isin 119876(3) otimes distributes across ⋁ from both sides that is 119902 otimes

(⋁ 119878) = ⋁119904isin119878(119902 otimes 119904) and (⋁ 119878) otimes 119902 = ⋁

119904isin119878(119904 otimes 119902) for

every 119902 isin 119876 and every 119878 sube 119876

Every quantale119876 has two residuations which are inducedby its binary operation otimes (called multiplication) and whichare defined by 1199021rarr 119897 1199022 = ⋁119902 isin 119876 | 119902 otimes 1199021 ⩽ 1199022 and1199021rarr 119903 1199022 = ⋁119902 isin 119876 | 1199021 otimes119902 ⩽ 1199022 (notice that ldquo119897rdquo (resp ldquo119903rdquo)stands for ldquomultiply on the leftrdquo (resp ldquorightrdquo)) respectivelyproviding a single residuation sdot rarr sdot in case of a commutativemultiplication (resulting in complete residuated lattices ofDenniston et al [29]) These operations have the standardproperties of poset adjunctions [50 Sections 0ndash3] or (order-preserving) Galois connections [37] (see Section 23 of thispaper) for example 1199021 ⩽ 1199022rarr 119897 1199023 iff 1199021 otimes 1199022 ⩽ 1199023 iff1199022 ⩽ 1199021rarr 119903 1199023 For convenience of the reader the followingtheorem recalls some of their other features which will beheavily used throughout the paper


Theorem 6 The residuations sdotrarr 119897sdot and sdotrarr 119903sdot of a quantale 119876have the following properties

(1) 119902rarr 119897 (⋀119878) = ⋀119904isin119878

(119902rarr 119897 119904) and 119902rarr 119903 (⋀119878) =

⋀119904isin119878(119902rarr 119903 119904) for every 119902 isin 119876 and every 119878 sube 119876 In

particular 119902rarr 119897 ⊤119876 = ⊤119876 and 119902rarr 119903 ⊤119876 = ⊤119876 forevery 119902 isin 119876

(2) (⋁119878)rarr 119897 119902 = ⋀119904isin119878

(119904rarr 119897 119902) and (⋁119878)rarr 119903 119902 =

⋀119904isin119878

(119904rarr 119903 119902) for every 119902 isin 119876 and every 119878 sube 119876 Inparticular perp119876rarr 119897 119902 = ⊤119876 and perp119876rarr 119903 119902 = ⊤119876 forevery 119902 isin 119876

(3) 1199021 ⩽ (1199021rarr 119897 1199022)rarr 119903 1199022 and 1199021 ⩽ (1199021rarr 119903 1199022)rarr 119897 1199022

for every 1199021 1199022 isin 119876(4) (1199021 otimes 1199022)rarr 119897 1199023 = 1199021rarr 119897 (1199022rarr 119897 1199023) and (1199021 otimes

1199022)rarr 119903 1199023 = 1199022rarr 119903 (1199021rarr 119903 1199023) for every 1199021 1199022 1199023 isin119876

Proof To give the flavor of the employedmachinery we showthe proofs of some of the claims of the items

Ad (3) 1199021rarr 119897 1199022 ⩽ 1199021rarr 119897 1199022 implies (1199021rarr 119897 1199022) otimes

1199021 ⩽ 1199022 implies 1199021 ⩽ (1199021rarr 119897 1199022)rarr 119903 1199022Ad (4) Given 119902 isin 119876 119902 ⩽ (1199021 otimes 1199022)rarr 119897 1199023 iff119902 otimes (1199021 otimes 1199022) ⩽ 1199023 iff 119902 otimes 1199021 ⩽ 1199022rarr 119897 1199023 iff 119902 ⩽

1199021rarr 119897 (1199022rarr 119897 1199023)Moreover 119902 ⩽ (1199021 otimes 1199022)rarr 119903 1199023 iff (1199021 otimes 1199022) otimes 119902 ⩽ 1199023 iff

1199022 otimes 119902 ⩽ 1199021rarr 119903 1199023 iff 119902 ⩽ 1199022rarr 119903 (1199021rarr 119903 1199023)

In the subsequent developments we will consider somespecial types of quantales mentioned in the following

Definition 7 CQuant is the variety of commutative quantaleswhich are quantales119876 with commutative multiplication thatis 1199021 otimes 1199022 = 1199022 otimes 1199021 for every 1199021 1199022 isin 119876

Definition 8 UQuant is the variety of unital quantales whichare quantales 119876 having an element 1119876 such that the triple(119876 otimes 1119876) is a monoid that is 119902 otimes 1119876 = 119902 = 1119876 otimes 119902 for every119902 isin 119876

Definition 9 SRSQuant (resp SLSQuant) is the varietyof strictly right-sided (resp left-sided) quantales which arequantales119876 such that 119902otimes⊤119876 = 119902 (resp⊤119876otimes119902 = 119902) for every119902 isin 119876 STSQuant is the variety of strictly two-sided quantalesthat is unital quantales whose unit is the top element

Definition 10 Frm is the variety of frames that is strictlytwo-sided quantales in which multiplication is the meetoperation

Definition 11 CBAlg is the variety of complete Booleanalgebras that is frames 119860 in which ⋀ are considered asprimitive operations (cf Definition 1) and which additionallyare equipped with the complement operation (minus)1015840 assigningto every 119886 isin 119860 an element 1198861015840 isin 119860 such that 119886 and 1198861015840 = perp119860 and119886 or 119886

1015840= ⊤119860

The varieties CBAlg Frm STSQuant UQuant andCSLat(⋁) provide a sequence of variety reducts in the senseof Definition 3 Additionally Frm is an extension of CQuant

The notations related to varieties in this paper followthe category-theoretic pattern (which is different from therespective one of the fuzzy community) From now onvarieties are denoted by A or B whereas those which extendthe variety CSLat(Ξ) of Ξ-semilattices with Ξ isin ⋁⋀

will be distinguished with the notation L S will standfor either subcategories of varieties or the subcategories oftheir dual categories The categorical dual of a variety A isdenoted by A119900119901 However the dual of Frm uses the alreadyaccepted notation Loc [48] Given a homomorphism 120593 thecorresponding morphism of the dual category is denoted by120593119900119901 and vice versa Every algebra119860 of a varietyA gives rise to

the subcategory S119860 of eitherA or A119900119901 whose only morphismis the identity map 119860

1119860997888997888rarr 119860

22 Powerset Operators One of the main tools in the subse-quent developments of the paper will be generalized versionsof the so-called forward and backward powerset operators

More precisely given a map 1198831

119891

997888rarr 1198832 there exists the

forward (resp backward) powerset operator P(1198831)119891rarr

997888997888997888rarr

P(1198832) (resp P(1198832)119891larr

997888997888rarr P(1198831)) defined by 119891rarr(119878) =

119891(119904) | 119904 isin 119878 (resp 119891larr(119879) = 119909 isin 1198831 | 119891(119909) isin 119879) The twooperators admit functorial extensions to varieties of algebras(For a thorough treatment of the powerset operator theorythe reader is referred to the articles of Rodabaugh [49 51ndash53]dealing with the lattice-valued case or to the papers of theauthor himself [21 54] dealing with a more general variety-based approach)

To begin with notice that given an algebra 119860 of a varietyA and a set 119883 there exists the 119860-powerset of 119883 namely theset 119860119883 of all maps 119883 120572

997888rarr 119860 Equipped with the pointwisealgebraic structure involving that of 119860 119860119883 provides analgebra of A In the paper we will often use specific elementsof this powerset algebraMore precisely every 119886 isin 119860 gives riseto the constant map 119883

119886

997888rarr 119860 with the value 119886 Moreover if Aextends the variety CSLat(⋁) of ⋁-semilattices then every

119878 sube 119883 and every 119886 isin 119860 give rise to the map 119883120594119886

119878

997888997888rarr 119860 (119886-characteristic map of 119878) which is defined by

120594119886

119878(119909) =

119886 119909 isin 119878

perp119860 otherwise(2)

Of particular importance will be the characteristic mapsof singletons that is 120594⊤119860

119909or 1205941119860

119909 the latter case assuming

additionally that A extends the variety UQuant of unitalquantales

Coming back to powerset operators the following exten-sion of the forward one has already appeared in a moregeneral way in [28] being different from the respectivelattice-valued approaches of Rodabaugh


Theorem 12 Given a variety L which extends CSLat(⋁)every subcategory S of L provides the functor Set times

S(minus)rarr

997888997888997888997888rarr CSLat(⋁) which is defined by ((1198831 1198711)(119891120593)

997888997888997888rarr

(1198832 1198712))rarr= 119871

1198831

1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)rarr (120572))(1199092) =

120593(⋁119891(1199091)=1199092

120572(1199091))

Proof To show that 1198711198831

1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 is ⋁-

preserving notice that given 119878 sube 1198711198831

1 for every

1199092 isin 1198832 ((119891 120593)rarr(⋁ 119878))(1199092) = 120593(⋁

119891(1199091)=1199092(⋁ 119878)(1199091)) =

120593(⋁119891(1199091)=1199092

⋁119904isin119878119904(1199091)) = 120593(⋁

119904isin119878⋁119891(1199091)=1199092

119904(1199091)) =

⋁119904isin119878120593(⋁

119891(1199091)=1199092119904(1199091)) = ⋁

119904isin119878((119891 120593)

rarr(119904))(1199092) =

(⋁119904isin119878(119891 120593)

rarr(119904))(1199092)

To show that the functor preserves composition

notice that given Set times S-morphisms (1198831 1198711)(119891120593)

997888997888997888rarr

(1198832 1198712) and (1198832 1198712)(11989110158401205931015840)

997888997888997888997888997888rarr (1198833 1198713) for every120572 isin 119871

1198831

1and every 1199093 isin 1198833 ((119891

1015840 120593

1015840)rarr

∘

(119891 120593)rarr(120572))(1199093) = 120593

1015840(⋁

1198911015840(1199092)=1199093((119891 120593)

rarr(120572))(1199092)) =

1205931015840(⋁

1198911015840(1199092)=1199093120593(⋁

119891(1199091)=1199092120572(1199091))) = 120593

1015840∘ 120593(⋁

1198911015840∘119891(1199091)=1199093120572(1199091)) =

((1198911015840∘119891 120593

1015840∘ 120593)

rarr(120572))(1199093) = (((119891

1015840 120593

1015840) ∘ (119891 120593))

rarr(120572))(1199093)

The case S = S119871 is denoted by (minus)rarr

119871and is called a

(variety-based) fixed-basis approach whereas all other casesare subsumed under (variety-based) variable-basis approachSimilar notations and naming conventions are applicableto all other variety-based extensions of powerset operatorsintroduced in this subsection and for the sake of brevity willnot be mentioned explicitly

There is another extension of the forward powersetoperator which uses the idea of Rodabaugh [52] (the readeris advised to recall the notation for lower and upper adjointmaps from the previous subsection) Before the actual resulthowever some words are due to our employed powersetoperator notations In all cases which are direct analogues ofthe classical powerset operators of the fuzzy community (egthose of Rodabaugh [51] and his followers) we use the solidarrow notation that is ldquo(minus)larrrdquo (resp ldquo(minus)rarr rdquo) for backward(resp forward) powerset operator In the cases which are notwell known to the researchers we use dashed arrow notationthat is ldquo(minus)rdquo (resp ldquo(minus)[rdquo) for backward (resp forward)powerset operator possibly adding some new symbols (egldquo⊣rdquo or ldquo⊢rdquo) to underline the involvement of upper or loweradjoint maps in the definition of the operator in question(eg (minus)⊣[ or (minus)⊢[) as in the following theorem

Theorem 13

(1) Given a variety L which extendsCSLat(⋀) every sub-

category S of L119900119901 gives rise to the functor SettimesS(minus)⊣[

997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊣[

=

≀1198711198831

1≀

(119891120593)⊣[

997888997888997888997888997888rarr ≀1198711198832

2≀ where ((119891 120593)

⊣[(120572))(1199092) =

120593119900119901⊣(⋁

119891(1199091)=1199092120572(1199091))

(2) Let L be a variety which extends CSLat(⋁) andlet S be a subcategory of L119900119901 such that for every

S-morphism 1198711

120593

997888rarr 1198712 the map |1198711|120593119900119901⊢

997888997888997888rarr |1198712| is ⋁-

preserving Then there exists the functor Set times S(minus)⊢[

997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊢[

=

1198711198831

1

(119891120593)⊢[

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)

⊢[(120572))(1199092) =

120593119900119901⊢(⋁

119891(1199091)=1199092120572(1199091))

Proof We rely on the fact that every⋀- (resp⋁-) semilatticeis actually a complete lattice and therefore is a ⋁- (resp⋀-) semilattice The proof then follows the steps of those ofTheorem 12 employing the fact that given CSLat(⋀)- (resp

CSLat(⋁)-) homomorphisms 1198711120593

997888rarr 1198712 and 1198712

1205931015840

997888rarr 1198713 itfollows that (1205931015840 ∘120593)⊣ = 120593⊣ ∘1205931015840

⊣

(resp (1205931015840 ∘120593)⊢ = 120593⊢ ∘1205931015840⊢

)

The respective generalization of the backward powersetoperator is even stronger (see eg [21]) in the sense that theemployed varieties do not need to be related to any kind oflattice-theoretic structure

Theorem 14 Given a variety A every subcategory S of A119900119901

induces the functor Set times S(minus)larr

997888997888997888rarr A119900119901 which is defined by

((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr

= 1198601198831

1

((119891120593)larr)119900119901

997888997888997888997888997888997888997888rarr 1198601198832

2 where

(119891 120593)larr(120572) = 120593

119900119901∘ 120572 ∘ 119891

Proof To show that1198601198832

2

(119891120593)larr

997888997888997888997888997888rarr 1198601198831

1is anA-homomorphism

notice that given 120582 isin ΛA and ⟨120572119894⟩119899120582isin (119860

1198832

2)119899120582 for every

1199091 isin 1198831 it follows that

((119891 120593)larr(120596

1198601198832

2

120582(⟨120572119894⟩119899120582

))) (1199091)

= 120593119900119901∘ (120596

1198601198832

2

120582(⟨120572119894⟩119899120582

)) ∘ 119891 (1199091)

= 120593119900119901∘ (120596

1198602

120582(⟨120572119894 ∘ 119891 (1199091)⟩119899120582

))

= 1205961198601

120582(⟨120593

119900119901∘ 120572119894 ∘ 119891 (1199091)⟩119899120582

)

= 1205961198601

120582(⟨((119891 120593)

larr(120572119894)) (1199091)⟩119899120582

)

= (1205961198601198831

1

120582(⟨(119891 120593)

larr(120572119894)⟩119899120582

)) (1199091)

(3)

To show that the functor preserves composition notice

that given Set times S-morphisms (1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602) and

(1198832 1198602)(11989110158401205931015840)

997888997888997888997888997888rarr (1198833 1198603) for every 120572 isin 1198601198833

3 (119891 120593)larr ∘

(1198911015840 120593

1015840)larr(120572) = (119891 120593)

larr(120593

1015840119900119901∘ 120572 ∘119891

1015840) = 120593

119900119901∘ 120593

1015840119900119901∘ 120572 ∘119891

1015840∘ 119891 =

(1205931015840∘ 120593)

119900119901∘ 120572 ∘ 119891

1015840∘ 119891 = (119891

1015840∘ 119891 120593

1015840∘ 120593)

larr(120572) = ((119891

1015840 120593

1015840) ∘

(119891 120593))larr(120572)

Similar to the case of variety-based forward powersetoperators we have the following modifications of the functorintroduced inTheorem 14

Theorem 15 Suppose that either (Ξ1 Ξ2 dagger) = (⋁⋀ ⊢) or(Ξ1 Ξ2 dagger) = (⋀⋁ ⊣) Given a variety L which extends


CSLat(Ξ1) every subcategory S of L provides the functor Settimes

S(minus)dagger

997888997888997888997888rarr (CSLat(Ξ2))119900119901 which is defined by ((1198831 1198711)

(119891120593)

997888997888997888rarr

(1198832 1198712))dagger

= ≀1198711198831

1≀

((119891120593)dagger)119900119901

997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger(120572) =

120593dagger∘ 120572 ∘ 119891

Proof The proof is a straightforward modification of that ofTheorem 14

In the paper we will use the following dualized versionsof the functors of Theorems 12 and 13

Theorem 16

(1) Given a variety L which extends CSLat(⋁) everysubcategory S of L119900119901 gives rise to the functor Set119900119901 times

S(minus)rarr119900

997888997888997888997888rarr (CSLat(⋁))119900119901 defined by ((1198831 1198711)(119891120593)

997888997888997888rarr

(1198832 1198712))rarr119900

= 1198711198831

1

((119891120593)rarr119900

)119900119901

997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)rarr119900

(120572))(1199091) = 120593119900119901(⋁

119891119900119901(1199092)=1199091120572(1199092))

(2) Given a variety L which extends CSLat(⋀) everysubcategory S of L gives rise to the functor Set119900119901 times

S(minus)⊣[119900


997888997888997888rarr

(1198832 L2))⊣[119900

= ≀1198711198831

1≀

((119891120593)⊣[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where

((119891 120593)⊣[119900

(120572))(1199091) = 120593⊣(⋁

119891119900119901(1199092)=1199091120572(1199092))

(3) Let L be a variety which extends CSLat(⋁) andlet S be a subcategory of L such that for every S-

morphism 1198711

120593

997888rarr 1198712 the map |1198712|120593⊢

997888997888rarr |1198711| is⋁-preserving Then there exists the functor Set119900119901 times

S(minus)⊢[119900


997888997888997888rarr

(1198832 1198712))⊢[119900

= 1198711198831

1

((119891120593)⊢[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)⊢[119900

(120572))(1199091) = 120593⊢(⋁

119891119900119901(1199092)=1199091120572(1199092))

Additionally the functors of Theorems 14 and 15 can bedualized as follows

Theorem 17

(1) Given a variety A every subcategory S of A induces

the functor Set119900119901 times S(minus)larr119900

997888997888997888997888rarr A which is defined by

((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr119900

= 1198601198831

1

(119891120593)larr119900

997888997888997888997888997888rarr 1198601198832

2

where (119891 120593)larr119900(120572) = 120593 ∘ 120572 ∘ 119891

119900119901(2) Suppose that either (Ξ1 Ξ2 dagger) = (⋁⋀ ⊢) or

(Ξ1 Ξ2 dagger) = (⋀⋁ ⊣) Given a variety L whichextends CSLat(Ξ1) every subcategory S of L119900119901 pro-

vides the functor Set119900119901 times S(minus)dagger119900

997888997888997888997888997888rarr CSLat(Ξ2)

which is defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))dagger119900

=

≀1198711198831

1≀

(119891120593)dagger119900

997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger119900

(120572) = 120593119900119901dagger

∘ 120572 ∘

119891119900119901

The reader should notice that the functors of Theorem 17are not standard in the fuzzy community

23 Galois Connections Thepaper relies heavily on the well-developedmachinery ofGalois connections For convenienceof the reader we recall from [32 37 50] their basic definitions

Definition 18 An order-preserving (resp -reversing) Galoisconnection is a tuple ((1198831 ⩽) 119891 119892 (1198832 ⩽)) where (1198831 ⩽)

(1198832 ⩽) are posets and1198831

119891

9994489994711198921198832 are maps such that for every

1199091 isin 1198831 and every 1199092 isin 1198832 1199091 ⩽ 119892(1199092) iff 119891(1199091) ⩽ 1199092 (resp1199091 ⩽ 119892(1199092) iff 1199092 ⩽ 119891(1199091))

As a consequence one easily gets the following well-known features of Galois connections

Theorem 19 Every order-preserving (resp order-reversing)Galois connection ((1198831 ⩽) 119891 119892 (1198832 ⩽)) has the followingproperties

(1) 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832(resp 11198831 ⩽ 119892 ∘ 119891 and

11198832⩽ 119891 ∘ 119892)

(2) Both 119891 and 119892 are order-preserving (resp order-reversing)

(3) 119891(⋁119878) = ⋁119904isin119878119891(119904) and 119892(⋀119879) = ⋀

119905isin119879119892(119905) (resp

119891(⋁119878) = ⋀119904isin119878119891(119904) and 119892(⋁119879) = ⋀

119905isin119879119892(119905)) provided

that the joins and meets in question exist in thedomains

(4) 119892 ∘ 119891 ∘ 119892 = 119892 and 119891 ∘ 119892 ∘ 119891 = 119891

Notice that the second item of Theorem 19 is responsiblefor the term ldquoorder-preservingrdquo (resp ldquoorder-reversingrdquo) inDefinition 18 Moreover in view of the results ofTheorem 19there exists the following equivalent definition of Galoisconnections

Definition 20 An order-preserving (resp order-reversing) Galois connection is provided by a tuple((1198831 ⩽) 119891 119892 (1198832 ⩽)) where (1198831 ⩽) (1198832 ⩽) are posets

and (1198831 ⩽)

119891

999448999471119892

(1198832 ⩽) are order-preserving (resp order-reversing) maps such that 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832 (resp11198831

⩽ 119892 ∘ 119891 and 11198832 ⩽ 119891 ∘ 119892)

To give the flavor of these notions we show someexamples of order-preserving Galois connections alreadyencountered in the paper An important example of order-reversing Galois connections will be encountered later on inthe paper

Example 21

(1) Every ⋁-semilattice (resp ⋀-semilattice) homo-morphism 1198601

120593

997888rarr 1198602 provides the order-preservingGalois connection (1198601 120593 120593

⊢ 1198602) (resp (1198602 120593

⊢

120593 1198601))

(2) Every element 119902 of a quantale 119876 provides the order-preservingGalois connections (119876 sdototimes119902 119902rarr 119897sdot 119876) and(119876 119902 otimes sdot 119902rarr 119903 sdot 119876)


(3) Every map 1198831

119891

997888rarr 1198832 provides the order-preservingGalois connection (P(1198831) 119891

rarr 119891

larrP(1198832))

(4) Given a variety L extending CSLat(⋁) every

Set times L-morphism (1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712)

provides the order-preserving Galois connection(119871

1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)

To conclude we would like to notice that this paperstudies the properties of the classical Galois connectionson lattice-valued powersets In contrast [38 40] employthe notion of fuzzy Galois connection which is particularlydesigned to fit the lattice-valued powerset framework Sucha modification is needed to preserve the main results validon crisp powersets The preservation in question comeshowever at the expense of a more complicated definition offuzzy Galois connections themselves which will be off ourtopic of study

3 Variety-Based Topological Systems andTheir Modifications

In [1] we introduced the concept of variety-based topologicalsystems as a generalization of Vickersrsquo topological systems[2] mentioned in the Introduction section Moreover in[22] we presented a modified category of variety-basedtopological systems to accommodate Vickersrsquo system local-ification procedure For convenience of the reader we recallboth definitions restating them however in a slightly moregeneral way to suit the framework of the current paper

Definition 22 Given a variety A a subcategory S of A119900119901 anda reduct B of A SB-TopSys is the category concrete overthe product category Set times B119900119901

times S which comprises thefollowing data

Objects Tuples 119863 = (pt119863Ω119863 Σ119863 ⊨) (SB-topological sys-tems or SB-systems) where (pt119863Ω119863 Σ119863) is a Set timesB119900119901

times S-object whereas pt119863 times Ω119863

⊨

997888rarr Σ119863 is a map (Σ119863-satisfactionrelation on (pt119863Ω119863)) such that Ω119863

⊨(119909minus)

997888997888997888997888rarr Σ119863 is a B-homomorphism for every 119909 isin pt119863

Morphisms 1198631

119891=(pt119891Ω119891Σ119891)997888997888997888997888997888997888997888997888997888997888997888rarr 1198632 are all those Set times B119900119901

times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632) whichsatisfy the property ⊨1(119909 (Ω119891)

119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)

for every 119909 isin pt1198631 and every 119887 isin Ω1198632 (SB-continuity)The following notational remark applies to all the cate-

gories of systems introduced in this paper

Remark 23 The case S = S119860 is called (variety-based) fixed-basis approach whereas all other cases are subsumed under(variety-based) variable-basis approach Moreover if A =

B then the notation SB-TopSys is truncated to S-TopSyswhereas if S = S119860 then the notation is shortened to119860B-TopSys In the latter case the objects (resp morphisms)are truncated to119863 = (pt119863Ω119863 ⊨) (resp 119891 = (pt119891Ω119891))

The next example gives the reader the intuition for thenew category showing that the latter incorporates many ofthe already existing concepts in the literature

Example 24

(1) If A = CBAlg then the category 2Frm-TopSys where2 = perp Τ is the two-element complete Booleanalgebra is isomorphic to the category TopSys oftopological systems of Vickers [2]

(2) If A = B = Frm then the category Loc-TopSysis the category of lattice-valued topological systems ofDenniston et al [8 9]

(3) If A = CQuant then the category 119860Set-TopSys isthe category 119860-IntSys of lattice-valued interchangesystems of Denniston et al [11 29 30] which is alsothe category119860-TransSys of lattice-valued transformersystems of Denniston et al [15] (notice that bothcategories are fixed-basis)

(4) If A = B = Set then the category 119860-TopSysis the category Chu119860 of Chu spaces over the set 119860of Pratt [13] In particular the objects of Chu2 arecalled formal contexts in Formal Concept Analysisof Ganter and Wille [31] whereas the objects of thecategory Chu119860 itself are called many-valued formalcontexts [31 Section 13] Additionally the respectivecontextmorphisms proposed byGanter andWille [31Chapter 7] are different from the morphisms of Chuspaces which will be dealt with in the subsequentsections of the paper

Moreover the following definition (extended and adaptedfor the current setting from [15]) illustrates the concept of atopological system morphism

Definition 25 Letting A = CBAlg GalCon (notice thatldquoGalConrdquo stands for ldquoGalois connectionsrdquo) is the full subcat-egory of the category 2Set-TopSys whose objects are those2Set-topological systems 119863 which satisfy the following twoconditions

(1) pt119863 = Ω119863 = 119883(2) ⊨ is a partial order on119883

The reader can easily see (or verify) that the objects of thecategory GalCon are posets and its morphisms are preciselythe order-preserving Galois connections which is reflectedin its name This category is essentially different from thecategory GAL of Galois connections of [32] in the sensethat while the latter category uses Galois connections as itsobjects the categoryGalCon uses them as morphisms It willbe the topic of our future research to consider the propertiesof the category GalCon (at least) up to the extent of therespective ones of the category GAL Notice however thatthe topic of the category GAL will find its proper place in thesubsequent developments of the paper

The reader should also be aware of the importantfact that the case of order-reversing Galois connections


(1198831 ⩽)

119891

999448999471119892(1198832 ⩽) (up to the knowledge of the author) does

not allow the previously mentioned system interpretationIndeed the already mentioned adjunction condition nowreads as 1199091 ⩽ 119892(1199092) iff 1199092 ⩽ 119891(1199091) for every 1199091 isin 1198831

and every 1199092 isin 1198832 With the help of the dual partial order⩽119889 on 1198832 we do get an order-preserving Galois connection

(1198831 ⩽)

119891

999448999471119892(1198832 ⩽

119889) It is unclear though how to incorporate

in the system setting another order-reversing Galois con-

nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is

how to define the morphism composition law of the possiblecategory (notice that (1198832 ⩽) and (1198832 ⩽

119889) are different as

posets ie the codomain of the first connection is not thedomain of the second one)

In the next step we introduce the modified category oftopological systems inspired by [22]

Definition 26 Given a variety A a subcategory S of Aand a reduct B of A SB-TopSys119898 (notice that ldquo119898rdquo standsfor ldquomodifiedrdquo) is the category concrete over the productcategory Set times B119900119901

times S which comprises the following data

Objects Tuples119863 = (pt119863Ω119863 Σ119863 ⊨) where (pt119863Ω119863 Σ119863)is a SettimesB119900119901

timesS-object andpt119863timesΩ119863 ⊨

997888rarr Σ119863 is amap such thatΩ119863

⊨(119909minus)


Morphisms1198631

119891=(pt119891Ω119891Σ119891)997888997888997888997888997888997888997888997888997888997888997888rarr 1198632 are SettimesB119900119901

timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632) such that Σ119891 ∘

⊨1(119909 (Ω119891)119900119901(119887)) = ⊨2(pt119891(119909) 119887) for every 119909 isin pt1198631 and

every 119887 isin Ω1198632

Since both a category and its dual have the same objectsthe only difference between the categories of systems intro-duced in Definitions 22 and 26 is in the direction of the Σ-component of their respective morphisms In particular withthe notational conventions of Remark 23 we get that all theitems of Example 24 except the second one can be restatedin terms of the category SB-TopSys119898 Moreover it is easy toprovide the conditions under which the system settings ofDefinitions 22 and 26 coincide

Definition 27 Given a subcategory S of A119900119901 SB-TopSys119894(resp S119900119901B -TopSys119894

119898) is the nonfull subcategory of the cat-

egory SB-TopSys (resp S119900119901

B -TopSys119898) which has the same

objects and whose morphisms 1198631

119891

997888rarr 1198632 are such that

Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901

997888997888997888997888rarr Σ1198632) is an S- (resp S119900119901-)isomorphism

Theorem 28 There exists the isomorphism SB-TopSys119894119865

997888rarr

S119900119901B -TopSys119894119898 which is defined by the formula 119865(1198631

119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632

Proof Straightforward computations

As a consequence we obtain that there is no differ-ence between the (fixed-basis) categories 119860B-TopSys and119860B-TopSys119898There exists however another obvious modifi-cation of the category of topological systems which has neverappeared in the literature before

Definition 29 Given a variety A a subcategory S of A119900119901and a reduct B of A SB-TopSys119886 (notice that ldquo119886rdquo standsfor ldquoalternativerdquo) is the category concrete over the productcategory Set times B times S which comprises the following data

ObjectsTuples119863 = (pt119863Ω119863 Σ119863 ⊨) where (pt119863Ω119863 Σ119863)is a Set times B times S-object whereas pt119863 times Ω119863

⊨

997888rarr Σ119863 is a mapsuch that Ω119863

⊨(119909minus)

997888997888997888997888rarr Σ119863 is a B-homomorphism for every119909 isin pt119863

Morphisms 1198631

119891=(pt119891Ω119891Σ119891)997888997888997888997888997888997888997888997888997888997888997888rarr 1198632 are Set times B times

S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)

such that ⊨1(119909 119887) = (Σ119891)119900119901∘ ⊨2(pt119891(119909) Ω119891(119887)) for every

119909 isin pt1198631 and every 119887 isin Ω1198631

It is important to emphasize that leaving (essentially)the same objects we change the definition of morphismsdramatically In particular we cannot obtain a single item ofExample 24 in the new framework

Similar to the previously mentioned classical systemsetting we can introduce a modified category of systems

Definition 30 Given a variety A a subcategory S of A and areduct B ofA SB-TopSys119886119898 is the category concrete over theproduct category Set times B times S which comprises the followingdata

Objects Tuples119863 = (pt119863Ω119863 Σ119863 ⊨) where (pt119863Ω119863 Σ119863)is a SettimesBtimesS-object and pt119863timesΩ119863 ⊨

997888rarr Σ119863 is a map such thatΩ119863

⊨(119909minus)


Morphisms 1198631

119891=(pt119891Ω119891Σ119891)997888997888997888997888997888997888997888997888997888997888997888rarr 1198632 are Set times B times S-morphisms

(pt1198631 Ω1198631 Σ1198631)119891


⊨1(119909 119887) = ⊨2(pt119891(119909) Ω119891(119887)) for every 119909 isin pt1198631 and every119887 isin Ω1198631

The explicit restatement of Theorem 28 for the setting ofthe categories SB-TopSys119886 SB-TopSys119886119898 is straightforwardand is therefore left to the reader

4 Lattice-Valued Formal Concept Analysis

For the required preliminaries in hand in this section weintroduce different approaches to fuzzification of basic toolsof Formal Concept Analysis (FCA) of Ganter and Wille [31]In particular we present three possible category-theoreticmachineries the first two of which stem from the approach ofDenniston et al [29] whereas the last one comes fromGanter


and Wille themselves For the sake of brevity we omittedevery motivational reason for the need of fuzziness in FCAwhich is clearly stated in for example [33] and also could befound at the beginning of Section 5 of [29] Ourmain interestis not the idea of FCA itself but rather a better category-theoretic framework for its successful development

41 Lattice-Valued Formal Contexts as Chu Spaces In thelast item of Example 24 we represented Chu spaces oversets of Pratt [13] as particular instances of variety-basedtopological systems Moreover we also mentioned that oneof the main building blocks of FCA that is (many-valued)formal contexts is in fact nothing else than Chu spaces(already noticed in the literature [55 56]) With these ideasin mind we introduce the following definition

Definition 31 Let L be an extension of the variety Quant ofquantales and let B = Set

(1) S-FC119862 (notice that ldquo119862rdquo stands for ldquoChurdquo) is a newnotation for the category SSet-TopSys whose objects((lattice-valued) formal contexts) are now denotedK = (119866119872 119871 119868) (in the notation of Definition 22K = 119863 which yields 119866 = pt 119863119872 = Ω119863119871 = Σ119863 and 119868 = ⊨) where 119866 (from GermanldquoGegenstanderdquo) is the set of context objects119872 (fromGerman ldquoMerkmalerdquo) is the set of context attributesand 119868 (fromGerman ldquoInzidenzrelationrdquo) is the contextincidence relationThe respectivemorphisms ((lattice-

valued) formal context morphisms) K1

119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)

997888997888997888997888997888997888997888rarr (11986621198722 1198712) (inthe notation of Definition 22 120572 = pt119891 120573 = Ω119891

and 120593 = Σ119891 and then for every 119892 isin 1198661 and every119898 isin 1198722 1198681(119892 120573

119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case

of L = CBAlg and S = S2 is called the crisp case

(2) S-FC119862

119898is a newnotation for the category SB-TopSys119898

The notational and naming remarks for objects andmorphisms of Definition 31 will apply to all the categoriesof (lattice-valued) formal contexts introduced in this paperMoreover the reader can easily apply Theorem 28 to singleout the isomorphic subcategories of the categories S-FC119862 andS-FC119862

119898

At the moment there is no agreement in the fuzzycommunity whether to build formal contexts over L orL119900119901 and therefore we just add the lower index ldquo(minus)119898rdquo inthe former case More precisely this paper is (up to theknowledge of the author) the first attempt to develop avariable-basis approach (a la [4]) to lattice-valued FCA themotivational article of Denniston et al [29] being fixed-basis

42 Lattice-Valued Formal Contexts in the Sense of GanterandWille In the previous subsection we presented a lattice-valued approach to FCAbased inChu spaces An experiencedreader however may be well aware of many-valued contextsof Ganter andWille themselves [31 Section 13] the theory ofwhich is already established in the literature In the following

we provide its category-theoretic analogue bringing togetherboth many-valued contexts and their morphisms

Definition 32 LetL be an extension ofQuant and letB = Set

(1) S-FC119866119882(notice that ldquo119866119882rdquo stands for ldquoGanter andWillerdquo) is a new notation for the category SB-TopSys119886

(2) S-FC119866119882

119898is a new notation for the category

SB-TopSys119886119898

It should be emphasized immediately that neither GanternorWille (up to the knowledge of the author) has considereda categorical approach to (many-valued) formal contexts thatis has united both formal contexts and their morphisms inone entity Additionally both (many-valued) formal contextsand their morphisms are treated in an algebraic way in [31](from where our notation for context morphisms which isdifferent from the system setting is partly borrowed) Theredoes exist some research on certain categories of contexts(eg [57ndash61]) which however does not study the categoricalfoundation for FCA but rather concentrates on a fixedcategorical framework and its related results It is the mainpurpose of this paper to investigate such possible category-theoretic foundations and their relationships to each other(leaving their related results to the further study on thetopic) In particular we consider a category of contextswhose morphisms are strikingly different from the standardframework of topological systems

43 Lattice-Valued Formal Contexts in the Sense of Dennistonet al In [29] Denniston et al have introduced anotherapproach to lattice-valued FCA which was based on thecategoryGAL of order-preservingGalois connections of [32]In the following we extend their approach in two respectsFirstly we employ the variety Quant of quantales instead ofCQuant of commutative ones of [29] (called there completecommutative residuated lattices) Secondly we provide avariable-basis approach (a la [4]) to the topic instead of thefixed-based one of [29]

To begin with we introduce some new notions andnotations (induced by the respective ones of [29]) which willbe used in the subsequent developments of the paper

Definition 33 Every lattice-valued formal contextK has thefollowing (lattice-valued) Birkhoff operators

(1) 119871119866 119867

997888rarr 119871119872 which is given by (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))

Given a lattice-valued context K the notation 119904 for theelements of 119871119866 (resp 119905 for the elements of 119871119872) that is lattice-valued sets of context objects (resp attributes) will be usedthroughout the paper

To give the reader more intuition we provide an exampleof the just introduced notions


Example 34 Every crisp context K (recall our crispnessremark from Definition 31(1)) provides the maps

(1) P(119866)119867

997888rarr P(119872) which is given by119867(119878) = 119898 isin 119872 |

119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870

997888rarr P(119866) which is given by 119870(119879) = 119892 isin 119866 |

119892 119868 119905 for every 119905 isin 119879

which are the classical Birkhoff operators generated by abinary relation

An important property of the maps of Definition 33 iscontained in the following result

Theorem 35 For every lattice-valued context K(119871

119866 119867119870 119871

119872) is an order-reversing Galois connection

Proof For convenience of the reader we provide the proofwhich is different from that of [29]

To show that the map 119867 is order-reversing notice thatgiven 1199041 1199042 isin 119871

119866 such that 1199041 ⩽ 1199042 and some119898 isin 119872 for every119892 isin 119866 it follows that 1199042(119892)rarr 119897 119868(119892119898) ⩽ 1199041(119892)rarr 119897 119868(119892119898)

(Theorem 6(2)) and therefore (119867(1199042))(119898) =

⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =

(119867(1199041))(119898) The case of119870 is similarTo show that 1119871119866 ⩽ 119870 ∘ 119867 notice that given 119904 isin 119871

119866

and 119892 isin 119866 it follows that (119870 ∘ 119867(119904))(119892) =

⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872

119904(119892) ⩾ 119904(119892) where (dagger) relies on the firstclaim of Theorem 6(3) The proof of 1119871119872 ⩽ 119867 ∘ 119870 uses thedual machinery and the second claim of Theorem 6(3)

We emphasize immediately that Theorem 35 is an ana-logue of the already obtained results of [38 Lemma 3] (for119871 a complete residuated lattice) and of [40 Proposition 39](for 119871 a unital quantale) in the framework of fuzzy Galoisconnections which is different (more demanding) from thesetting of Galois connections of the current paper (seeadditionally the remark after Theorem 42) There also existsan extension of Theorem 35 to a more general context of(commutative right-distributive complete) extended-orderalgebras 119871 (whose multiplication in some cases need not beassociative) [62 Propositions 39 55] and additionally toeven more general case which involves relations instead ofmaps (the so-called 119871-Galois triangles) [63]

Everything is now in hand to define another lattice-valued extension of formal contexts of FCA

Definition 36 Given a variety L which extends Quant anda subcategory S of L S-FC119863119872119877 (notice that ldquoDMRrdquo standsfor ldquoDenniston Melton and Rodabaughrdquo) is the categoryconcrete over the product category Set times Set119900119901 whichcomprises the following data

Objects Lattice-valued formal contexts K (in the sense ofDefinition 31) such that 119871 is an object of S

Morphisms K1

119891=(120572120573)

997888997888997888997888997888997888rarr K2 are Set times Set119900119901-morphisms

(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722

2) which make the diagrams

11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute

It should be emphasized that Definition 36 is inspired bythe respective one of [29] which in its turn comes from thedefinition of the category GAL of order-preserving Galoisconnections of [32] Moreover two important features of thecategory S-FC119863119872119877 are worth mentioning Firstly it neverdepends on the morphisms of the category S mentionedat the beginning of its definition (S-objects however areemployed) which is reflected well enough in its truncatedground category Secondly its underlying functor to theground category Set times Set119900119901 is defined on objects as |K| =

(119871119866 119871

119872) which is a huge difference from the already intro-

duced categories of formal contexts (based on the machineryof topological systems) the underlying functor of which isjust |K| = (119866119872 119871)

By analogy with the already presented categories oflattice-valued formal contexts we introduce another versionof the category of Definition 36 Unlike the case of thecategories S-FC119862 and S-FC119866119882 where the change concernsthe direction of the algebraic part of their morphismsthis time we change the set-theoretic morphism part andtherefore use the notation ldquo(minus)119886rdquo (ldquoalternativerdquo) instead ofldquo(minus)119898rdquo (ldquomodifiedrdquo)

Definition 37 Given a variety L which extends Quant anda subcategory S of L S-FC119863119872119877

119886is the category concrete over

the product category SettimesSet which comprises the followingdata


Morphisms K1

119891=(120572120573)

997888997888997888997888997888997888rarr K2 are Set times Set-morphisms

(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute

In the following we adapt the technique of Theorem 28to single out the possible isomorphic subcategories of thecategories S-FC119863119872119877 and S-FC119863119872119877

119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877

119886119894) is the nonfull sub-

category of the category S-FC119863119872119877 (resp S-FC119863119872119877

119886) which

has the same objects and whose morphisms K1

119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722

2) is an isomorphism

in Set that is bijective

Theorem 39 There exists the isomorphism S-FC119863119872119877

119894

119865

997888rarr

S-FC119863119872119877

119886119894 which is defined by the formula 119865(K1

119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2

The reader should be aware that the fixed-basis approach(ie the case of S = S119871) will (in general) provide noniso-morphic categories 119871-FC119863119872119877 and 119871-FC119863119872119877

119886 since both of

them are not dependant on S-morphisms More precisely itis possible to obtain the following result

Theorem 40 Suppose that 119871 is not a singleton and addition-ally its quantale multiplication is not a constant map to perp119871Then the categories 119871-FC119863119872119877 and 119871-FC119863119872119877

119886are nonisomor-

phic

Proof It is easy to see that the context K = (119883119883 119868) where119883 is the empty set and 119883 times 119883

119868

997888rarr 119871 is the unique possiblemap provides a terminal object in the category119871-FC119863119872119877

119886The

category 119871-FC119863119872119877 however does not have a terminal objectwhich can be shown as follows

Suppose the context K119879 = (119866119879119872119879 119868119879) is terminal in119871-FC119863119872119877 Define a context K1015840

= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is

a nonempty set and 1198681015840 = ⊤119871 (the constant map with value

⊤119871) One gets two equal constant maps 1198711198831015840 1198671015840=1198701015840=⊤119871

997888997888997888997888997888997888997888997888rarr 1198711198831015840

Since there exists an 119871-FC119863119872119877-morphism K1015840119891=(120572120573)

997888997888997888997888997888997888rarr K119879the diagrams

119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)

commute It follows that the map 119870119879 in the right-handdiagram is constant (since 1198701015840 has that property) Thus if

119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840

maps everything to⊤119871 then every map 1198711198831015840 1205721015840

997888rarr

119871119866119879 taking ⊤119871 to the constant value of the map 119870119879 gives

another 119871-FC119863119872119877-morphism K10158401198911015840

997888997888rarr K119879 The uniquenessof such morphism necessitates 119871119866119879 to be a singleton andthen 119866119879 is the empty set (since 119871 is not a singleton) Asa consequence we get that the map 119867119879 is constant as well(with the value ⊤119871) Define now another context K10158401015840

=

(11988310158401015840 119883

10158401015840 119868

10158401015840) where11988310158401015840 is a nonempty set and 11986810158401015840 = perp119871 Since

the quantale multiplication of 119871 is not the constant map toperp119871 we get a nonconstant map11986710158401015840 (11986710158401015840

(perp119871) = ⊤119871 =11986710158401015840(⊤119871))

and therefore a contradiction since the left-hand side of theprevious two diagrams factorizes then (with the help of some

119871-FC119863119872119877-morphismK1015840101584011989110158401015840

997888997888rarr K119879) the nonconstant map11986710158401015840

through the constant map119867119879

The case when 119871 is a singleton clearly provides theisomorphic categories 119871-FC119863119872119877 and 119871-FC119863119872119877

119886 The case

when 119871 is equipped with the constant perp119871-valued quantalemultiplication results in all contextmaps119867119870 being constant(with the value ⊤119871 which should additionally be preservedby the corresponding maps 120572 and 120573119900119901 (resp 120573)) which doesnot fit the machinery of Theorem 40 and will be treated inour subsequent articles on lattice-valued FCA We shouldemphasize however immediately that the just mentionedtwo cases are not that interesting (ie are usually skipped bythe researchers) and therefore we can rightly conclude thatthe approaches of Definitions 36 and 37 are fundamentallydifferent which provides a (partial) justification for ouralternative setting of the category 119871-FC119863119872119877

119886

The difference between the categories S-FC119863119872119877 andS-FC119863119872119877

119886is similar to that between the categories S-FC119862

and S-FC119866119882 (which served as our main motivation forintroducing the approach of S-FC119863119872119877

119886 whichwasmentioned

but never studied in [29])

44 Some Properties of Lattice-Valued Birkhoff Operators Itis a well-known (and easy to establish) fact that all order-reversing Galois connections between crisp powersets arisefrom a pair of crisp Birkhoff operators of Example 34 In thefollowing we show that the lattice-valued case destroys partlythis essential property (the reader is advised to recall thenotational remarks concerning particular maps of the pow-erset algebras mentioned at the beginning of Section 22)

Theorem 41 If L extends UQuant then every lattice-valuedcontextK satisfies the following

(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871

119898))(119892) for every 119892 isin 119866 and

every119898 isin 119872(2) (119867(119886otimes1205941119871

119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871The previously mentioned items are equivalent to the

following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871

Proof Ad (1) The item follows from the fact that(119867(120594

1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871

119898))(119892) where (dagger) uses Theorem 6(2)


Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871

119892))(119898) where (dagger) (resp (daggerdagger))

uses item (4) (resp item (1)) of Theorem 6Ad (3) (119870(1205941119871

119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)

where (dagger) (resp (daggerdagger)) uses item (4) (resp item (1)) ofTheorem 6

Ad (1) (2) (3) rArr (119886) (119887) Use (1) in the right-hand sideof both (2) and (3)

Ad (119886) (119887) rArr (1) (2) (3) Put 119886 = 1119871 in any of (119886) or (119887)and get (1) Now use (1) in the right-hand side of both (119886)and (119887) to get (2) and (3) respectively

Theorem 41 paves the way for a partial generalizationof the previously mentioned result on generating order-reversing Galois connections between powersets

Theorem 42 Let 119866119872 be sets and let 119871 be a unital quantaleFor every order-reversing Galois connection (119871119866 120572 120573 119871119872) thefollowing statements are equivalent

(1) There exists a map 119866 times 119872119868

997888rarr 119871 such that 120572 = 119867 and120573 = 119870

(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871


every119898 isin 119872(b) (120572(119886 otimes 120594

1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every

119892 isin 119866119898 isin 119872 and 119886 isin 119871(c) (120573(1205941119871

119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every

119892 isin 119866119898 isin 119872 and 119886 isin 119871(3) (a) (120572(119886 otimes 120594

1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every

119892 isin 119866119898 isin 119872 and 119886 isin 119871(b) (120573(1205941119871

119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every

119892 isin 119866119898 isin 119872 and 119886 isin 119871

Proof (1) rArr (2) and (2) hArr (3) It follows fromTheorem 41(2) rArr (1) Define the required map 119866 times 119872

119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871

119866 and119898 isin 119872 it follows that (119867(119904))(119898) = ⋀

119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes

1119871 = 119904(1198921015840) for every 1198921015840 isin 119866) whereas given 119905 isin 119871


119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871

119898otimes 119905(119898))))(119892) = (120573(119905))(119892) where (dagger) uses

Theorem 19(3)

Theorems 41 and 42 show that in the lattice-valuedsetting every map 119866 times 119872

119868

997888rarr 119871 provides an order-reversingGalois connection whereas the converse way is possibleunder certain restrictions only As a simple counterexam-ple consider the case when 119871 is the unit interval I =

([0 1] ⋁⋀ 1) and both 119866 and 119872 are singletons We cantherefore assume that both I119866 and I119872 are the same as I Theorder-reversing involution map I

120572

997888rarr I 120572(119886) = 1 minus 119886 is a partof the order-reversing Galois connection (I 120572 120572 I) It is easyto see that the condition of for example Theorem 42(3)(a)amounts to 120572(119886) = 119886 rarr 120572(1) for every 119886 isin I (notice that thequantale in question is commutative) However for 119886 = 12we obtain that 120572(12) = 12 = 0 = 12 rarr 0 = 12 rarr 120572(1)

We also would like to emphasize here that [40] uses adifferent approach to Galois connections (called polarities ofcertain types) which has the advantage of still preservingthe one-to-one correspondence between them and Birkhoffoperators even in the lattice-valued case [40 Theorem 311]The underlying framework of such polarities however ismore demanding than that of this paper (eg requiresenriched categories) and therefore will be not consideredhere Also notice that [64 Proposition 71] provides ananalogue of Theorem 42 for the case of strictly two-sidedcommutative quantales considering however lattice-valuedversions of the powerset operators generated by a relation andextending those given at the beginning of Section 22 andtherefore dealing with order-preserving Galois connectionsMore precisely in the notation of [64] every formal contextK = (119866119872 119871 119868) (where 119871 is additionally assumed tobe a strictly two-sided commutative quantale) provides thefollowing generalizations of the standard powerset operators

(1) 119871119866 119868otimes

997888997888rarr 119871119872 given by (119868otimes(119904))(119898) = ⋁

119892isin119866119868(119892119898) otimes 119904(119892)

(forward powerset operator)

(2) 119871119872119868rarr

minus

997888997888rarr 119871119866 given by (119868rarr

minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr

119905(119898) (backward powerset operator)

In particular (119871119866 119868otimes 119868rarrminus 119871

119872) is an order-preserving Galois

connection (recall Definition 18) It is the main purpose of[64 Proposition 71] to clarify the conditions when an order-preserving Galois connection (119871119866 120572 120573 119871119872) has the previousform that is is generated by a map 119866 times119872

119868

997888rarr 119871We end the subsection with another simple (but useful)

property of lattice-valued Birkhoff operators

Lemma43 Given a lattice-valued formal contextK such that119868 = ⊤119871 it follows that 119867 = ⊤119871 and 119870 = ⊤119871 In particular inthe crisp case119867 = 119872 and 119870 = 119866

Proof It is enough to consider the case of119867 (that of119870 beingsimilar) Given 119904 isin 119871

119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871

45 Lattice-Valued Formal Concepts Protoconcepts and Pre-concepts In the developments of [29] Denniston et al use


lattice-valued extensions of the notions of formal conceptprotoconcept and preconcept of for example [36] In thefollowing we recall their respective definitions and showsome of their properties in our more general frameworkwhich will be used later on in the paper

Definition 44 Let K be a lattice-valued formal context andlet 119904 isin 119871119866 119905 isin 119871119872 The pair (119904 119905) is called a

(1) (lattice-valued) formal concept of K provided that119867(119904) = 119905 and119870(119905) = 119904

(2) (lattice-valued) formal protoconcept of K providedthat119870 ∘ 119867(119904) = 119870(119905) (equivalently119867 ∘ 119870(119905) = 119867(119904))

(3) (lattice-valued) formal preconcept ofK provided that119904 ⩽ 119870(119905) (equivalently 119905 ⩽ 119867(119904))

Using the properties of Galois connections mentionedin Theorem 19 one can easily get that every concept isa protoconcept and every protoconcept is a preconceptwhereas the converse implications in general are not trueMoreover the next result provides simple properties of thenotions of Definition 44 with respect to themorphisms of thepreviously mentioned categories S-FC119863119872119877 and S-FC119863119872119877

119886

Theorem 45 Let K1

119891

997888rarr K2 be a morphism of the categoryS-FC119863119872119877 and let 1199041 isin 119871

1198661

1 1199052 isin 119871

1198722

2

(1) (1199041 120573119900119901(1199052)) is a formal concept ofK1 and1198672∘1198702(1199052) =

1199052 iff (120572(1199041) 1199052) is a formal concept of K2 and 1198701 ∘

1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed

points that is1198702 ∘1198672 ∘120572(119904) = 120572(119904) implies1198701 ∘1198671(119904) =

119904 for every 119904 isin 11987111986611 and1198671 ∘1198701 ∘120573

119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a

formal concept of K1 iff (120572(1199041) 1199052) is a formal conceptofK2

(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then

(1199041 120573119900119901(1199052)) is a formal concept ofK1 iff (120572(1199041) 1199052) is a

formal concept ofK2(4) (1199041 120573

119900119901(1199052)) is a formal protoconcept of K1 iff

(120572(1199041) 1199052) is a formal protoconcept ofK2

(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-

preserving then (1199041 120573119900119901(1199052)) is a formal preconcept ofK1 iff (120572(1199041) 1199052) is a formal preconcept ofK2

If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1) is injective then 120572

(resp 120573) reflects fixed points in the sense of (2)

Proof Ad (1) ldquorArrrdquo Since both 1199041 = 1198701 ∘ 120573119900119901(1199052) and1198671(1199041) =

120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover

1198701 ∘1198671(1199041) = 1198701 ∘120573119900119901(1199052) = 1199041 ldquolArrrdquo Since both1198672 ∘120572(1199041) = 1199052

and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052

Ad (2) ldquorArrrdquo120573119900119901(1199052) = 1198671(1199041) yields1198671∘1198701∘120573119900119901(1199052) = 1198671∘

1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =

1199052 since 120573119900119901 reflects fixed points Now employ item (1) ldquolArrrdquo

1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =

1198702(1199052) = 120572(1199041) and therefore1198701∘1198671(1199041) = 1199041 since120572 reflectsfixed points Now employ item (1)

Ad (3) It follows from (2) and the last claim of thetheorem

Ad (4) ldquorArrrdquo Since 1198701 ∘ 1198671(1199041) = 1198701 ∘ 120573119900119901(1199052) we get

1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =

120572 ∘1198701 ∘ 120573119900119901(1199052) = 1198702(1199052) ldquolArrrdquo Since1198672 ∘1198702(1199052) = 1198672 ∘ 120572(1199041)

we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)

Ad (5) ldquorArrrdquo 1199041 ⩽ 1198701 ∘ 120573119900119901(1199052) implies 120572(1199041) ⩽ 120572 ∘ 1198701 ∘

120573119900119901(1199052) = 1198702(1199052) (120572 is order-preserving) ldquolArrrdquo 1199052 ⩽ 1198672 ∘ 120572(1199041)

yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573

119900119901 is order-preserving)

For the last claim notice that given 119904 isin 11987111986611such that1198702 ∘

1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘

120572(119904) = 1198702∘1198672∘120572(119904) = 120572(119904) and thus1198701∘1198671(119904) = 119904 (injectivityof 120572) Additionally given 119905 isin 1198711198722

2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573

119900119901(119905) and thus1198672 ∘ 1198702(119905) = 119905 (injectivity

of 120573119900119901)

Theorem 46 Let K1

119891

997888rarr K2 be a morphism of the categoryS-FC119863119872119877

119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then

(1199041 1199051) is a formal concept of K1 iff (120572(1199041) 120573(1199051)) is aformal concept ofK2

(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then

(1199041 1199051) is a formal protoconcept ofK1 iff (120572(1199041) 120573(1199051))is a formal protoconcept ofK2

(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is order-preserving

and moreover one of the maps is order-reflecting then(1199041 1199051) is a formal preconcept ofK1 iff (120572(1199041) 120573(1199051)) isa formal preconcept ofK2

Proof Ad (1) ldquorArrrdquo Since 1198671(1199041) = 1199051 and 1198701(1199051) = 1199041 1198672 ∘

120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)

(no injectivity condition is employed) ldquolArrrdquo Since1198672∘120572(1199041) =

120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)

and 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051) = 120572(1199041) imply 1198671(1199041) = 1199051 and1198701(1199051) = 1199041 respectively (both 120572 and 120573 are injective)

Ad (2) ldquorArrrdquo Since1198701 ∘1198671(1199041) = 1198701(1199051)1198702 ∘1198672 ∘ 120572(1199041) =

1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)

(no injectivity condition is employed) ldquolArrrdquo If 11987111986611

120572

997888rarr 1198711198662

2is

injective then 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) implies 120572 ∘ 1198701 ∘

1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)


Ad (3) ldquorArrrdquo If 11987111986611

120572

997888rarr 1198711198662

2is order-preserving then 1199041 ⩽

1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is

order-preserving then 1199051 ⩽ 1198671(1199041) yields 120573(1199051) ⩽ 120573∘1198671(1199041) =

1198672 ∘120572(1199041) ldquolArrrdquo If 11987111986611

120572

997888rarr 1198711198662

2is order-reflecting then 120572(1199041) ⩽

1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is

order-reflecting then 120573(1199051) ⩽ 1198672 ∘ 120572(1199041) = 120573 ∘ 1198671(1199041) yields1199051 ⩽ 1198671(1199041)

The two previous theorems provide crucial tools fordefining certain functors between the categories of lattice-valued formal contexts which is the main topic of the nextsection

5 Functorial Relationships betweenthe Categories ofLattice-Valued Formal Contexts

The previous section introduced several categories (partlymotivated by the respective ones of [29]) the objects ofwhich are lattice-valued formal contexts It is the purpose ofthis section to consider functorial relationships between thenew categories the functors in question being again partlyinspired by the respective approach of [29] As a result wenot only extend (the change from commutative quantalesto noncommutative ones and the shift from fixed-basis tovariable-basis) and streamline the machinery of Dennistonet al but also provide several new functors (induced by newcategories) thereby clarifying the relationships between dif-ferent frameworks for doing lattice-valued FCA and bringingto light their respective (dis)advantages

51 S-FC119862 and S-FC119862

119898versus S-FC119863119872119877 and S-FC119863119872119877

119886 In this

subsection we consider possible functorial links between thecategories S-FC119862 and S-FC119862

119898 on one side and the categories

S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877

119886 In this subsection we con-

struct a functorial embedding of a particular subcategory ofthe category S-FC119862

119898into the category S-FC119863119872119877

119886 We begin

with singling out the subcategory in question

Definition 47 S-FC119862

119898lowastis the (nonfull) subcategory of the

category S-FC119862

119898 with the same objects whose morphisms

K1

119891

997888rarr K2 are such that the maps 1198661

120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721

are surjective whereas the S-morphism 1198711

120593

997888rarr 1198712 preserves⋀ rarr 119897 and rarr 119903

Notice that the notation (minus)lowast will usually be used for theconstructed subcategories in this paper the scope of each (minus)lowastbeing limited to its respective subsection More importantconstructions however will be distinguished respectively

With the category of Definition 47 in hand we canconstruct the following functor (the reader is advised to recallour powerset operator notations from Section 22)

Theorem 48 There exists the functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2

Proof To show that the functor is correct on morphisms weverify commutativity of the diagrams

11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)

For the left-hand diagram notice that given 119904 isin 1198711198661

1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdaggerdagger)

= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)

where (dagger) relies on Theorem 6(2) (daggerdagger) uses the fact thatK1

119891

997888rarr K2 is a morphism of S-FC119862

119898 whereas (daggerdaggerdagger) employs

the definition of the category S-FC119862

119898lowast


For the right-hand diagram notice that given 119905 isin 11987111987211

and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)

where (dagger) (daggerdagger) use the definition of the category S-FC119862

119898lowast

whereas (daggerdaggerdagger) employs the fact thatK1

119891

997888rarr K2 is amorphismof the category S-FC119862

119898

In the notation ldquo119867119862119898119863119886rdquo ldquo119862119898rdquo shows its domain andldquo119863119886rdquo (short for ldquo119863119872119877119886rdquo) stands for its codomain Similarnotational conventions will be used in the remainder of thispaper but will not be mentioned explicitly again

In the following we make the new functor into anembedding The result depends on a particular subcategoryof its domain

Definition 49 Suppose that the variety L of which S is asubcategory extends the variety UQuant of unital quantalesS-FC119862

119898lowastlowast(resp S-FC119862

119898lowast∙) is the full subcategory of the

category S-FC119862

119898lowast whose objects K = (119866119872 119871 119868) are such

that 119866 is nonempty (resp 119872 is nonempty) and moreover1119871 = perp119871

The notation (minus)lowastlowast (resp (minus)lowast∙) will be used for manyof the subcategories constructed in this paper the scope ofeach being limited to its respective subsection Additionallymore important constructions will be distinguished in adifferent way It is also important to emphasize that thealgebra requirement of Definition 49 is satisfied by the crispcase (see Definition 31(1))

Theorem 50 The restriction 119867lowast

119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)

gives an (in general nonfull) embedding of the latter categoryinto S-FC119863119872119877

119886

Proof We consider the case of the functor 119867lowast

119862119898119863119886first and

show its faithfulness Given K1

1198911

9994909994731198912

K2 in S-FC119862

119898lowastlowastwith

119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)

larr119900To show that 1205721 = 1205722 (which we will refer to as 120572) notice

that given 1198921 isin 1198661

((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)

perp1198711 otherwise

)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)

for every 1198922 isin 1198662 As a consequence we obtain that 12059411198712

1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712

1205722(1198921)and therefore

1205721(1198921) = 1205722(1198921) by our assumption that 11198712 = perp1198712

To show that 1205731199001199011= 120573

119900119901

2= 120573

119900119901 notice that given1198982 isin 1198722((1205731 1205931)

larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) it immediately follows that

120573119900119901

1(1198982) = 120573

119900119901

2(1198982) by our assumption that 11198712 = perp1198712


To show that 1205931 = 1205932 = 120593 notice that given 1198861 isin 1198711we can choose some 1198921 isin 1198661 (1198661 is nonempty) Then((120572 1205931)

rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)

Turning the attention to the functor 119867∙

119862119898119863119886 one follows

the same steps to obtain 1205721 = 1205722 = 120572 and 1205731199001199011= 120573

119900119901

2= 120573

119900119901The only difference is in the proof of 1205931 = 1205932 = 120593 Given1198861 isin 1198711 we choose some 1198982 isin 1198722 (1198722 is nonempty) Itfollows then that ((120573 1205931)

larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =

1205931(1198861) and therefore 1205931(1198861) = ((120573 1205931)larr119900(1198861))(1198982) =

((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)

For the nonfullness claim we restrict the setting to thecrisp case (L = CBAlg and S = S2) and consider thecontext K = (119866119872 119868) where 119866 = 119872 = |2| and 119868 =

119866 times119872 By Lemma 43 it follows thatP(119866)119867=119872

997888997888997888997888rarr P(119872) and

P(119872)119870=119866

997888997888997888rarr P(119866) We define a formal context morphism

K119891

997888rarr K of the category 2-FC119863119872119877

119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872

997888997888997888997888rarr P(119872) Since the setting fits both the functor119867

lowast

119862119898119863119886and 119867∙

119862119898119863119886 we consider the case of the former one

Suppose there exists some K1198911015840

997888997888rarr K in 2-FC119862

119898lowastlowastsuch that

119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572

1015840)rarr (crisp forward

powerset operator) and therefore 1205721015840(perp) = (1205721015840)rarr(perp) =

120572(perp) = 119866 which is an obvious contradiction

512 From S-FC119862

119898to S-FC119863119872119877 In this subsection we con-


119898into the category S-FC119863119872119877 We begin

again with singling out the subcategory in question



category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr

1198721 are surjective whereas the S-morphism 1198711

120593

997888rarr 1198712 is anisomorphism

With the new definition in hand we can construct thefollowing functor


119898lowast

119867119862119898119863997888997888997888997888rarr

S-FC119863119872119877 which is given by 119867119862119898119863(K1

119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)

where (dagger) relies on the fact that K1

119891

997888rarr K2 is a morphismof the category S-FC119862

119898 whereas (daggerdagger) employs the definition

of the category S-FC119862

119898lowast(notice that the inverse of 120593 is easily

seen to be 120593⊢)For the right-hand diagram notice that given 119905 isin 1198711198722

2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)

where (dagger) uses the definition of the category S-FC119862

119898lowast whereas

(daggerdagger) relies on the fact that K1

119891

997888rarr K2 is a morphism of thecategory S-FC119862

119898

The category S-FC119862


119898lowast∙) of Definition 49

provides the following result


119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)


Proof Follow the steps of the proof of Theorem 50 wherethe backward powerset operator (120573 120593)larr119900 is substituted by theforward powerset operator (120573 120593)⊢[119900

513 From S-FC119862 to S-FC119863119872119877


struct a functorial embedding of a particular subcategory ofthe category S-FC119862 into the category S-FC119863119872119877

119886 We start by

defining the subcategory in question


lowastis the (nonfull) subcategory of the

category S-FC119862 with the same objects whose morphisms


K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are

surjective whereas the S119900119901-morphism 1198712

120593119900119901

997888997888rarr 1198711 is injective

and moreover |1198711|120593119900119901⊢

997888997888997888rarr |1198712| preserves⋁ rarr 119897 and rarr 119903

For the new category in hand in the following weconstruct its respective functor


lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886

which is given by119867119862119863119886(K1

119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)

where (dagger) uses Definition 54 (notice that 120593119900119901⊢ is a left inverse

to 120593119900119901) whereas (daggerdagger) relies on the fact that K1

119891

997888rarr K2 is amorphism of S-FC119862


and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)

where (dagger) uses Definition 54 whereas (daggerdagger) relies on the factthatK1

119891



In the following we make the new functor into anembedding

Definition 56 Suppose that the variety L of which S is asubcategory extends UQuant S-FC119862

lowastlowast(resp S-FC119862

lowast∙) is the

(nonfull) subcategory of the category S-FC119862

lowast comprising the

following data

Objects S-FC119862

lowast-objectsK for which119866 is nonempty (resp119872

is nonempty) and moreover 1119871 = perp119871

Morphisms S-FC119862

lowast-morphisms K1

119891

997888rarr K2 for which

|1198711|120593119900119901⊢

997888997888997888rarr |1198712| is 1-preserving

The reader should notice the essential difference in thesetting of Definition 49 namely an additional condition onthe morphisms of the new category

Theorem 57 The restriction119867lowast

119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886to the category S-FC119862


lowast∙)


119886

Proof Follow the proof of Theorem 50 with the respectivechanges in the powerset operators

514 From S-119865C119862 to S-FC119863119872119877 In this subsection we con-struct a functorial embedding of a particular subcategory ofthe category S-FC119862 into the category S-FC119863119872119877 As before webegin with singling out the subcategory in question


lowastis the (nonfull) subcategory of the cat-

egory S-FC119862 with the same objects and whose morphisms

K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721

are surjective whereas the S119900119901-morphism 1198712

120593119900119901


With the new category in hand we are ready to define anew functor as follows


lowast

119867119862119863997888997888997888rarr S-FC119863119872119877

which is given by 119867119862119863(K1

119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891

997888rarr K2 is a morphism of S-FC119862For the right-hand diagram notice that given 119905 isin 1198711198722

2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)

where (dagger) uses the fact that K1

119891

997888rarr K2 is a morphism ofS-FC119862 whereas (daggerdagger) employs Definition 58

Similar to Definition 49 one obtains the categoriesS-FC119862

lowastlowastand S-FC119862

lowast∙ Moreover similar to Theorem 50 one

gets the following result


119862119863(resp 119867∙

119862119863) of the functor

S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an

(in general nonfull) embedding of the latter category intoS-FC119863119872119877

515 From S-FC119863119872119877 to S-FC119862 In this subsection we con-struct functorial embeddings of several subcategories of thecategory S-FC119863119872119877 into the category S-FC119862 (most of themare motivated by [29]) We begin with defining the firstsubcategory in question The reader should note that all thesubcategories constructed in this subsection are fixed-basis

Definition 61 Given an L-algebra 119871 119871-FC119863119872119877

i is the (nonfull)subcategory of the category 119871-FC119863119872119877 with the same objects

whose morphisms K1

119891

997888rarr K2 have the property that the

maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901

997888997888rarr 1198711198721 are injective

The reader should notice that ldquoirdquo in 119871-FC119863119872119877

i stands forldquoinjectiverdquo


Theorem 62 There exists the functor 119871-FC119863119872119877

i

119867i119863119862

997888997888997888rarr S-FC119862

which is given by119867i119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where

119895 (119904 119905) = ⊤119871 (119904 119905) is a formal concept ofK119895


Proof We have to show that the functor is correct onmorphisms Given 1199041 isin 119871

1198661 and 1199052 isin 1198711198722 it follows that

1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573

119900119901(1199052)) is a formal concept ofK1

iff (120572(1199041) 1199052) is a formal concept of K2 iff 2(120572(1199041) 1199052) = ⊤119871where the second ldquoiffrdquo uses Theorem 45(3)

In the following we make the functor ofTheorem 62 intoan embedding To achieve the goal we impose a restrictionon the chosen L-algebra 119871

Definition 63 An L-algebra 119871 is called quasistrictly right-sided (qsrs-algebra for short) provided that 119886 ⩽ (⊤119871rarr 119897119886) otimes

⊤119871 for every 119886 isin 119871

Since every quantale119876has the property (⊤119876rarr 119897119902) otimes ⊤119876 ⩽

119902 for every 119902 isin 119876 (use Example 21(2)) every qsrs-algebra 119871satisfies (⊤119871rarr 119897119886)otimes⊤119871 = 119886 for every 119886 isin 119871 Additionally if119876is a strictly right-sided quantale (recall Definition 9) then forevery 119902 isin 119876 ⊤119871rarr 119897119902 = 119902 yields (⊤119871rarr 119897119902) otimes ⊤119871 = ⊤119871rarr 119897119902 =

119902 As a result every L-algebra 119871 extending a strictly right-sided quantale 119876 is a qsrs-algebra but not vice versa

Theorem 64 For every qsrs-algebra 119871 the functor

119871-FC119863119872119877

i

119867i119863119862

997888997888997888rarr S-FC119862 is a nonfull embedding

Proof It will be enough to show that the functor 119867i119863119862

is injective on objects Given K1 K2 in 119871-FC119863119872119877

i

such that 119867i119863119862(K1) = 119867

i119863119862(K2) it follows that

(1198711198661 119871

1198721 119871 1) = (1198711198662 119871

1198722 119871 2) and therefore1198661 = 1198662 = 119866 1198721 = 1198722 = 119872 and 1 = 2 To show that1198681 = 1198682 we assume that 1198681 = 1198682 Then there exist some 119892 isin 119866

and119898 isin 119872 such that 1198681(119892119898) 997408 1198682(119892119898) As a consequencewe obtain that perp119871 = ⊤119871 The properties of Galois connectionsof Theorem 19 imply that 1198671 ∘ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) is a formal concept

of K1 As a consequence 1(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) is a formal concept of K2 It follows

that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903

1198682(119892119898) As a result 1198681(119892119898)(dagger)

⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽

1198682(119892119898) where (dagger) uses the definition of qsrs-algebrasand therefore 1198681(119892119898) ⩽ 1198682(119892119898) which contradicts ourassumption 1198681(119892119898) 997408 1198682(119892119898)

To show that the functor is nonfull we let L = Quantand 119871 = ([0 1] ⋁⋀) that is taking the unit interval [0 1]with its standard lattice-theoretic structure Since 119871 is clearlystrictly two-sided it is quasistrictly right-sided Define a map|119871|

120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)

It follows that 119871120593

997888rarr 119871 is an STSQuant-isomorphism which isdifferent from 1119871 Define now a lattice-valued formal contextK as 119866 = 119872 = lowast and 119868 = ⊤119871 It is easy to see that

119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888rarr 119867i119863119862(K) is an S-FC119862-morphism

since 120593(119886) = ⊤119871 (resp 120593(119886) = perp119871) iff 119886 = ⊤119871 (resp 119886 = perp119871)Additionally the morphism is clearly not in the image of thefunctor119867i

119863119862

To continue we provide the second of the claimedembeddings As usual we begin by singling out a particularfixed-basis subcategory of the category S-FC119863119872119877

Definition 65 Given anL-algebra119871119871-FC119863119872119877

119900119903119901is the (nonfull)

subcategory of the category 119871-FC119863119872119877 with the same objects

whose morphisms K1

119891

997888rarr K2 satisfy the property that the

maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901

997888997888rarr 1198711198721 are order-preserving

The reader should notice that ldquo119900119903119901rdquo in 119871-FC119863119872119877

119900119903119901stands

for ldquoorder-preservingrdquo


119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where

119895 (119904 119905) = ⊤119871 (119904 119905) is a formal preconcept ofK119895



1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal

preconcept of K1 iff (120572(1199041) 1199052) is a formal preconcept ofK2 iff 2(120572(1199041) 1199052) = ⊤119871 where the second ldquoiffrdquo usesTheorem 45(5)

The machinery of Theorem 64 does well enough in thismodified case as well


119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862


Proof The proof starts precisely as that of Theorem 64 Toshow that 1198681 = 1198682 we again assume the existence of some 119892 isin119866 and119898 isin 119872 such that 1198681(119892119898) 997408 1198682(119892119898)The properties of


Galois connections (Theorem 19) provide120594⊤119871119892

⩽ 1198701∘1198671(120594⊤119871

119892)

which implies that (120594⊤119871119892 1198671(120594

⊤119871

119892)) is a formal preconcept of

K1 It follows that 1(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892)) is a formal

preconcept ofK2 and as a result120594⊤119871

119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar

to the proof of Theorem 64 one obtains that 1198681(119892119898) ⩽

1198682(119892119898) thereby getting a contradiction with the assumption1198681(119892119898) 997408 1198682(119892119898)

The nonfullness part follows the steps of the proof ofTheorem 64

To continue we show yet another one of the claimedembeddings which never requires singling out a particularsubcategory of the category 119871-FC119863119872119877 (but is still fixed-basis)

Theorem 68 There exists the functor 119871-FC119863119872119877119867119863119862997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where

119895 (119904 119905) = ⊤119871 (119904 119905) is a formal protoconcept ofK119895



1198661 and 1199052 isin L1198722 it followsthat 1(1199041 120573

119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-

toconcept of K1 iff (120572(1199041) 1199052) is a formal protoconcept ofK2 iff 2(120572(1199041) 1199052) = ⊤119871 where the second ldquoiffrdquo usesTheorem 45(4)

The machinery of Theorem 64 does well even in thisextended setting

Theorem 69 For every qsrs-algebra 119871 the functor119871-FC119863119872119877


Proof The proof starts precisely as that of Theorem 64 Toshow that 1198681 = 1198682 we again assume the existence of some119892 isin 119866 and 119898 isin 119872 such that 1198681(119892119898) 997408 1198682(119892119898) Since1198701 ∘ 1198671(120594

⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871


protoconcept of K1 It follows that 1(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871

and therefore 2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))

is a formal protoconcept of K2 and as a result 120594⊤119871

119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871

119892) Similar to the proof ofTheorem 64

one obtains that 1198681(119892119898) ⩽ 1198682(119892119898) thereby getting acontradiction with the assumption 1198681(119892119898) 997408 1198682(119892119898)


Theorems 71 and 72 show the last of the claimed embed-dings this embedding does require a particular subcategoryof 119871-FC119863119872119877


119903119891119901is the (non-

full) subcategory of the category 119871-FC119863119872119877 with the same

objects whose morphisms K1

119891

997888rarr K2 satisfy the property

that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901

997888997888rarr 1198711198721 reflect fixed points


119903119891119901stands

for ldquoreflect fixed pointsrdquo


119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573



The embedding property of the just introduced functor iseasy to get


119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862


Proof The proof follows the steps of that ofTheorem 64

In view of the last claim of Theorem 45 the category119871-FC119863119872119877

i is a (nonfull) subcategory of the category119871-FC119863119872119877

119903119891119901

Thus the functor119867i119863119872

is the restriction of the functor119867119903119891119901

119863119872

to the category 119871-FC119863119872119877

i As a last remark we notice that in every unital quantale

119876 (1119876rarr 119897119902)otimes1119876 = 1119876rarr 119897119902 = 119902 for every 119902 isin 119876 In particularall the results in this subsection can be done in a moresimple way assuming that L extends the variety UQuant ofunital quantales and then making use of the unit 1119871 insteadof the top element ⊤119871 of the given L-algebra 119871 In viewof the previously mentioned property of unital quantalesthe respective functors are then embeddings without anyadditional requirement on the L-algebra 119871

516 From S-FC119863119872119877 to S-FC119862

119898 The reader can easily show

(as a simple exercise) that the functors from the previoussubsection can be used in case of the categories S-FC119863119872119877 andS-FC119862

119898as well providing similar results In particular all the

nonfullness examples work also in the new setting

52 S-FC119866119882 and S-FC119866119882


119886 In

this subsection we consider possible functorial links betweenthe categories S-FC119866119882 and S-FC119866119882

119898 on one side and the

categories S-FC119863119872119877 and S-FC119863119872119877

119886 on the otherTheobtained

functors can be easily made into embeddings employingthe machinery of Section 51 related to the functors fromthe constructed subcategories of the category S-FC119862(resp


S-FC119862

119898) to the category S-FC119863119872119877 (resp S-FC119863119872119877

119886) In

particular assuming that L extends the variety UQuant ofunital quantales whereas S-FC119866119882

daggerlowastis the domain of the newly

constructed functor (see the results of following subsections)where dagger is ether ldquordquo or ldquo119898rdquo we can define the following twosubcategories


daggerlowastlowast(resp S-FC119866119882

daggerlowast∙) is the full subcat-

egory of the category S-FC119866119882

daggerlowast whose objects K are such


Employing the technique of Section 51 we can easilyget that the restriction of the respective constructed functorto the previously mentioned two categories provides twoembeddings To save the space therefore wewill notmentionthe just discussed developments explicitly

521 From S-FC119866119882

119898to S-FC119863119872119877

119886 In this subsection we

construct a functor from a particular subcategory of thecategory S-FC119866119882


119886 To begin with

we provide the definition of the subcategory in question



category S-FC119866119882


K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593


For the new category in hand we can construct thefollowing functor


119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)

where (dagger) uses Definition 74 whereas (daggerdagger) employs the fact

thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882

119898to S-FC119863119872119877 In this subsection we


119898to the category S-FC119863119872119877 To begin with






K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr

S-FC119863119872119877 which is defined by the formula 119867119866119898119863(K1

119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891

997888rarr K2 is a morphism ofS-FC119866119882

119898 whereas (daggerdagger) employs Definition 76


and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)

where (dagger) uses Definition 76 whereas (daggerdagger) uses the fact thatK1

119891


119898

523 From S-FC119866119882 to S-FC119863119872119877


construct a functor from a particular subcategory of thecategory S-FC119866119882 to the category S-FC119863119872119877

119886 As before we

start with singling out the subcategory in question




K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901


and moreover the map |1198711|120593119900119901⊢

997888997888997888rarr |1198712| preserves ⋁ rarr 119897 andrarr 119903


lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877

119886 which is defined by 119867119866119863119886(K1

119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891


524 From S-FC119866119882 to S-FC119863119872119877 In this subsection weconstruct a functor from a particular subcategory of thecategory S-FC119866119882 to the category S-FC119863119872119877 As before webegin with singling out the subcategory in question




K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr

S-FC119863119872119877 which is defined by 119867119866119863(K1

119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882


construct functorial embeddings of several subcategoriesof the category S-FC119863119872119877


119898 We

begin with singling out the first subcategory in questionThe reader should notice that similar to Section 515 all thesubcategories constructed in the current one are fixed-basis


119886i is the (nonfull)subcategory of the category 119871-FC119863119872119877

119886 with the same objects

whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882

119898 which is defined by the formula 119867

i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where

119895 (119904 119905) = ⊤119871 (119904 119905) is a formal concept of K119895




1(1199041 1199051) = ⊤119871 iff (1199041 1199051) is a formal concept of K1 iff(120572(1199041) 120573(1199051)) is a formal concept of K2 iff 2(120572(1199041) 120573(1199051)) =⊤119871 where the second ldquoiffrdquo uses Theorem 46(1)

To make the functor into an embedding one uses themachinery of Theorem 64


119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882

119898is a nonfull embedding

Proof Follow the steps of the proof of Theorem 64

To continue we provide the second of the claimedembeddings We begin with singling out the subcategory ofthe category S-FC119863119872119877

119886 which is suitable for the occasion


119886i120572 (resp119871-FC119863119872119877

119886i120573 ) is the (nonfull) subcategory of the category

119871-FC119863119872119877

119886 with the same objects whose morphisms K1

119891

997888rarr

K2 have the property that the map 1198711198661 120572

997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr

1198711198722) is injective

Theorem 86 Let dagger be either 120572 or 120573 There exists the functor

119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882

119898 which is given by 119867idagger

119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where

119895 (119904 119905) = ⊤119871 (119904 119905) is a formal protoconcept of K119895



1198661 and 1199051 isin 1198711198721 it follows

that 1(1199041 1199051) = ⊤119871 iff (1199041 1199051) is a formal protoconcept


of K1 iff (120572(1199041) 120573(1199051)) is a formal protoconcept of K2

iff 2(120572(1199041) 120573(1199051)) = ⊤119871 where the second ldquoiffrdquo usesTheorem 46(2)

Theorem 87 Let dagger be either 120572 or 120573 For every qsrs-algebra 119871

the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882



To continue we show the last one of the claimed embed-dings


119886119900119890120572(resp

119871-FC119863119872119877

119886119900119890120573) is the (nonfull) subcategory of the category

119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr

1198711198722) is an order embedding (ie both order-preserving and

order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882

119898 which is given by 119867119900119890dagger

119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where

119895 (119904 119905) = ⊤119871 (119904 119905) is a formal preconcept of K119895




that 1(1199041 1199051) = ⊤119871 iff (s1 1199051) is a formal preconceptof K1 iff (120572(1199041) 120573(1199051)) is a formal preconcept of K2



the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882



As a last remark we notice that the case of L extending thevariety UQuant simplifies the previously mentioned results(see the remark at the end of Section 515)

53 S-FC119866119882 and S-FC119866119882


119898 In this

subsection we consider possible functorial links betweenthe categories S-FC119866119882 and S-FC119866119882



119898 on the other

531 From S-FC119866119882

119898to S-FC119862


struct a functorial isomorphism between particular subcat-egories of the categories S-FC119866119882

119898and S-FC119862

119898 respectively


119898lowast(resp S-FC119862

119898lowast) is the (nonfull) sub-

category of the category S-FC119866119882

119898(resp S-FC119862

119898) with the

same objects whose morphisms K1

119891

997888rarr K2 have the

property that the map 1198721

120573

997888rarr 1198722 (resp 1198722

120573119900119901

997888997888rarr 1198721) isbijective

The categories of Definition 91 give rise to the followingfunctors

Theorem 92 There exist two functors

(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862

119898lowast which is given by

119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2

for which119867119866119898119862119898 ∘ 119867119862119898119866119898 = 1S-FC119862119898lowast

and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast

Proof To show that the functor 119867119866119898119862119898 is correct on mor-phisms notice that given1198921 isin 1198661 and1198982 isin 1198722 it follows that120593 ∘ 1198681(1198921 120573

minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)

Additionally to show that the functor 119867119862119898119866119898 is correct onmorphisms notice that given1198921 isin 1198661 and1198981 isin 1198721 it followsthat 120593 ∘ 1198681(1198921 1198981) = 120593 ∘ 1198681(1198921 120573

119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))

532 From S-FC119866119882 to S-FC119862 In this subsection we con-struct an isomorphism between particular subcategories ofthe categories S-FC119866119882 and S-FC119862 respectively


lowast(resp S-FC119862

lowast) is the (nonfull) subcat-

egory of the category S-FC119866119882 (resp S-FC119862) with the same

objects whose morphismsK1

119891

997888rarr K2 have the property that

the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901

997888997888rarr 1198721) is bijective



(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862

lowast which is given by 119867119866119862(K1

119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2

for which119867119866119862 ∘ 119867119862119866 = 1S-FC119862lowastand119867119862119866 ∘ 119867119866119862 = 1S-FC119866119882

lowast

Proof To show that the functor119867119866119862 is correct onmorphismsnotice that given 1198921 isin 1198661 and 1198982 isin 1198722 it follows that1198681(1198921 120573

minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘

1198682(120572(1198921) 1198982) Additionally to show that the functor 119867119862119866

is correct on morphisms notice that given 1198921 isin 1198661 and1198981 isin 1198721 it follows that 1198681(1198921 1198981) = 1198681(1198921 120573

119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))


54 119871-FC119862 versus 119871-FC119863119872119877 In this subsection we makeour framework closer to that of Denniston et al [29] thatis consider a fixed-basis approach to lattice-valued FCAWe construct several functors between the categories 119871-FC119862

and 119871-FC119863119872119877 and show that they fail to provide an adjointsituation between the categories in question

541 From 119871-FC119862 to 119871-FC119863119872119877 This subsection provides afunctor from the category 119871-FC119862 to the category 119871-FC119863119872RMore precisely Theorems 52 and 53 (or also Theorems 59and 60) give rise to the following result


lowastis the (nonfull)

subcategory of the category 119871-FC119862 with the same objects and

with morphisms K1

119891

997888rarr K2 such that the maps 1198661

120572

997888rarr 1198662

1198722

120573119900119901

997888997888rarr 1198721 are surjective


lowast

119867119871119862119871119863997888997888997888997888rarr

119871-FC119863119872119877 which is given by 119867119871119862119871119863(K1

119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2 If ⊤119871 = perp119871 then 119867119871119862119871119863 is anonfull embedding

Proof It is enough to show that the functor is faithful (we donot require 119871 to have the unit)

Given some 119871-FC119862-morphisms K1

1198911

9994909994731198912

K2 such that

119867119871119862119871119863(1198911) = 119867119871119862119871119863(1198912) it follows that (1205721rarr

119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr

119871 To show that 1205721 = 1205722 notice that given 1198921 isin 1198661

(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)

perp119871 otherwise= 120594

⊤119871

1205721(1198921)(1198922)

(41)

for every 1198922 isin 1198662 As a consequence 120594⊤119871

1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871

1205722(1198921) The assumption ⊤119871 = perp119871 then gives

1205721(1198921) = 1205722(1198921) Similar technique can be employed to showthat 120573119900119901

1= 120573

119900119901

2

542 From 119871-FC119863119872119877 to 119871-FC119862 This subsection shows func-torial embeddings of some subcategories of the category119871-FC119863119872119877 into the category 119871-FC119862The results of Section 515(with their respective notations) suggest the following

Theorem 97 Every qsrs-algebra 119871 provides the followingnonfull embeddings

(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(4) 119871-FC119863119872119877119867119871119863119871119862997888997888997888997888rarr 119871-FC119862 given by 119867119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



Proof The only thing that requires verification is nonfullnessof the embeddings We provide the explicit proof for the lastitem which can be easily adapted for each of the remainingones

Consider the crisp case L = CBAlg 119871 = 2 and let119866 = 119872 = |119871| 119868 = 119866 times 119872 We define (cf Lemma 43) the119871-FC119863119872119877-object K = (119866119872 119868) where 119868 = ⊤119871 and thus119867 = ⊤119871 and 119870 = ⊤119871 For every 119878 isin P(119866) and every119879 isin P(119872) 119867 ∘ 119870(119879) = 119872 = 119867(119878) and thus (119878 119879) isa formal protoconcept of K Thus = ⊤119871 and therefore

every pair of mapsP(119866)120572

997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then

an 119871-FC119862-morphism (119867119871119863119871119862(K) = (P(119866)P(119872) ))(120572120573)

997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)

997888997888997888rarr |K| is not an 119871-FC119863119872119877-morphism

543 119871-FC119862 versus 119871-FC119863119872119877 The previous two subsectionsgave the functors 119871-FC119862

lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901


It is easy to see that there exist the restrictions

119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the

previously mentioned two functors A natural question couldbe whether the pair makes an adjoint situationThe followingresult however provides a negative answer

Theorem 99 In general 119867lowast

119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left

adjoint to119867lowast

119871119863119871119862(resp119867lowast

119871119862119871119863)

Proof Suppose that 119867lowast

119871119862119871119863is a left adjoint to 119867lowast

119871119863119871119862 Then

every object K of 119871-FC119862

lowasthas an 119867lowast

119871119863119871119862-universal map that

is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871

119872 )) and therefore 119868(119892 120573119900119901(119905)) = (120572(119892) 119905) for every

119892 isin 119866 and every 119905 isin 119871119872 It follows that 119868(119892 120573119900119901(119905)) isin

perp119871 ⊤119871 Define 119866 = 119892 119872 = 119898 119871 = ([0 1] ⋁⋀) (theunit interval with the standard lattice-theoretic structure)and 119868 = 12 There exists then a unique map 119871

119872

997888rarr

119872 and moreover one obtains that 12 = 119868(119892 (12)) =

(120572(119892) 12) isin 0 1 which is an obvious contradictionSuppose that119867lowast

119871119863119871119862is a left adjoint to119867lowast

119871119862119871119863 Then every

object K of 119871-FC119863119872119877


119871119862119871119863-universal map that is

an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871

119872 )) and thus 119868(119892 120573119900119901(119905)) = (120572(119892) 119905) for every 119892 isin 119866

and every 119905 isin 119871119872 Similar to the aforementioned we get a

contradiction

55 119871-FC119862 versus 119871-FC119863119872119877

119886 The reader should notice that

the results of Section 51 can be easily restricted to the caseof the category 119871-FC119863119872119877

119886 In particular each of Theorems 48

and 55 provides the following result




with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2

Due to the already considerable size of the paper therespective fixed-basis restrictions of the variable-basis settingof the previous subsections are left to the reader

6 Conclusion Open Problems

In this paper we provided different approaches to a possiblelattice-valued generalization of Formal Concept Analysis(FCA) of Ganter and Wille [31] motivated by the recent

study of Denniston et al [29] on this topic In particular wehave constructed several categories whose objects are lattice-valued extensions of formal contexts of FCA and whosemorphisms reflect the crisp setting of Pratt [13] the lattice-valued setting of Denniston et al [29] and the many-valuedsetting of Ganter and Wille [31] themselves In the next stepwe considered many possible functors between the newlydefined categories As a consequence we embedded each ofthe constructed categories into its respective counterpartsThe crucial difference of this paper from the motivating oneof Denniston et al [29] can be briefly formulated as follows

(i) The underlying lattices of lattice-valued formal con-texts are extensions of quantales instead of restrictedto precisely commutative quantales as in [29]

(ii) All the categories of lattice-valued formal contextsand almost all of their respective functors are madevariable-basis (in the sense of [4]) instead of beingfixed-basis as in [29]

(iii) Unlike [29] we consider the setting of many-valuedformal contexts of Ganter and Wille [31]

In the wake of the constructed functors of this paperwe can single out the following important properties of theapproach of Denniston et al [29] which is based on thecategory of Galois connections of [32]

Firstly their approach falls rather out of the standardsettings of Pratt as well as Ganter and Wille On one handthere exists an obvious isomorphism between the subcate-gory 2-PGAL of the categoryGAL [32] of Galois connectionson crisp powersets and the crisp category 2-FC119863119872119877

119886(cf

Definitions 31(1) and 37) of this paper (essentially due toDen-niston et al [29])On the other hand the result ofTheorem 42shows that given a unital quantale 119871 the categories 119871-PGALand 119871-FC119863119872119877

119886are in general nonisomorphic The variable-

basis approach complicates the case even more Speakingmetamathematically while the notion of formal conceptof FCA does involve an order-reversing Galois connectionbetween the respective powersets in its definition its mainessence appears to come from a binary relation on two setsthat is the set of objects (119866) and the set of their respectiveattributes (119872) which says whether a given object 119892 isin 119866

has a given property 119898 isin 119872 Thus these are not the Galoisconnections but their generating relations which seem toplay the main role Moreover Theorem 99 of the paper statesthat while a particular subcategory of the category of lattice-valued formal contexts in the sense of Pratt is isomorphic to anonfull subcategory of the category ofDenniston et al it doesnot provide a (co)reflective subcategory of the latter categorywith respect to the constructed functors (at the momentthough we do not know whether one can define differentfunctors which do give rise to a (co)reflective subcategory inquestion)

Secondly the category of lattice-valued formal contextsbased on the category of Galois connections tends to havethe nature of a fixed-basis category More precisely whilemaking the setting of Denniston et al variable-basis in thecurrent paper we were unable to involve the morphisms of


the underlying algebras of lattice-valued formal contexts in itsdefinition thereby providing a quasi-variable-basis setting

We would like to end the section with several openproblems related to the setting of this paper

61 Proper Way of Building Lattice-Valued FCA The previ-ously mentioned discussion on the difference between thesettings of binary relations and Galois connections in thelattice-valued case motivates the following open problem

Problem 1 Is it possible to build a lattice-valued approach toFCA which is based on order-reversing Galois connectionson lattice-valued powersets which are not generated bylattice-valued relations on their respective sets of objects andtheir attributes

This problem seems especially important becausewe haveestablished that using order-reversing Galois connections ismore general than using relations It may even seem thatusing order-reversing Galois connections with the corre-sponding Birkhoff operators is more robust and better thanusing relations Therefore it seems important to be ableto understand how to use the Birkhoff operators when acorresponding relation may not exist The answer may notdepend on defining a relation The answer may depend onbeing able to do formal concept analysis without relations

Notice that although the notions of formal conceptprotoconcept andpreconcept (cfDefinition 44) can be easilydefined with the help of just a Galois connection in hand it istheir respective interpretation that poses the main problemMore precisely what will be then in place of the crucialterm ldquoan object 119892 has an attribute 119898rdquo It is our opinionthat such an approach can still be developed substitutingthe required single lattice-valued relation 119866 times 119872

119868

997888rarr 119871 by apair of lattice-valued relations 1198681(119892119898) = (120572(120594

1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573

119871119872 is a given order-

reversing Galois connection which violates the conditions ofTheorem 42 (notice that the case of a commutative quantaleie sdotrarr 119897sdot = sdotrarr 119903sdot simplifies but does not save the situation)In view of the discussion a particular instance of Problem 1is as follows

Problem 2 What is the interpretation of the relations 1198681 and1198682 and what kind of lattice-valued FCA can be developedthrough them

An ad hoc approach could be to read 1198681(119892119898) as thedegree to which ldquoan object 119892 has an attribute 119898rdquo and to read1198682(119892119898) as the degree to which ldquoan attribute 119898 is shared byan object 119892rdquo that is distinguishing between the classicallyindistinguishable expressions ldquoto have an attributerdquo and ldquoanattribute is being sharedrdquo

62 Adjoint Situations between Possible Approaches to Lattice-Valued FCA In the paper we have constructed functorialembeddings of several categories of lattice-valued formalcontexts into each other However we failed to construct

a single adjoint situation between the categories in questionbeing able just to obtain a negative result in this respect(Theorem 99) In view of the discussion our next problemcould be formulated as follows

Problem 3 Construct (if possible) an adjoint situationbetween the categories of lattice-valued formal contextsintroduced in this paper

Notice that the problem has a close relation to thediscussion of the previous subsection More precisely anadjoint situation between the categories of lattice-valuedformal contexts in the sense of Pratt and Denniston et alrespectively can provide a partial answer to the problem ofdispensing with binary relations in lattice-valued FCA andsubstituting them by order-reversing Galois connections

63 Proper Morphisms of Lattice-Valued Formal ContextsIn the paper we have dealt with both lattice-valued formalcontexts and their respectivemorphismsWhereas the formerhave already been treated in the fuzzy literature the caseof the latter is still not sufficiently clear In particular wehave provided three possible approaches to (lattice-valued)formal context morphisms that is that of Pratt that ofDenniston et al and that of Ganter and Wille Taking apartthe metamathematical discussion on the fruitfulness of anyof the previously mentioned approaches the last problem ofthe paper can be stated then as follows

Problem 4 What is the most suitable (if any) way of definingthe morphisms of lattice-valued FCA

All the previously mentioned open problems will beaddressed in our subsequent articles on the topic of lattice-valued FCA An interested reader is kindly invited to sharethe effort

Acknowledgments

The author is grateful to Professor W Tholen and ProfessorJ Steprans who invited him to spend the first six months ofthe year 2012 at theDepartment ofMathematics and Statisticsof York University (Toronto ON Canada) during whichperiod the paper was prepared The author would also liketo thank J T Denniston A Melton (Kent State University)and S E Rodabaugh (Youngstown State University) for goingthrough the draft version of the paper pointing out its variousdeficiencies and suggesting their adjustments This researchwas supported by the ESF Project no CZ1072300200051ldquoAlgebraic Methods in Quantum Logicrdquo of the Masaryk Uni-versity in Brno Czech Republic S of L S-FC119863119872119877

119886following

References

[1] S A Solovyov ldquoVariable-basis topological systems versusvariable-basis topological spacesrdquo Soft Computing vol 14 no10 pp 1059ndash1068 2010

[2] S Vickers Topology via Logic Cambridge University PressCambridge UK 1989


[3] J T Denniston and S E Rodabaugh ldquoFunctorial relation-ships between lattice-valued topology and topological systemsrdquoQuaestiones Mathematicae vol 32 no 2 pp 139ndash186 2009

[4] S E Rodabaugh ldquoCategorical foundations of variable-basisfuzzy topologyrdquo in Mathematics of Fuzzy Sets Logic Topologyand Measure Theory U Hohle and S E Rodabaugh Eds vol3 of The Handbooks of Fuzzy Sets Series pp 273ndash388 KluwerAcademic Publishers Dordrecht The Netherlands 1999

[5] C Guido ldquoAttachment between fuzzy points and fuzzy setsrdquoin Abstracts of the 30th Linz Seminar on Fuzzy Set Theory UBodenhofer B De Baets E P Klement and S Saminger-PlatzEds pp 52ndash54 Johannes Kepler Universitat Linz Austria2009

[6] C Guido ldquoFuzzy points and attachmentrdquo Fuzzy Sets andSystems vol 161 no 16 pp 2150ndash2165 2010

[7] U Hohle ldquoA note on the hypergraph functorrdquo Fuzzy Sets andSystems vol 131 no 3 pp 353ndash356 2002

[8] J TDennistonAMelton and S E Rodabaugh ldquoLattice-valuedtopological systemsrdquo in Abstracts of the 30th Linz Seminar onFuzzy SetTheory U Bodenhofer B De Baets E P Klement andS Saminger-Platz Eds pp 24ndash31 Johannes Kepler UniversitatLinz Austria 2009

[9] J T Denniston A Melton and S E Rodabaugh ldquoInterweav-ing algebra and topology Lattice-valued topological systemsrdquoFuzzy Sets and Systems vol 192 pp 58ndash103 2012

[10] T Kubiak and A Sostak ldquoFoundations of the theory of (119871119872)-fuzzy topological spacesrdquo in Abstracts of the 30th Linz Seminaron Fuzzy Set Theory U Bodenhofer B De Baets E P Klementand S Saminger-Platz Eds pp 70ndash73 Johannes Kepler Univer-sitat Linz Austria 2009

[11] J TDennistonAMelton and S E Rodabaugh ldquoLattice-valuedpredicate transformers and interchange systemsrdquo in Abstractsof the 31st Linz Seminar on Fuzzy Set Theory P Cintula EP Klement and L N Stout Eds pp 31ndash40 Johannes KeplerUniversitat Linz Austria 2010

[12] E W Dijkstra A Discipline of Programming Prentice-HallEnglewood Ill USA 1976

[13] V Pratt ldquoChu spacesrdquo in School on Category Theory andApplications Lecture Notes of Courses Coimbra Portugal vol21 of Textos Matematicas Series B pp 39ndash100 Universidadede Coimbra Departamento deMatematica Coimbra Portugal1999

[14] M Barr lowast-Autonomous Categories With An Appendix By Po-Hsiang Chu Springer-Verlag New York NY USA 1979

[15] J T Denniston A Melton and S E Rodabaugh ldquoLattice-valued predicate transformers Chu games and lattice-valuedtransformer systemsrdquo Fuzzy Sets and Systems In press

[16] D Aerts ldquoFoundations of quantum physics a general realisticand operational approachrdquo International Journal of TheoreticalPhysics vol 38 no 1 pp 289ndash358 1999

[17] D Aerts E Colebunders A Van Der Voorde and B VanSteirteghem ldquoState property systems and closure spaces a studyof categorical equivalencerdquo International Journal of TheoreticalPhysics vol 38 no 1 pp 359ndash385 1999

[18] D Aerts E Colebunders A Van Der Voorde and B VanSteirteghem ldquoOn the amnestic modification of the category ofstate property systemsrdquo Applied Categorical Structures vol 10no 5 pp 469ndash480 2002

[19] S A Solovyov ldquoOn a generalization of the concept of stateproperty systemrdquo Soft Computing pp 1ndash12 2011

[20] S A Solovyov ldquoHypergraph functor and attachmentrdquo FuzzySets and Systems vol 161 no 22 pp 2945ndash2961 2010

[21] S A Solovyov ldquoCategorical foundations of variety-based topol-ogy and topological systemsrdquo Fuzzy Sets and Systems vol 192pp 176ndash200 2012

[22] S A Solovyov ldquoLocalification of variable-basis topologicalsystemsrdquo Quaestiones Mathematicae vol 34 no 1 pp 11ndash332011

[23] J C Kelly ldquoBitopological spacesrdquo Proceedings London Mathe-matical Society vol 13 no 3 pp 71ndash89 1963

[24] C J Mulvey and J W Pelletier ldquoOn the quantisation of pointsrdquoJournal of Pure and Applied Algebra vol 159 no 2-3 pp 231ndash295 2001

[25] C J Mulvey and J W Pelletier ldquoOn the quantisation of spacesrdquoJournal of Pure and Applied Algebra vol 175 no 1ndash3 pp 289ndash325 2002

[26] S Solovyov ldquoComposite variety-based topological theoriesrdquoFuzzy Sets and Systems vol 195 pp 1ndash32 2012

[27] S A Solovyov ldquoGeneralized fuzzy topology versus non-commutative topologyrdquo Fuzzy Sets and Systems vol 173 no 1pp 100ndash115 2011

[28] S Solovyov ldquoLattice-valuedsoft algebrasrdquo Soft Computing Inpress

[29] J T Denniston A Melton and S E Rodabaugh ldquoFormalconcept analysis and lattice-valuedChu systemsrdquoFuzzy Sets andSystems vol 216 pp 52ndash90 2013

[30] J T Denniston A Melton and S E Rodabaugh ldquoFormalconcept analysis and lattice-valued interchange systemsrdquo inAbstracts of the 32nd Linz Seminar on Fuzzy Set Theory DDubois M Grabisch R Mesiar and E P Klement Eds pp41ndash47 Johannes Kepler Universitat Linz Austria 2011

[31] B Ganter and R Wille Formale Begriffsanalyse MathematischeGrundlagen Springer Berlin Germany 1996

[32] J M McDill A C Melton and G E Strecker ldquoA categoryof Galois connectionsrdquo in Proceedings of the Conference onCategoryTheory and Computer Science vol 283 of Lecture Notesin Computer Science pp 290ndash300 Edinburgh UK 1987

[33] R B Belohlavek Fuzzy Relational Systems Foundations andPrinciples Kluwer Academic Publishers New York NY USA2002

[34] D Kruml and J Paseka ldquoAlgebraic and categorical aspects ofquantalesrdquo Handbook of Algebra vol 5 pp 323ndash362 2008

[35] K I Rosenthal Quantales and Their Applications vol 234of Pitman Research Notes in Mathematics Addison WesleyLongman Boston Mass USA 1990

[36] B Vormbrock and R Wille ldquoSemiconcept and protoconceptalgebras the basic theoremsrdquo in Formal Concept AnalysisFoundations and Applications B Ganter G Stumme and RWille Eds vol 3626 of Lecture Notes in Computer Science pp34ndash48 Springer Berlin Germany 2005

[37] M Erne J Koslowski A Melton and G E Strecker ldquoA primeron Galois connectionsrdquo Annals of the New York Academy ofSciences vol 704 pp 103ndash125 1993

[38] R Belohlavek ldquoFuzzy Galois connectionsrdquoMathematical LogicQuarterly vol 45 no 4 pp 497ndash504 1999

[39] G Birkhoff Lattice Theory American Mathematical SocietyPublications New York NY USA 1940

[40] J G Garcıa I Mardones-Perez M A de Prada Vicente and DZhang ldquoFuzzy Galois connections categoricallyrdquoMathematicalLogic Quarterly vol 56 no 2 pp 131ndash147 2010


[41] J Adamek H Herrlich and G E Strecker Abstract and Con-crete Categories The Joy of Cats Dover Publications MineolaNY USA 2009

[42] H Herrlich and G E Strecker Category Theory vol 1 ofSigma Series in Pure Mathematics Heldermann Verlag BerlinGermany 3rd edition 2007

[43] S Mac Lane Categories For the Working MathematicianSpringer-Verlag New York NY USA 2nd edition 1998

[44] P M Cohn Universal Algebra D Reidel Publishing CompanyDordrecht The Netherlands 1981

[45] G Richter Kategorielle Algebra Akademie-Verlag Berlin Ger-many 1979

[46] B Banaschewski and E Nelson ldquoTensor products and bimor-phismsrdquoCanadianMathematical Bulletin vol 19 no 4 pp 385ndash402 1976

[47] E GManesAlgebraicTheories Springer-Verlag NewYork NYUSA 1976

[48] P T Johnstone Stone Spaces Cambridge University PressCambridge UK 1982

[49] S E Rodabaugh ldquoRelationship of algebraic theories to powersettheories and fuzzy topological theories for Lattice-valuedmath-ematicsrdquo International Journal of Mathematics and Mathemati-cal Sciences vol 2007 Article ID 43645 71 pages 2007

[50] G Gierz K H Hofmann K Keimel et al Continuous Latticesand Domains Cambridge University Press Cambridge UK2003

[51] S E Rodabaugh ldquoPowerset operator based foundation forpoint-set lattice-theoretic (poslat) fuzzy set theories and topolo-giesrdquo Quaestiones Mathematicae vol 20 no 3 pp 463ndash5301997

[52] S E Rodabaugh ldquoPowerset operator foundations for poslatfuzzy set theories and topologiesrdquo in Mathematics of FuzzySets Logic Topology and Measure Theory U Hohle and SE Rodabaugh Eds vol 3 of The Handbooks of Fuzzy SetsSeries pp 91ndash116 Kluwer Academic Publishers DordrechtTheNetherlands 1999

[53] S E Rodabaugh ldquoRelationship of algebraic theories to power-sets over objects in Set and Settimes Crdquo Fuzzy Sets and Systems vol161 no 3 pp 453ndash470 2010

[54] S Solovyov ldquoPowerset operator foundations for catalg fuzzy settheoriesrdquo Iranian Journal of Fuzzy Systems vol 8 no 2 pp 1ndash462011

[55] S Krajci ldquoA categorical view at generalized concept latticesrdquoKybernetika vol 43 no 2 pp 255ndash264 2007

[56] G Q Zhang ldquoChu spaces concept lattices and domainsrdquoin Proceedings of the 9th Conference on the MathematicalFoundations of Programming Semantics (Montreal rsquo03) vol 83of Electronic Notes in Theoretical Computer Science 2004

[57] K Deiters and M Erne ldquoSums products and negations ofcontexts and complete latticesrdquo Algebra Universalis vol 60 no4 pp 469ndash496 2009

[58] M Erne Categories of Contexts University of HannoverHanover Germany 1990

[59] B Ganter ldquoRelational Galois connectionsrdquo in Proceedings of the5th International Conference on Formal Concept Analysis S OKuznetsov Ed vol 4390 of Lecture Notes in Computer Sciencepp 1ndash17 2007

[60] H Mori ldquoFunctorial properties of formal concept analysisrdquo inProceedings of the 15th International Conference on ConceptualStructures U Priss Ed vol 4604 of Lecture Notes in ComputerScience pp 505ndash508 2007

[61] L Shen and D Zhang ldquoThe concept lattice functorsrdquo Interna-tional Journal of Approximate Reasoning vol 54 no 1 pp 166ndash183 2013

[62] A Frascella ldquoFuzzyGalois connections underweak conditionsrdquoFuzzy Sets and Systems vol 172 no 1 pp 33ndash50 2011

[63] M E Della Stella and C Guido ldquoL-relations and Galoistrianglesrdquo inAlgebraic Semantics for Uncertainty and VaguenessPalazzo Genovese Salerno Italy 2011

[64] A Frascella and C Guido ldquoTransporting many-valued setsalong many-valued relationsrdquo Fuzzy Sets and Systems vol 159no 1 pp 1ndash22 2008

Submit your manuscripts athttpwwwhindawicom

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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Guido to a new level [20] (eg made it variable-basis in thesense of Rodabaugh [4] as well as extended the machinery ofHohle [7] to construct its right adjoint functor) Moreoverin [21] we provided a thorough categorical extension of thesystem spatialization procedure to the variety-based settingthe respective localification procedure being elaborated in[22] Additionally certain aspects of the new theory bearinglinks to bitopological spaces of Kelly [23] as well as tononcommutative topology of Mulvey and Pelletier [24 25]were treated extensively in [26 27] respectively

Seeing the fruitfulness of their newly introduced notionsboth Denniston et al and the author of this paper indepen-dently turned their attention to possible applications of thearising system framework to the areas beyond topology Inparticular in [28] we started the theory of lattice-valuedsoft universal algebra whereas Denniston et al presentedin [29 30] a challenging categorical outlook on a certainlattice-valued extension of Formal Concept Analysis (FCA)of Ganter and Wille [31] Similar to the case of lattice-valuedtopological systems their research motivated us to take adeeper look into the proposed topic of lattice-valued FCAthereby streamlining the motivating approach of Dennistonet al and also extending some of their results

Modifying slightly the definition of the category of Galoisconnections of McDill et al [32] Denniston et al [29]consider its relationships to the already mentioned categoryof interchange systems the objects of both categories beingessentially formal contexts of FCA The main result is acategorical embedding of each category into its counterpartthe ultimate conclusion though amounting to the categoriesin question being rather different that is the two viewpointson FCA are not categorically isomorphic The rest of thepaper provides a lattice-valued fixed-basis extension of thecrisp achievements based on the lattice-valued FCA ofBelohlavek [33] Moreover towards the end of the paperwith the help of the constructed embeddings the authorscompare the categories of (lattice-valued) formal contextsand (lattice-valued) topological systems inside the categoryof (lattice-valued) interchange systems making a significant(and rather striking) metamathematical conclusion on theirbeing disjoint

The just mentioned approach of Denniston et al canbe extended taking into consideration the following factsFirstly it never gives much attention to the links betweeninterchange systems and Chu spaces thereby making nouse of the well-developed machinery of the latter Secondlywhile pursuing their approach to many-valued FCA theauthors never mention the already existing many-valuedformal contexts of Ganter and Wille [31] which (beingsimilar to Chu spaces) essentially extend their own Thirdlybeing categorically oriented the paper introduces two typesof formal context morphisms in terms of Chu transformsand Galois connections respectively but misses the well-established context morphisms of FCA which are differentfrom those of the paper Lastly (andmost important) turningto fuzzification of their notions the authors concentrateon the fixed-basis lattice-valued approach notwithstandingthat the more promising variable-basis setting (initiated byone of them) has already gained its popularity in the fuzzy

community currently being a kind of de facto standard forfuzzy structures

The main purposes of this paper are to extend the resultsof Denniston et al and to clarify the relationships betweenChu spaces many-valued formal contexts of FCA lattice-valued interchange systems and Galois connections Inparticular we introduce three categories (each of themhavinga twofold modification arising from its respective variable-basis) whose objects are lattice-valued versions of formalcontexts of FCA but whose morphisms are quite differenteach instance reflecting the approach of Pratt Ganter andWille andDenniston et al respectively Following the resultsof [29] we embed (nonfully) each of the categories into itstwo counterparts showing however that the constructedembeddings fail to provide a categorical isomorphism (evenan adjoint situation) between any two of the presentedthree approaches As a consequence we obtain three dif-ferent categorical outlooks on FCA which are done in thevariable-basis lattice-valued framework of [4] Moreoverthe respective underlying lattices of the fuzzified formalcontexts are quantales [34 35] instead of complete residuatedlattices of Denniston et al that is the heavy assumption oncommutativity of the quantale multiplication is dropped Inparticular we show lattice-valued generalizations of formalconcept preconcept and protoconcept of [36] which arebased on an arbitrary (namely non-commutative) quantale(actually an algebra having a quantale as its reduct)

Similar to Denniston et al we develop the requiredmachinery of quantale-valued Galois connections whichextends that of Erne et al [37] and which is motivatedby the commutative approach of fuzzy Galois connectionsof Belohlavek [38] For example we provide yet anothergeneralization to the quantale setting of the well-knownfact that every order-reversing Galois connection betweenpowersets arises as a pair of Birkhoff operators with respectto a binary relation [39] The generalization though appearsto be somewhat truncated in the sense that while lattice-valued binary relations still give rise to order-reversingGaloisconnections the converseway is not always available ([38 40]do restore the full result even in the lattice-valued case butrely on the more demanding notion of fuzzy Galois con-nection) Moreover we define a category of order-preservingGalois connections as a certain subcategory of the categoryof Chu spaces in which (unlike McDill et al and Dennistonet al) Galois connections play the role of morphisms ratherthan objects thereby providing particular instances of Chutransforms We fail to provide a similar machinery for theorder-reversing case

At the end of the paper we will notice that the approachto FCA through order-reversing Galois connections takenup by Denniston et al does not suit exactly the underlyingideas of FCA of Ganter and Wille which are formulatedin terms of the incidence relations of formal contexts Themain sticking point here is the previously mentioned factthat the important crisp case one-to-one correspondencebetween order-reversing Galois connections on powersetsand pairs of Birkhoff operators breaks down in the lattice-valued case (quantales) As a consequence the incidencerelation viewpoint provides one with a Galois connection








120582)120582isinΛ)


120582

997888997888rarr


1198601

120582)120582isinΛ)

120593

997888rarr (1198602 (1205961198602

120582)120582isinΛ) is a map 1198601

120593


1198601 1198602

1198601198991205821 119860

1198991205822

1205961198601120582

1205961198602120582

120593

120593

119899120582

(1)





minus







120593


|1198602|120593⊢


⊢(1198862) for every 1198861 isin


⊢⩽ 11198602

and11198601





120593


997888997888rarr






119904isin119878(119904 otimes 119902) for





(1) 119902rarr 119897 (⋀119878) = ⋀119904isin119878

(119902rarr 119897 119904) and 119902rarr 119903 (⋀119878) =



(2) (⋁119878)rarr 119897 119902 = ⋀119904isin119878

(119904rarr 119897 119902) and (⋁119878)rarr 119903 119902 =

⋀119904isin119878
















1015840= ⊤119860





1119860997888997888rarr 119860



119891



997888997888997888rarr

P(1198832) (resp P(1198832)119891larr





119886



119878


120594119886

119878(119909) =

119886 119909 isin 119878



119909or 1205941119860






S(minus)rarr


997888997888997888rarr

(1198832 1198712))rarr= 119871

1198831

1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)rarr (120572))(1199092) =

120593(⋁119891(1199091)=1199092

120572(1199091))


1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 is ⋁-


1 for every

1199092 isin 1198832 ((119891 120593)rarr(⋁ 119878))(1199092) = 120593(⋁

119891(1199091)=1199092(⋁ 119878)(1199091)) =

120593(⋁119891(1199091)=1199092

⋁119904isin119878119904(1199091)) = 120593(⋁

119904isin119878⋁119891(1199091)=1199092

119904(1199091)) =

⋁119904isin119878120593(⋁

119891(1199091)=1199092119904(1199091)) = ⋁

119904isin119878((119891 120593)

rarr(119904))(1199092) =

(⋁119904isin119878(119891 120593)

rarr(119904))(1199092)



997888997888997888rarr

(1198832 1198712) and (1198832 1198712)(11989110158401205931015840)


1198831

1and every 1199093 isin 1198833 ((119891

1015840 120593

1015840)rarr

∘

(119891 120593)rarr(120572))(1199093) = 120593

1015840(⋁

1198911015840(1199092)=1199093((119891 120593)

rarr(120572))(1199092)) =

1205931015840(⋁

1198911015840(1199092)=1199093120593(⋁

119891(1199091)=1199092120572(1199091))) = 120593

1015840∘ 120593(⋁

1198911015840∘119891(1199091)=1199093120572(1199091)) =

((1198911015840∘119891 120593

1015840∘ 120593)

rarr(120572))(1199093) = (((119891

1015840 120593

1015840) ∘ (119891 120593))

rarr(120572))(1199093)





Theorem 13



997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊣[

=

≀1198711198831

1≀

(119891120593)⊣[

997888997888997888997888997888rarr ≀1198711198832

2≀ where ((119891 120593)

⊣[(120572))(1199092) =

120593119900119901⊣(⋁

119891(1199091)=1199092120572(1199091))


S-morphism 1198711

120593

997888rarr 1198712 the map |1198711|120593119900119901⊢

997888997888997888rarr |1198712| is ⋁-


997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊢[

=

1198711198831

1

(119891120593)⊢[

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)

⊢[(120572))(1199092) =

120593119900119901⊢(⋁

119891(1199091)=1199092120572(1199091))



997888rarr 1198712 and 1198712

1205931015840


⊣

(resp (1205931015840 ∘120593)⊢ = 120593⊢ ∘1205931015840⊢

)





((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr

= 1198601198831

1

((119891120593)larr)119900119901

997888997888997888997888997888997888997888rarr 1198601198832

2 where

(119891 120593)larr(120572) = 120593

119900119901∘ 120572 ∘ 119891


2

(119891120593)larr

997888997888997888997888997888rarr 1198601198831



1198832

2)119899120582 for every


((119891 120593)larr(120596

1198601198832

2

120582(⟨120572119894⟩119899120582

))) (1199091)

= 120593119900119901∘ (120596

1198601198832

2

120582(⟨120572119894⟩119899120582

)) ∘ 119891 (1199091)

= 120593119900119901∘ (120596

1198602

120582(⟨120572119894 ∘ 119891 (1199091)⟩119899120582

))

= 1205961198601

120582(⟨120593

119900119901∘ 120572119894 ∘ 119891 (1199091)⟩119899120582

)

= 1205961198601

120582(⟨((119891 120593)

larr(120572119894)) (1199091)⟩119899120582

)

= (1205961198601198831

1

120582(⟨(119891 120593)

larr(120572119894)⟩119899120582

)) (1199091)

(3)



997888997888997888rarr (1198832 1198602) and

(1198832 1198602)(11989110158401205931015840)


3 (119891 120593)larr ∘

(1198911015840 120593

1015840)larr(120572) = (119891 120593)

larr(120593

1015840119900119901∘ 120572 ∘119891

1015840) = 120593

119900119901∘ 120593

1015840119900119901∘ 120572 ∘119891

1015840∘ 119891 =

(1205931015840∘ 120593)

119900119901∘ 120572 ∘ 119891

1015840∘ 119891 = (119891

1015840∘ 119891 120593

1015840∘ 120593)

larr(120572) = ((119891

1015840 120593

1015840) ∘

(119891 120593))larr(120572)





S(minus)dagger


(119891120593)

997888997888997888rarr

(1198832 1198712))dagger

= ≀1198711198831

1≀

((119891120593)dagger)119900119901

997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger(120572) =

120593dagger∘ 120572 ∘ 119891



Theorem 16


S(minus)rarr119900


997888997888997888rarr

(1198832 1198712))rarr119900

= 1198711198831

1

((119891120593)rarr119900

)119900119901

997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)rarr119900

(120572))(1199091) = 120593119900119901(⋁

119891119900119901(1199092)=1199091120572(1199092))


S(minus)⊣[119900


997888997888997888rarr

(1198832 L2))⊣[119900

= ≀1198711198831

1≀

((119891120593)⊣[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where

((119891 120593)⊣[119900

(120572))(1199091) = 120593⊣(⋁

119891119900119901(1199092)=1199091120572(1199092))


morphism 1198711

120593

997888rarr 1198712 the map |1198712|120593⊢


S(minus)⊢[119900


997888997888997888rarr

(1198832 1198712))⊢[119900

= 1198711198831

1

((119891120593)⊢[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)⊢[119900

(120572))(1199091) = 120593⊢(⋁

119891119900119901(1199092)=1199091120572(1199092))


Theorem 17




((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr119900

= 1198601198831

1

(119891120593)larr119900

997888997888997888997888997888rarr 1198601198832

2

where (119891 120593)larr119900(120572) = 120593 ∘ 120572 ∘ 119891




997888997888997888997888997888rarr CSLat(Ξ2)


997888997888997888rarr (1198832 1198712))dagger119900

=

≀1198711198831

1≀

(119891120593)dagger119900

997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger119900

(120572) = 120593119900119901dagger

∘ 120572 ∘

119891119900119901




(1198832 ⩽) are posets and1198831

119891





(1) 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832(resp 11198831 ⩽ 119892 ∘ 119891 and

11198832⩽ 119891 ∘ 119892)


(3) 119891(⋁119878) = ⋁119904isin119878119891(119904) and 119892(⋀119879) = ⋀

119905isin119879119892(119905) (resp

119891(⋁119878) = ⋀119904isin119878119891(119904) and 119892(⋁119879) = ⋀

119905isin119879119892(119905)) provided


(4) 119892 ∘ 119891 ∘ 119892 = 119892 and 119891 ∘ 119892 ∘ 119891 = 119891



and (1198831 ⩽)

119891

999448999471119892


⩽ 119892 ∘ 119891 and 11198832 ⩽ 119891 ∘ 119892)


Example 21


120593


⊢ 1198602) (resp (1198602 120593

⊢

120593 1198601))



(3) Every map 1198831

119891


rarr 119891

larrP(1198832))



997888997888997888rarr (1198832 1198712)


1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)








⊨


⊨(119909minus)


Morphisms 1198631


times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891


119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)






Example 24











(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















120582)120582isinΛ)


120582

997888997888rarr


1198601

120582)120582isinΛ)

120593

997888rarr (1198602 (1205961198602

120582)120582isinΛ) is a map 1198601

120593


1198601 1198602

1198601198991205821 119860

1198991205822

1205961198601120582

1205961198602120582

120593

120593

119899120582

(1)





minus







120593


|1198602|120593⊢


⊢(1198862) for every 1198861 isin


⊢⩽ 11198602

and11198601





120593


997888997888rarr






119904isin119878(119904 otimes 119902) for





(1) 119902rarr 119897 (⋀119878) = ⋀119904isin119878

(119902rarr 119897 119904) and 119902rarr 119903 (⋀119878) =



(2) (⋁119878)rarr 119897 119902 = ⋀119904isin119878

(119904rarr 119897 119902) and (⋁119878)rarr 119903 119902 =

⋀119904isin119878
















1015840= ⊤119860





1119860997888997888rarr 119860



119891



997888997888997888rarr

P(1198832) (resp P(1198832)119891larr





119886



119878


120594119886

119878(119909) =

119886 119909 isin 119878



119909or 1205941119860






S(minus)rarr


997888997888997888rarr

(1198832 1198712))rarr= 119871

1198831

1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)rarr (120572))(1199092) =

120593(⋁119891(1199091)=1199092

120572(1199091))


1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 is ⋁-


1 for every

1199092 isin 1198832 ((119891 120593)rarr(⋁ 119878))(1199092) = 120593(⋁

119891(1199091)=1199092(⋁ 119878)(1199091)) =

120593(⋁119891(1199091)=1199092

⋁119904isin119878119904(1199091)) = 120593(⋁

119904isin119878⋁119891(1199091)=1199092

119904(1199091)) =

⋁119904isin119878120593(⋁

119891(1199091)=1199092119904(1199091)) = ⋁

119904isin119878((119891 120593)

rarr(119904))(1199092) =

(⋁119904isin119878(119891 120593)

rarr(119904))(1199092)



997888997888997888rarr

(1198832 1198712) and (1198832 1198712)(11989110158401205931015840)


1198831

1and every 1199093 isin 1198833 ((119891

1015840 120593

1015840)rarr

∘

(119891 120593)rarr(120572))(1199093) = 120593

1015840(⋁

1198911015840(1199092)=1199093((119891 120593)

rarr(120572))(1199092)) =

1205931015840(⋁

1198911015840(1199092)=1199093120593(⋁

119891(1199091)=1199092120572(1199091))) = 120593

1015840∘ 120593(⋁

1198911015840∘119891(1199091)=1199093120572(1199091)) =

((1198911015840∘119891 120593

1015840∘ 120593)

rarr(120572))(1199093) = (((119891

1015840 120593

1015840) ∘ (119891 120593))

rarr(120572))(1199093)





Theorem 13



997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊣[

=

≀1198711198831

1≀

(119891120593)⊣[

997888997888997888997888997888rarr ≀1198711198832

2≀ where ((119891 120593)

⊣[(120572))(1199092) =

120593119900119901⊣(⋁

119891(1199091)=1199092120572(1199091))


S-morphism 1198711

120593

997888rarr 1198712 the map |1198711|120593119900119901⊢

997888997888997888rarr |1198712| is ⋁-


997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊢[

=

1198711198831

1

(119891120593)⊢[

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)

⊢[(120572))(1199092) =

120593119900119901⊢(⋁

119891(1199091)=1199092120572(1199091))



997888rarr 1198712 and 1198712

1205931015840


⊣

(resp (1205931015840 ∘120593)⊢ = 120593⊢ ∘1205931015840⊢

)





((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr

= 1198601198831

1

((119891120593)larr)119900119901

997888997888997888997888997888997888997888rarr 1198601198832

2 where

(119891 120593)larr(120572) = 120593

119900119901∘ 120572 ∘ 119891


2

(119891120593)larr

997888997888997888997888997888rarr 1198601198831



1198832

2)119899120582 for every


((119891 120593)larr(120596

1198601198832

2

120582(⟨120572119894⟩119899120582

))) (1199091)

= 120593119900119901∘ (120596

1198601198832

2

120582(⟨120572119894⟩119899120582

)) ∘ 119891 (1199091)

= 120593119900119901∘ (120596

1198602

120582(⟨120572119894 ∘ 119891 (1199091)⟩119899120582

))

= 1205961198601

120582(⟨120593

119900119901∘ 120572119894 ∘ 119891 (1199091)⟩119899120582

)

= 1205961198601

120582(⟨((119891 120593)

larr(120572119894)) (1199091)⟩119899120582

)

= (1205961198601198831

1

120582(⟨(119891 120593)

larr(120572119894)⟩119899120582

)) (1199091)

(3)



997888997888997888rarr (1198832 1198602) and

(1198832 1198602)(11989110158401205931015840)


3 (119891 120593)larr ∘

(1198911015840 120593

1015840)larr(120572) = (119891 120593)

larr(120593

1015840119900119901∘ 120572 ∘119891

1015840) = 120593

119900119901∘ 120593

1015840119900119901∘ 120572 ∘119891

1015840∘ 119891 =

(1205931015840∘ 120593)

119900119901∘ 120572 ∘ 119891

1015840∘ 119891 = (119891

1015840∘ 119891 120593

1015840∘ 120593)

larr(120572) = ((119891

1015840 120593

1015840) ∘

(119891 120593))larr(120572)





S(minus)dagger


(119891120593)

997888997888997888rarr

(1198832 1198712))dagger

= ≀1198711198831

1≀

((119891120593)dagger)119900119901

997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger(120572) =

120593dagger∘ 120572 ∘ 119891



Theorem 16


S(minus)rarr119900


997888997888997888rarr

(1198832 1198712))rarr119900

= 1198711198831

1

((119891120593)rarr119900

)119900119901

997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)rarr119900

(120572))(1199091) = 120593119900119901(⋁

119891119900119901(1199092)=1199091120572(1199092))


S(minus)⊣[119900


997888997888997888rarr

(1198832 L2))⊣[119900

= ≀1198711198831

1≀

((119891120593)⊣[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where

((119891 120593)⊣[119900

(120572))(1199091) = 120593⊣(⋁

119891119900119901(1199092)=1199091120572(1199092))


morphism 1198711

120593

997888rarr 1198712 the map |1198712|120593⊢


S(minus)⊢[119900


997888997888997888rarr

(1198832 1198712))⊢[119900

= 1198711198831

1

((119891120593)⊢[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)⊢[119900

(120572))(1199091) = 120593⊢(⋁

119891119900119901(1199092)=1199091120572(1199092))


Theorem 17




((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr119900

= 1198601198831

1

(119891120593)larr119900

997888997888997888997888997888rarr 1198601198832

2

where (119891 120593)larr119900(120572) = 120593 ∘ 120572 ∘ 119891




997888997888997888997888997888rarr CSLat(Ξ2)


997888997888997888rarr (1198832 1198712))dagger119900

=

≀1198711198831

1≀

(119891120593)dagger119900

997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger119900

(120572) = 120593119900119901dagger

∘ 120572 ∘

119891119900119901




(1198832 ⩽) are posets and1198831

119891





(1) 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832(resp 11198831 ⩽ 119892 ∘ 119891 and

11198832⩽ 119891 ∘ 119892)


(3) 119891(⋁119878) = ⋁119904isin119878119891(119904) and 119892(⋀119879) = ⋀

119905isin119879119892(119905) (resp

119891(⋁119878) = ⋀119904isin119878119891(119904) and 119892(⋁119879) = ⋀

119905isin119879119892(119905)) provided


(4) 119892 ∘ 119891 ∘ 119892 = 119892 and 119891 ∘ 119892 ∘ 119891 = 119891



and (1198831 ⩽)

119891

999448999471119892


⩽ 119892 ∘ 119891 and 11198832 ⩽ 119891 ∘ 119892)


Example 21


120593


⊢ 1198602) (resp (1198602 120593

⊢

120593 1198601))



(3) Every map 1198831

119891


rarr 119891

larrP(1198832))



997888997888997888rarr (1198832 1198712)


1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)








⊨


⊨(119909minus)


Morphisms 1198631


times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891


119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)






Example 24











(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1) 119902rarr 119897 (⋀119878) = ⋀119904isin119878

(119902rarr 119897 119904) and 119902rarr 119903 (⋀119878) =



(2) (⋁119878)rarr 119897 119902 = ⋀119904isin119878

(119904rarr 119897 119902) and (⋁119878)rarr 119903 119902 =

⋀119904isin119878
















1015840= ⊤119860





1119860997888997888rarr 119860



119891



997888997888997888rarr

P(1198832) (resp P(1198832)119891larr





119886



119878


120594119886

119878(119909) =

119886 119909 isin 119878



119909or 1205941119860






S(minus)rarr


997888997888997888rarr

(1198832 1198712))rarr= 119871

1198831

1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)rarr (120572))(1199092) =

120593(⋁119891(1199091)=1199092

120572(1199091))


1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 is ⋁-


1 for every

1199092 isin 1198832 ((119891 120593)rarr(⋁ 119878))(1199092) = 120593(⋁

119891(1199091)=1199092(⋁ 119878)(1199091)) =

120593(⋁119891(1199091)=1199092

⋁119904isin119878119904(1199091)) = 120593(⋁

119904isin119878⋁119891(1199091)=1199092

119904(1199091)) =

⋁119904isin119878120593(⋁

119891(1199091)=1199092119904(1199091)) = ⋁

119904isin119878((119891 120593)

rarr(119904))(1199092) =

(⋁119904isin119878(119891 120593)

rarr(119904))(1199092)



997888997888997888rarr

(1198832 1198712) and (1198832 1198712)(11989110158401205931015840)


1198831

1and every 1199093 isin 1198833 ((119891

1015840 120593

1015840)rarr

∘

(119891 120593)rarr(120572))(1199093) = 120593

1015840(⋁

1198911015840(1199092)=1199093((119891 120593)

rarr(120572))(1199092)) =

1205931015840(⋁

1198911015840(1199092)=1199093120593(⋁

119891(1199091)=1199092120572(1199091))) = 120593

1015840∘ 120593(⋁

1198911015840∘119891(1199091)=1199093120572(1199091)) =

((1198911015840∘119891 120593

1015840∘ 120593)

rarr(120572))(1199093) = (((119891

1015840 120593

1015840) ∘ (119891 120593))

rarr(120572))(1199093)





Theorem 13



997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊣[

=

≀1198711198831

1≀

(119891120593)⊣[

997888997888997888997888997888rarr ≀1198711198832

2≀ where ((119891 120593)

⊣[(120572))(1199092) =

120593119900119901⊣(⋁

119891(1199091)=1199092120572(1199091))


S-morphism 1198711

120593

997888rarr 1198712 the map |1198711|120593119900119901⊢

997888997888997888rarr |1198712| is ⋁-


997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊢[

=

1198711198831

1

(119891120593)⊢[

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)

⊢[(120572))(1199092) =

120593119900119901⊢(⋁

119891(1199091)=1199092120572(1199091))



997888rarr 1198712 and 1198712

1205931015840


⊣

(resp (1205931015840 ∘120593)⊢ = 120593⊢ ∘1205931015840⊢

)





((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr

= 1198601198831

1

((119891120593)larr)119900119901

997888997888997888997888997888997888997888rarr 1198601198832

2 where

(119891 120593)larr(120572) = 120593

119900119901∘ 120572 ∘ 119891


2

(119891120593)larr

997888997888997888997888997888rarr 1198601198831



1198832

2)119899120582 for every


((119891 120593)larr(120596

1198601198832

2

120582(⟨120572119894⟩119899120582

))) (1199091)

= 120593119900119901∘ (120596

1198601198832

2

120582(⟨120572119894⟩119899120582

)) ∘ 119891 (1199091)

= 120593119900119901∘ (120596

1198602

120582(⟨120572119894 ∘ 119891 (1199091)⟩119899120582

))

= 1205961198601

120582(⟨120593

119900119901∘ 120572119894 ∘ 119891 (1199091)⟩119899120582

)

= 1205961198601

120582(⟨((119891 120593)

larr(120572119894)) (1199091)⟩119899120582

)

= (1205961198601198831

1

120582(⟨(119891 120593)

larr(120572119894)⟩119899120582

)) (1199091)

(3)



997888997888997888rarr (1198832 1198602) and

(1198832 1198602)(11989110158401205931015840)


3 (119891 120593)larr ∘

(1198911015840 120593

1015840)larr(120572) = (119891 120593)

larr(120593

1015840119900119901∘ 120572 ∘119891

1015840) = 120593

119900119901∘ 120593

1015840119900119901∘ 120572 ∘119891

1015840∘ 119891 =

(1205931015840∘ 120593)

119900119901∘ 120572 ∘ 119891

1015840∘ 119891 = (119891

1015840∘ 119891 120593

1015840∘ 120593)

larr(120572) = ((119891

1015840 120593

1015840) ∘

(119891 120593))larr(120572)





S(minus)dagger


(119891120593)

997888997888997888rarr

(1198832 1198712))dagger

= ≀1198711198831

1≀

((119891120593)dagger)119900119901

997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger(120572) =

120593dagger∘ 120572 ∘ 119891



Theorem 16


S(minus)rarr119900


997888997888997888rarr

(1198832 1198712))rarr119900

= 1198711198831

1

((119891120593)rarr119900

)119900119901

997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)rarr119900

(120572))(1199091) = 120593119900119901(⋁

119891119900119901(1199092)=1199091120572(1199092))


S(minus)⊣[119900


997888997888997888rarr

(1198832 L2))⊣[119900

= ≀1198711198831

1≀

((119891120593)⊣[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where

((119891 120593)⊣[119900

(120572))(1199091) = 120593⊣(⋁

119891119900119901(1199092)=1199091120572(1199092))


morphism 1198711

120593

997888rarr 1198712 the map |1198712|120593⊢


S(minus)⊢[119900


997888997888997888rarr

(1198832 1198712))⊢[119900

= 1198711198831

1

((119891120593)⊢[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)⊢[119900

(120572))(1199091) = 120593⊢(⋁

119891119900119901(1199092)=1199091120572(1199092))


Theorem 17




((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr119900

= 1198601198831

1

(119891120593)larr119900

997888997888997888997888997888rarr 1198601198832

2

where (119891 120593)larr119900(120572) = 120593 ∘ 120572 ∘ 119891




997888997888997888997888997888rarr CSLat(Ξ2)


997888997888997888rarr (1198832 1198712))dagger119900

=

≀1198711198831

1≀

(119891120593)dagger119900

997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger119900

(120572) = 120593119900119901dagger

∘ 120572 ∘

119891119900119901




(1198832 ⩽) are posets and1198831

119891





(1) 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832(resp 11198831 ⩽ 119892 ∘ 119891 and

11198832⩽ 119891 ∘ 119892)


(3) 119891(⋁119878) = ⋁119904isin119878119891(119904) and 119892(⋀119879) = ⋀

119905isin119879119892(119905) (resp

119891(⋁119878) = ⋀119904isin119878119891(119904) and 119892(⋁119879) = ⋀

119905isin119879119892(119905)) provided


(4) 119892 ∘ 119891 ∘ 119892 = 119892 and 119891 ∘ 119892 ∘ 119891 = 119891



and (1198831 ⩽)

119891

999448999471119892


⩽ 119892 ∘ 119891 and 11198832 ⩽ 119891 ∘ 119892)


Example 21


120593


⊢ 1198602) (resp (1198602 120593

⊢

120593 1198601))



(3) Every map 1198831

119891


rarr 119891

larrP(1198832))



997888997888997888rarr (1198832 1198712)


1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)








⊨


⊨(119909minus)


Morphisms 1198631


times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891


119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)






Example 24











(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











S(minus)rarr


997888997888997888rarr

(1198832 1198712))rarr= 119871

1198831

1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)rarr (120572))(1199092) =

120593(⋁119891(1199091)=1199092

120572(1199091))


1

(119891120593)rarr

997888997888997888997888997888rarr 1198711198832

2 is ⋁-


1 for every

1199092 isin 1198832 ((119891 120593)rarr(⋁ 119878))(1199092) = 120593(⋁

119891(1199091)=1199092(⋁ 119878)(1199091)) =

120593(⋁119891(1199091)=1199092

⋁119904isin119878119904(1199091)) = 120593(⋁

119904isin119878⋁119891(1199091)=1199092

119904(1199091)) =

⋁119904isin119878120593(⋁

119891(1199091)=1199092119904(1199091)) = ⋁

119904isin119878((119891 120593)

rarr(119904))(1199092) =

(⋁119904isin119878(119891 120593)

rarr(119904))(1199092)



997888997888997888rarr

(1198832 1198712) and (1198832 1198712)(11989110158401205931015840)


1198831

1and every 1199093 isin 1198833 ((119891

1015840 120593

1015840)rarr

∘

(119891 120593)rarr(120572))(1199093) = 120593

1015840(⋁

1198911015840(1199092)=1199093((119891 120593)

rarr(120572))(1199092)) =

1205931015840(⋁

1198911015840(1199092)=1199093120593(⋁

119891(1199091)=1199092120572(1199091))) = 120593

1015840∘ 120593(⋁

1198911015840∘119891(1199091)=1199093120572(1199091)) =

((1198911015840∘119891 120593

1015840∘ 120593)

rarr(120572))(1199093) = (((119891

1015840 120593

1015840) ∘ (119891 120593))

rarr(120572))(1199093)





Theorem 13



997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊣[

=

≀1198711198831

1≀

(119891120593)⊣[

997888997888997888997888997888rarr ≀1198711198832

2≀ where ((119891 120593)

⊣[(120572))(1199092) =

120593119900119901⊣(⋁

119891(1199091)=1199092120572(1199091))


S-morphism 1198711

120593

997888rarr 1198712 the map |1198711|120593119900119901⊢

997888997888997888rarr |1198712| is ⋁-


997888997888997888997888rarr

CSLat(⋁) defined by ((1198831 1198711)(119891120593)

997888997888997888rarr (1198832 1198712))⊢[

=

1198711198831

1

(119891120593)⊢[

997888997888997888997888997888rarr 1198711198832

2 where ((119891 120593)

⊢[(120572))(1199092) =

120593119900119901⊢(⋁

119891(1199091)=1199092120572(1199091))



997888rarr 1198712 and 1198712

1205931015840


⊣

(resp (1205931015840 ∘120593)⊢ = 120593⊢ ∘1205931015840⊢

)





((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr

= 1198601198831

1

((119891120593)larr)119900119901

997888997888997888997888997888997888997888rarr 1198601198832

2 where

(119891 120593)larr(120572) = 120593

119900119901∘ 120572 ∘ 119891


2

(119891120593)larr

997888997888997888997888997888rarr 1198601198831



1198832

2)119899120582 for every


((119891 120593)larr(120596

1198601198832

2

120582(⟨120572119894⟩119899120582

))) (1199091)

= 120593119900119901∘ (120596

1198601198832

2

120582(⟨120572119894⟩119899120582

)) ∘ 119891 (1199091)

= 120593119900119901∘ (120596

1198602

120582(⟨120572119894 ∘ 119891 (1199091)⟩119899120582

))

= 1205961198601

120582(⟨120593

119900119901∘ 120572119894 ∘ 119891 (1199091)⟩119899120582

)

= 1205961198601

120582(⟨((119891 120593)

larr(120572119894)) (1199091)⟩119899120582

)

= (1205961198601198831

1

120582(⟨(119891 120593)

larr(120572119894)⟩119899120582

)) (1199091)

(3)



997888997888997888rarr (1198832 1198602) and

(1198832 1198602)(11989110158401205931015840)


3 (119891 120593)larr ∘

(1198911015840 120593

1015840)larr(120572) = (119891 120593)

larr(120593

1015840119900119901∘ 120572 ∘119891

1015840) = 120593

119900119901∘ 120593

1015840119900119901∘ 120572 ∘119891

1015840∘ 119891 =

(1205931015840∘ 120593)

119900119901∘ 120572 ∘ 119891

1015840∘ 119891 = (119891

1015840∘ 119891 120593

1015840∘ 120593)

larr(120572) = ((119891

1015840 120593

1015840) ∘

(119891 120593))larr(120572)





S(minus)dagger


(119891120593)

997888997888997888rarr

(1198832 1198712))dagger

= ≀1198711198831

1≀

((119891120593)dagger)119900119901

997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger(120572) =

120593dagger∘ 120572 ∘ 119891



Theorem 16


S(minus)rarr119900


997888997888997888rarr

(1198832 1198712))rarr119900

= 1198711198831

1

((119891120593)rarr119900

)119900119901

997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)rarr119900

(120572))(1199091) = 120593119900119901(⋁

119891119900119901(1199092)=1199091120572(1199092))


S(minus)⊣[119900


997888997888997888rarr

(1198832 L2))⊣[119900

= ≀1198711198831

1≀

((119891120593)⊣[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where

((119891 120593)⊣[119900

(120572))(1199091) = 120593⊣(⋁

119891119900119901(1199092)=1199091120572(1199092))


morphism 1198711

120593

997888rarr 1198712 the map |1198712|120593⊢


S(minus)⊢[119900


997888997888997888rarr

(1198832 1198712))⊢[119900

= 1198711198831

1

((119891120593)⊢[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)⊢[119900

(120572))(1199091) = 120593⊢(⋁

119891119900119901(1199092)=1199091120572(1199092))


Theorem 17




((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr119900

= 1198601198831

1

(119891120593)larr119900

997888997888997888997888997888rarr 1198601198832

2

where (119891 120593)larr119900(120572) = 120593 ∘ 120572 ∘ 119891




997888997888997888997888997888rarr CSLat(Ξ2)


997888997888997888rarr (1198832 1198712))dagger119900

=

≀1198711198831

1≀

(119891120593)dagger119900

997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger119900

(120572) = 120593119900119901dagger

∘ 120572 ∘

119891119900119901




(1198832 ⩽) are posets and1198831

119891





(1) 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832(resp 11198831 ⩽ 119892 ∘ 119891 and

11198832⩽ 119891 ∘ 119892)


(3) 119891(⋁119878) = ⋁119904isin119878119891(119904) and 119892(⋀119879) = ⋀

119905isin119879119892(119905) (resp

119891(⋁119878) = ⋀119904isin119878119891(119904) and 119892(⋁119879) = ⋀

119905isin119879119892(119905)) provided


(4) 119892 ∘ 119891 ∘ 119892 = 119892 and 119891 ∘ 119892 ∘ 119891 = 119891



and (1198831 ⩽)

119891

999448999471119892


⩽ 119892 ∘ 119891 and 11198832 ⩽ 119891 ∘ 119892)


Example 21


120593


⊢ 1198602) (resp (1198602 120593

⊢

120593 1198601))



(3) Every map 1198831

119891


rarr 119891

larrP(1198832))



997888997888997888rarr (1198832 1198712)


1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)








⊨


⊨(119909minus)


Morphisms 1198631


times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891


119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)






Example 24











(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











S(minus)dagger


(119891120593)

997888997888997888rarr

(1198832 1198712))dagger

= ≀1198711198831

1≀

((119891120593)dagger)119900119901

997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger(120572) =

120593dagger∘ 120572 ∘ 119891



Theorem 16


S(minus)rarr119900


997888997888997888rarr

(1198832 1198712))rarr119900

= 1198711198831

1

((119891120593)rarr119900

)119900119901

997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)rarr119900

(120572))(1199091) = 120593119900119901(⋁

119891119900119901(1199092)=1199091120572(1199092))


S(minus)⊣[119900


997888997888997888rarr

(1198832 L2))⊣[119900

= ≀1198711198831

1≀

((119891120593)⊣[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr ≀1198711198832

2≀ where

((119891 120593)⊣[119900

(120572))(1199091) = 120593⊣(⋁

119891119900119901(1199092)=1199091120572(1199092))


morphism 1198711

120593

997888rarr 1198712 the map |1198712|120593⊢


S(minus)⊢[119900


997888997888997888rarr

(1198832 1198712))⊢[119900

= 1198711198831

1

((119891120593)⊢[119900

)119900119901

997888997888997888997888997888997888997888997888997888rarr 1198711198832

2 where

((119891 120593)⊢[119900

(120572))(1199091) = 120593⊢(⋁

119891119900119901(1199092)=1199091120572(1199092))


Theorem 17




((1198831 1198601)(119891120593)

997888997888997888rarr (1198832 1198602))larr119900

= 1198601198831

1

(119891120593)larr119900

997888997888997888997888997888rarr 1198601198832

2

where (119891 120593)larr119900(120572) = 120593 ∘ 120572 ∘ 119891




997888997888997888997888997888rarr CSLat(Ξ2)


997888997888997888rarr (1198832 1198712))dagger119900

=

≀1198711198831

1≀

(119891120593)dagger119900

997888997888997888997888997888997888rarr ≀1198711198832

2≀ where (119891 120593)dagger119900

(120572) = 120593119900119901dagger

∘ 120572 ∘

119891119900119901




(1198832 ⩽) are posets and1198831

119891





(1) 11198831 ⩽ 119892 ∘ 119891 and 119891 ∘ 119892 ⩽ 11198832(resp 11198831 ⩽ 119892 ∘ 119891 and

11198832⩽ 119891 ∘ 119892)


(3) 119891(⋁119878) = ⋁119904isin119878119891(119904) and 119892(⋀119879) = ⋀

119905isin119879119892(119905) (resp

119891(⋁119878) = ⋀119904isin119878119891(119904) and 119892(⋁119879) = ⋀

119905isin119879119892(119905)) provided


(4) 119892 ∘ 119891 ∘ 119892 = 119892 and 119891 ∘ 119892 ∘ 119891 = 119891



and (1198831 ⩽)

119891

999448999471119892


⩽ 119892 ∘ 119891 and 11198832 ⩽ 119891 ∘ 119892)


Example 21


120593


⊢ 1198602) (resp (1198602 120593

⊢

120593 1198601))



(3) Every map 1198831

119891


rarr 119891

larrP(1198832))



997888997888997888rarr (1198832 1198712)


1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)








⊨


⊨(119909minus)


Morphisms 1198631


times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891


119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)






Example 24











(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(3) Every map 1198831

119891


rarr 119891

larrP(1198832))



997888997888997888rarr (1198832 1198712)


1198831

1 (119891 120593)

rarr (119891 120593)

⊢ 119871

1198832

2)








⊨


⊨(119909minus)


Morphisms 1198631


times S-

morphisms (pt1198631 Ω1198631 Σ1198631)119891


119900119901(119887)) = (Σ119891)

119900119901∘⊨2(pt119891(119909) 119887)






Example 24











(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1198831 ⩽)

119891




(1198831 ⩽)

119891

999448999471119892(1198832 ⩽



nection (1198832 ⩽)

1198911015840

9994489994711198921015840(1198833 ⩽) (or (1198832 ⩽)

1198911015840

9994489994711198921015840

(1198833 ⩽119889)) that is










⊨(119909minus)


Morphisms1198631


timesS-morphisms

(pt1198631 Ω1198631 Σ1198631)119891



every 119887 isin Ω1198632







119891


Σ1198631

Σ119891

997888997888rarr Σ1198632 (resp Σ1198631

(Σ119891)119900119901



997888rarr


119891

997888rarr 1198632) =

1198631

(pt119891Ω119891((Σ119891)119900119901)minus1)997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr 1198632





⊨


⊨(119909minus)


Morphisms 1198631


S-morphisms (pt1198631 Ω1198631 Σ1198631)119891

997888rarr (pt1198632 Ω1198632 Σ1198632)








⊨(119909minus)


Morphisms 1198631


(pt1198631 Ω1198631 Σ1198631)119891












119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119891

997888rarr K2 are

now triples (11986611198721 1198711)119891=(120572120573120593)



119900119901(119898)) = 120593

119900119901∘ 1198682(120572(119892) 119898)) The case


(2) S-FC119862



119898






(2) S-FC119866119882


SB-TopSys119886119898





(1) 119871119866 119867


⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898))

(2) 119871119872 119870


⋀119898isin119872

(119905(119898)rarr 119903 119868(119892119898))





(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1) P(119866)119867


119904 119868 119898 for every 119904 isin 119878

(2) P(119872)119870


119892 119868 119905 for every 119905 isin 119879




119866 119867119870 119871






⋀119892isin119866

(1199042(119892)rarr 119897 119868(119892119898)) ⩽ ⋀119892isin119866

(1199041(119892)rarr 119897 119868(119892119898)) =


119866


⋀119898isin119872

((119867(119904))(119898)rarr 119903 119868(119892119898)) = ⋀119898isin119872

((⋀1198921015840isin119866

(119904(1198921015840)rarr 119897

119868(1198921015840 119898)))rarr 119903 119868(119892119898)) ⩾ ⋀

119898isin119872((119904(119892)rarr 119897 119868(119892119898))rarr 119903

119868(119892119898))

(dagger)

⩾ ⋀119898isin119872






Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

120572

120572120573119900119901

120573119900119901

(4)

commute


(119871119866 119871








Morphisms K1

119891=(120572120573)


(1198711198661

1 119871

1198721

1)

(120572120573)

997888997888997888rarr (1198711198662

2 119871

1198722


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211

11987111987222

11987111987222

1198671 1198701 1198702

120572

120572

120573

120573

1198672 (5)

commute


119886


Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Definition 38 S-FC119863119872119877

119894(resp S-FC119863119872119877



119886) which


119891

997888rarr K2 are

such that11987111987222

120573119900119901

997888997888rarr 1198711198721

1(resp1198711198721

1

120573

997888rarr 1198711198722




119894

119865

997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

(120572(120573119900119901)minus1)

997888997888997888997888997888997888997888997888rarr K2






phic


119868


119886The



= (1198831015840 119883

1015840 119868

1015840) where 1198831015840 is



997888997888997888997888997888997888997888997888rarr 1198711198831015840



119871119883998400

119871119883998400

119871119883998400

119871119883998400

119871119872119879

119871119872119879119871119866119879

119871119866119879

119867998400 119870998400119867119879119870119879

120572

120572

120573119900119901

120573119900119901

(6)


119871119872119879

1205731015840119900119901

997888997888997888rarr 1198711198831015840


997888rarr




=

(11988310158401015840 119883

10158401015840 119868



(perp119871) = ⊤119871 =11986710158401015840(⊤119871))


119871-FC119863119872119877-morphismK1015840101584011989110158401015840




119886 The case


119886








(1) (119867(1205941119871119892))(119898) = (119870(120594

1119871



119892))(119898) = 119886rarr 119897(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(3) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119870(120594

1119871

119898))(119892) for every 119892 isin 119866


following ones

(a) (119867(119886otimes1205941119871119892))(119898) = 119886rarr 119897(119870(120594

1119871

119898))(119892) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871(b) (119870(1205941119871

119898otimes119886))(119892) = 119886rarr 119903(119867(120594

1119871

119892))(119898) for every 119892 isin 119866

119898 isin 119872 and 119886 isin 119871


1119871

119892))(119898) = ⋀

1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898))

(dagger)

= 1119871rarr 119897

119868(119892119898) = 119868(119892119898) = 1119871rarr 119903 119868(119892119898)(dagger)

= ⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903

119868(1198921198981015840)) = (119870(120594

1119871



Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ad (2) (119867(119886 otimes 1205941119871

119892))(119898) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892)(119892

1015840)rarr 119897

119868(1198921015840 119898)) = ⋀

1198921015840isin119866((119886 otimes 120594

1119871

119892(119892

1015840))rarr 119897 119868(119892

1015840 119898))

(dagger)

= ⋀1198921015840isin119866

(119886rarr 119897

(1205941119871

119892(119892

1015840)rarr 119897 119868(119892

1015840 119898)))

(daggerdagger)

= 119886rarr 119897(⋀1198921015840isin119866(120594

1119871

119892(119892

1015840)rarr 119897

119868(1198921015840 119898))) = 119886rarr 119897(119867(120594

1119871



119898otimes 119886))(119892) = ⋀

1198981015840isin119872((120594

1119871

119898otimes 119886)(119898

1015840)rarr 119903

119868(1198921198981015840)) = ⋀

1198981015840isin119872((120594

1119871

119898(119898

1015840) otimes 119886)rarr 119903 119868(119892119898

1015840))

(dagger)

=

⋀1198981015840isin119872

(119886rarr 119903(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840)))

(daggerdagger)

= 119886rarr 119903

(⋀1198981015840isin119872

(1205941119871

119898(119898

1015840)rarr 119903 119868(119892119898

1015840))) = 119886rarr 119903(119870(120594

1119871

119898))(119892)








(2) (a) (120572(1205941119871119892))(119898) = (120573(120594

1119871



1119871

119892))(119898) = 119886rarr 119897(120572(120594

1119871

119892))(119898) for every


119898otimes 119886))(119892) = 119886rarr 119903(120573(120594

1119871

119898))(119892) for every


1119871

119892))(119898) = 119886rarr 119897(120573(120594

1119871

119898))(119892) for every


119898otimes 119886))(119892) = 119886rarr 119903(120572(120594

1119871

119892))(119898) for every



119868

997888rarr 119871 by119868(119892119898) = (120572(120594

1119871

119892))(119898)

(119886)

= (120573(1205941119871

119898))(119892) Given 119904 isin 119871


119892isin119866(119904(119892)rarr 119897 119868(119892119898)) =

⋀119892isin119866

(119904(119892)rarr 119897(120572(1205941119871

119892))(119898))

(119887)

= ⋀119892isin119866

(120572(119904(119892) otimes 1205941119871

119892))(119898)

(dagger)

=

(120572(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892)))(119898) = (120572(119904))(119898) (notice that

(⋁119892isin119866

(119904(119892) otimes 1205941119871

119892))(119892

1015840) = ⋁

119892isin119866(119904(119892)(119892

1015840) otimes 120594

1119871

119892(119892

1015840)) = 119904(119892

1015840) otimes



119898isin119872(119905(119898)rarr 119903 119868(119892119898)) =

⋀119898isin119872

(119905(119898)rarr 119903(120573(1205941119871

119898))(119892))

(119888)

= ⋀119898isin119872

(120573(1205941119871

119898otimes 119905(119898)))(119892)

(dagger)

=

(120573(⋁119898isin119872

(1205941119871


Theorem 19(3)


119868



120572



(1) 119871119866 119868otimes


119892isin119866119868(119892119898) otimes 119904(119892)


(2) 119871119872119868rarr

minus


minus(119905))(119892) = ⋀

119898isin119872119868(119892119898) rarr





119868





119866 for every 119898 isin 119872 (119867(119904))(119898) =

⋀119892isin119866

(119904(119892)rarr 119897 119868(119892119898)) = ⋀119892isin119866

(119904(119892)rarr 119897 ⊤119871) = ⋀119892isin119866

⊤119871 =

⊤119871 that is119867(119904) = ⊤119871









119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119886

Theorem 45 Let K1

119891


1198661

1 1199052 isin 119871

1198722

2



1198671(1199041) = 1199041

(2) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1reflect fixed



119900119901(119905) = 120573

119900119901(119905) implies

1198672 ∘ 1198702(119905) = 119905 for every 119905 isin 1198711198722

2 then (1199041 120573119900119901(1199052)) is a


(3) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198722

2

120573119900119901

997888997888rarr 1198711198721

1are injective then





(5) If both 1198711198661

1

120572

997888rarr 1198711198662

2and 119871

1198722

2

120573119900119901

997888997888rarr 1198711198721

1are order-


If 11987111986611

120572

997888rarr 1198711198662

2(resp 1198711198722

2

120573119900119901

997888997888rarr 1198711198721




120573119900119901(1199052) we get 120572(1199041) = 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 1198702(1199052)1198672 ∘ 120572(1199041) =

1198672 ∘ 120572 ∘ 1198701 ∘ 120573119900119901(1199052) = 1198672 ∘ 1198702(1199052) = 1199052 and moreover


and 120572(1199041) = 1198702(1199052) we get 120573119900119901(1199052) = 120573

119900119901∘1198672 ∘120572(1199041) = 1198671(1199041)

1198701 ∘ 120573119900119901(1199052) = 1198701 ∘ 120573

119900119901∘ 1198672 ∘ 120572(1199041) = 1198701 ∘ 1198671(1199041) = 1199041 and

moreover1198672 ∘ 1198702(1199052) = 1198672 ∘ 120572(1199041) = 1199052


1198701 ∘ 1198671(1199041) = 1198671(1199041) = 120573119900119901(1199052) and therefore1198672 ∘ 1198702(1199052) =


1198702(1199052) = 120572(1199041) yields 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 1198672 ∘ 1198702(1199052) =




1198702 ∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) =


we get1198671 ∘ 1198701 ∘ 120573119900119901(1199052) = 120573

119900119901∘ 1198672 ∘ 120572 ∘ 1198701 ∘ 120573

119900119901(1199052) = 120573

119900119901∘

1198672 ∘ 1198702(1199052) = 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041)



yields 120573119900119901(1199052) ⩽ 120573119900119901∘ 1198672 ∘ 120572(1199041) = 1198671(1199041) (120573



1198672 ∘120572(119904) = 120572(119904) it follows that 120572∘1198701 ∘1198671(119904) = 120572∘1198701 ∘120573119900119901∘1198672 ∘


2such that1198671 ∘ 1198701 ∘ 120573

119900119901(119905) =

120573119900119901(119905) it follows that120573119900119901∘1198672∘1198702(119905) = 120573

119900119901∘1198672∘120572∘1198701∘120573

119900119901(119905) =

1198671 ∘ 1198701 ∘ 120573119900119901(119905) = 120573


of 120573119900119901)

Theorem 46 Let K1

119891


119886and let 1199041 isin 119871

1198661

1 1199051 isin 119871

1198721

1

(1) If both 11987111986611

120572

997888rarr 1198711198662

2and 1198711198721

1

120573

997888rarr 1198711198722

2are injective then


(2) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722

2is injective then


(3) If one of 11987111986611

120572

997888rarr 1198711198662

2 1198711198721

1

120573

997888rarr 1198711198722




120572(1199041) = 120573 ∘1198671(1199041) = 120573(1199051) and1198702 ∘ 120573(1199051) = 120572 ∘1198701(1199051) = 120572(1199041)


120573(1199051) and 1198702 ∘ 120573(1199051) = 120572(1199041) 120573 ∘ 1198671(1199041) = 1198672 ∘ 120572(1199041) = 120573(1199051)



1198702 ∘ 120573 ∘ 1198671(1199041) = 120572 ∘ 1198701 ∘ 1198671(1199041) = 120572 ∘ 1198701(1199051) = 1198702 ∘ 120573(1199051)


120572

997888rarr 1198711198662

2is


1198671(1199041) = 1198702 ∘ 120573 ∘ 1198671(1199041) = 1198702 ∘ 1198672 ∘ 120572(1199041) = 1198702 ∘ 120573(1199051) =

120572 ∘ 1198701(1199051) and thus 1198701 ∘ 1198671(1199041) = 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is


1198701(1199051) = 1198672 ∘ 120572 ∘ 1198701(1199051) = 1198672 ∘ 1198702 ∘ 120573(1199051) = 1198672 ∘ 120572(1199041) =

120573 ∘ 1198671(1199041) and thus1198671 ∘ 1198701(1199051) = 1198671(1199041)



120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572

997888rarr 1198711198662


1198701(1199051) implies 120572(1199041) ⩽ 120572∘1198701(1199051) = 1198702 ∘120573(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is



120572

997888rarr 1198711198662


1198702 ∘ 120573(1199051) = 120572 ∘ 1198701(1199051) implies 1199041 ⩽ 1198701(1199051) If 1198711198721

1

120573

997888rarr 1198711198722

2is





51 S-FC119862 and S-FC119862


119886 In this



S-FC119863119872119877 and S-FC119863119872119877

119886 on the other

511 From S-FC119862

119898to S-FC119863119872119877




119886 We begin




category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593





119898lowast

119867119862119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

larr119900)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)larr119900

(120573 120593)larr119900

1198671 1198672 1198701 1198702 (7)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)rarr(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= 120593 ∘ (1198671 (119904)) (120573119900119901(1198982))

= ((120573 120593)larr119900

∘ 1198671 (119904)) (1198982)

(8)


119891




119898lowast



and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 (119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981)

997888rarr119903 1198681 (1198921 1198981)))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593 ∘ 119905 ∘ 120573119900119901)) (1198922) = (1198702 ∘ (120573 120593)

larr119900(119905)) (1198922)

(9)


119898lowast


119891


119898






category S-FC119862





119862119898119863119886(resp 119867∙

119862119898119863119886) of the

functor S-FC119862

119898lowast

119867119862119898119863119886997888997888997888997888997888rarr S-FC119863119872119877

119886to S-FC119862


119898lowast∙)


119886


119862119898119863119886first and


1198911

9994909994731198912

K2 in S-FC119862


119867lowast

119862119898119863119886(1198911) = 119867

lowast

119862119898119863119886(1198912) it follows that (1205721 1205931)

rarr=

(1205722 1205932)rarr (1205731 1205931)

larr119900= (1205732 1205932)



((1205721 1205931)rarr(120594

11198711

1198921)) (1198922) = 1205931( ⋁

1205721(1198921015840

1)=1198922

12059411198711

1198921(119892

1015840

1))

= 1205931 (11198711

1198922 = 1205721 (1198921)


)

= 11198712

1198922 = 1205721 (1198921)


= 12059411198712

1205721(1198921)(1198922)

(10)


1205721(1198921)=

(1205721 1205931)rarr(120594

11198711

1198921) = (1205722 1205932)

rarr(120594

11198711

1198921) = 120594

11198712



To show that 1205731199001199011= 120573

119900119901

2= 120573


larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) = 1205931 ∘ 12059411198711

120573119900119901

1(1198982)

∘ 120573119900119901

1(1198982) = 11198712

Additionally

((1205732 1205932)larr119900

(12059411198711

120573119900119901

1 (1198982))) (1198982)

= 1205932 ∘ 12059411198711

120573119900119901

1 (1198982)∘ 120573

119900119901

2(1198982)

= 11198712

120573119900119901

2(1198982) = 120573

119900119901

1(1198982)


(11)

Since ((1205731 1205931)larr119900(120594

11198711

120573119900119901

1(1198982)

))(1198982) =

((1205732 1205932)larr119900(120594

11198711

120573119900119901

1(1198982)


120573119900119901

1(1198982) = 120573

119900119901




rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











rarr(1198861))(120572(1198921)) = 1205931(⋁120572(1198921015840

1)=120572(1198921)

1198861(1198921015840

1)) =

1205931(1198861) implies 1205931(1198861) = ((120572 1205931)rarr(1198861))(120572(1198921)) =

((120572 1205932)rarr(1198861))(120572(1198921)) = 1205932(1198861)


119862119898119863119886 one follows


119900119901

2= 120573


larr119900(1198861))(1198982) = 1205931 ∘ 1198861 ∘ 120573

119900119901(1198982) =


((120573 1205932)larr119900(1198861))(1198982) = 1205932(1198861)



997888997888997888997888rarr P(119872) and

P(119872)119870=119866


K119891


119886byP(119866)

120572=119866

997888997888997888rarr P(119866) and

P(119872)120573=119872


lowast

119862119898119863119886and 119867∙



997888997888rarr K in 2-FC119862


119867lowast

119862119898119863119886(119891

1015840) = 119891 If follows that 120572 = (120572




512 From S-FC119862







category S-FC119862


K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr


120593




119898lowast

119867119862119898119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)rarr((120573120593)

⊢[119900)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

(120572 120593)rarr

(120572 120593)rarr(120573 120593)⊢119900

(120573 120593)⊢119900

1198671 1198672 1198701 1198702 (12)


1and

1198981 isin 1198721

((120573 120593)⊢[119900

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)rarr(119904)) (1198982))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982) )))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982)))))

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982))))

(dagger)

= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))))


= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 120593⊢( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921)

997888rarr119897 120593 ∘ 1198681 (1198921 1198981))))

(daggerdagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(13)


119891






2and

1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢[119900(119905)) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)⊢[119900

(119905)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)⊢[119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593⊢∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593⊢∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982)

997888rarr119903 120593 ∘ 1198681 (1198921 120573119900119901(1198982)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593 ∘ 120593⊢∘ 119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(14)




119891


119898






119862119898119863(resp 119867∙

119862119898119863) of the

functor S-FC119862

119898lowast

119867119862119898119863997888997888997888997888rarr S-FC119863119872119877 to S-FC119862


119898lowast∙)



513 From S-FC119862 to S-FC119863119872119877



119886 We start by






K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










K1

119891


120572

997888rarr 11986621198722

120573119900119901

997888997888rarr 1198721 are


120593119900119901


and moreover |1198711|120593119900119901⊢




lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877

119886


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)⊢119900

)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622

11987111987211

11987111987211 119871

1198722

1198722

2

1198712

(120573 120593)⊢119900

(120573 120593)⊢119900

(120572 120593)⊢

1198671

(120572 120593)⊢

1198672 1198701 1198702 (15)


1and

1198982 isin 1198722

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982)

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(dagger)

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr1198971198682 (1198922 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(dagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982))))

= (120593119900119901⊢

∘ 1198671 (119904) ∘ 120573119900119901) (1198982)

= ((120573 120593)⊢119900

∘ 1198671 (119904)) (1198982)

(16)



119891



and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982)

997888rarr119903 120593119900119901∘ 1198682 (1198922 1198982))))

(dagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(dagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 119905 ∘ 120573119900119901(1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (120593119900119901⊢

∘ 119905 ∘ 120573119900119901)) (1198922)

= (1198702 ∘ (120573 120593)⊢119900

(119905)) (1198922)

(17)


119891






lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













lowast∙) is the



following data

Objects S-FC119862




lowast-morphisms K1

119891


|1198711|120593119900119901⊢




119862119863119886(resp119867∙

119862119863119886) of the functor

S-FC119862

lowast

119867119862119863119886997888997888997888997888rarr S-FC119863119872119877



lowast∙)


119886






K1

119891


120572

997888rarr 1198662 1198722

120573119900119901

997888997888rarr 1198721


120593119900119901




lowast

119867119862119863997888997888997888rarr S-FC119863119872119877


119891

997888rarr K2) = K1

((120572120593)⊢[

((120573120593)rarr119900

)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

1198712

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)rarr119900

(120573 120593)rarr119900

1198671 1198672 1198701 1198702

1198722

(18)


1and

1198981 isin 1198721

((120573 120593)rarr119900

∘ 1198672 ∘ (120572 120593)⊢[

(119904)) (1198981)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(1198672 ∘ (120572 120593)⊢[

(119904)) (1198982))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922)

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr1198971198682 (1198922 1198982))))

(dagger)

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))

997888rarr119897 1198682 (1198922 1198982))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 1198982) ))))

= 120593119900119901( ⋁

120573119900119901(1198982)=1198981

( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (120572 (1198921) 1198982) )))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921)

997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982) )

(daggerdagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(120593119900119901∘120593

119900119901⊢∘119904 (1198921)997888rarr119897 1198681 (1198921 120573

119900119901(1198982)))

(dagger)

= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 120573119900119901(1198982)))


= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= ⋁

120573119900119901(1198982)=1198981

⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(19)


119891


2and

1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)rarr119900

(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)rarr119900

(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)rarr119900

(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901( ⋁

120573119900119901(1198982)=1198981

119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(( ⋁

120573119900119901(1198982)=1198981

120593119900119901∘ 119905 (1198982))

997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

( ⋀

120573119900119901(1198982)=1198981

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 1198681 (1198921 120573119900119901(1198982)))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982)

997888rarr119903 120593119900119901∘ 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

∘120593119900119901( ⋁

120572(1198921)=1198922

( ⋀

1198982isin1198722

(119905 (1198982)

997888rarr119903 1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (120572 (1198921) 1198982))

= ⋁

120572(1198921)=1198922

⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(20)


119891







119862119863(resp 119867∙


S-FC119862

lowast

119867119862119863997888997888997888rarr S-FC119863119872119877 to S-FC119862


lowast∙) gives an





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901






i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











i

119867i119863119862

997888997888997888rarr S-FC119862


119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199052)) = ⊤119871 iff (1199041 120573





⊤119871 for every 119886 isin 119871





119871-FC119863119872119877

i

119867i119863119862




i

such that 119867i119863119862(K1) = 119867


(1198711198661 119871

1198721 119871 1) = (1198711198662 119871



⊤119871

119892) = 1198671(120594

⊤119871

119892)

and therefore (1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871



119892)1198671(120594

⊤119871

119892)) = ⊤119871

and thus 2(1198701 ∘ 1198671(120594⊤119871

119892)1198671(120594

⊤119871

119892)) = ⊤119871 that is (1198701 ∘

1198671(120594⊤119871

119892)1198671(120594

⊤119871


that 120594⊤119871

119892⩽ 1198701 ∘ 1198671(120594

⊤119871

119892) = 1198702 ∘ 1198671(120594

⊤119871

119892) and

therefore ⊤119871 = 120594⊤119871

119892(119892) ⩽ (1198701 ∘ 1198671(120594

⊤119871

119892))(119892) = (1198702 ∘

1198671(120594⊤119871

119892))(119892) = ⋀

1198981015840isin119872((1198671(120594

⊤119871

119892))(119898

1015840)rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⋀1198921015840isin119866

(120594⊤119871

119892(119892

1015840)rarr 1198971198681(119892

1015840 119898

1015840)))rarr 1199031198682(119892119898

1015840)) =

⋀1198981015840isin119872

((⊤119871rarr 1198971198681(1198921198981015840))rarr 1199031198682(119892119898

1015840))⩽ (⊤119871rarr 1198971198681(119892119898))rarr 119903


⩽ (⊤119871rarr 1198971198681(119892119898)) otimes ⊤119871 ⩽



120593

997888rarr |119871| by

120593 (119886) =

3

2sdot 119886 119886 isin [0

1

2]

1

2sdot 119886 +

1

2 119886 isin (

1

2 1]

(22)



119867i119863119862(K)

119891=(1119871119866 1119871119872

120593119900119901)



119863119862



119900119903119901is the (nonfull)


whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119900119903119901stands



119900119903119901

119867119900119903119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119900119903119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where




1198661 and 1199052 isin 1198711198722 it fol-

lows that 1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal




119871-FC119863119872119877

119900119903119901

119867119900119903119901

119863119862





⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⩽ 1198701∘1198671(120594⊤119871

119892)


⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871 and therefore

2(120594⊤119871

119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871



119892⩽ 1198702∘1198671(120594

⊤119871

119892) Similar







119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





119900119901(1199051)) = ⊤119871 iff (1199041 120573

119900119901(1199051)) is a formal pro-






⊤119871

119892) = 1198701 ∘ 1198671(120594

⊤119871

119892) (120594⊤119871

119892 1198671(120594

⊤119871



119892 1198671(120594

⊤119871

119892)) = ⊤119871


119892 1198671(120594

⊤119871

119892)) = ⊤119871 that is (120594

⊤119871

119892 1198671(120594

⊤119871

119892))


119892⩽ 1198702 ∘

1198672(120594⊤119871

119892) = 1198702 ∘1198671(120594

⊤119871






119903119891119901is the (non-



119891


that the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119903119891119901stands



119903119891119901

119867119903119891119901

119863119862

997888997888997888rarr S-FC119862

which is given by119867119903119891119901

119863119862(K1

119891

997888rarr K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr

(1198711198662 119871

1198722 119871 2) where





1(1199041 120573119900119901(1199051)) = ⊤119871 iff (1199041 120573





119871-FC119863119872119877

119903119891119901

119867119903119891119901

119863119862





119903119891119901



119863119872




516 From S-FC119863119872119877 to S-FC119862





52 S-FC119866119882 and S-FC119866119882


119886 In







S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S-FC119862


119886) In











521 From S-FC119866119882

119898to S-FC119863119872119877










K1

119891


120572

997888rarr 1198662 and1198721

120573

997888rarr 1198722


120593




119898lowast

119867119866119898119863119886997888997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr

K2) = K1

((120572120593)rarr(120573120593)

rarr)

997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671

(120573 120593)rarr

(120573 120593)rarr

(120572 120593)rarr

(120572 120593)rarr

1198672 1198701 1198702 (26)


1and

1198982 isin 1198722

((120573 120593)rarr∘ 1198671 (119904)) (1198982)

= 120593( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981)))

= ⋁

120573(1198981)=1198982

⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

(dagger)

= ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= (1198672 ∘ (120572 120593)rarr(119904)) (1198982)

(27)


thatK1

119891


119898


and1198922 isin 1198662

(1198702 ∘ (120573 120593)rarr(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)rarr(119905)) (1198982) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593 ∘ 119905 (1198981))997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))


= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= ⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 119905 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(dagger)

= 120593( ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981)))

= 120593( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)rarr∘ 1198701 (119905)) (1198922)

(28)


thatK1

119891


119886

522 From S-FC119866119882









K1

119891


120572

997888rarr 1198662 and 1198721

120573

997888rarr


120593



119898lowast

119867119866119898119863997888997888997888997888rarr


119891

997888rarr

K2) = K1

((120572120593)rarr((120573120593)

⊢)119900119901

)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611 119871

11986622

11987111986611 119871

11986622119871

11987211

11987111987211

11987111987222

11987111987222

(120573 120593)⊢

(120573 120593)⊢

1198671 1198672 1198701 1198702

(120572 120593)rarr

(120572 120593)rarr

(29)


1and

1198981 isin 1198721

((120573 120593)⊢

∘ 1198672 ∘ (120572 120593)rarr(119904)) (1198981)

= 120593⊢∘ (1198672 ((120572 120593)

rarr(119904))) ∘ 120573 (1198981)

= 120593⊢(⋀

1198922isin1198662

(((120572 120593)rarr(119904)) (1198922)997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

(120593( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

=120593⊢(⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593 ∘ 119904 (1198921))997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593⊢(⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593 ∘ 119904 (1198921)997888rarr119897 1198682 (1198922 120573 (1198981)))))

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= 120593⊢( ⋀

1198921isin1198661

(120593 ∘ 119904 (1198921) 997888rarr119897 120593 ∘ 1198681 (1198921 1198981)))

(daggerdagger)

= 120593⊢∘ 120593( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))

= (1198671 (119904)) (1198981)

(30)


119891




and1198922 isin 1198662

((120572 120593)rarr∘ 1198701 ∘ (120573 120593)

⊢(119905)) (1198922)

= ((120572 120593)rarr∘ (1198701 (120593

⊢∘ 119905 ∘ 120573))) (1198922)

= 120593( ⋁

120572(1198921)=1198922

(1198701 (120593⊢∘ 119905 ∘ 120573)) (1198921))

= 120593( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593⊢∘ 119905 ∘ 120573 (1198981)997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593 ∘ 120593⊢∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593 ∘ 1198681 (1198921 1198981))

(daggerdagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(120593∘120593⊢∘119905∘120573 (1198981)997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (120572 (1198921) 120573 (1198981)))


= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= ⋁

120572(1198921)=1198922

⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(119905 ∘ 120573 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982))

= (1198702 (119905)) (1198922)

(31)


119891


119898

523 From S-FC119866119882 to S-FC119863119872119877



119886 As before we





K1

119891


120572

997888rarr 11986621198721

120573

997888rarr 1198722 are


120593119900119901





lowast

119867119866119863119886997888997888997888997888rarr

S-FC119863119872119877


119891

997888rarr K2) =

K1

((120572120593)⊢[

(120573120593)⊢[

)

997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622

1198711

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢

(120573 120593)⊢

(120573 120593)⊢

1198721

(32)


1and

1198982 isin 1198722

((120573 120593)⊢[

∘ 1198671 (119904)) (1198982)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

(1198671 (119904)) (1198981))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120573(1198981)=1198982

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901

∘1198682 (120572 (1198921) 1198982))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 1198982)))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (120572 (1198921) 1198982))

(daggerdagger)

= ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (120572 (1198921) 1198982))

= ⋀

1198922isin1198662

⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921) 997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(( ⋁

120572(1198921)=1198922

120593119900119901⊢

∘ 119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))997888rarr119897 1198682 (1198922 1198982))

= ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 1198982))

= ((1198672 ∘ (120572 120593)⊢[) (119904)) (1198982)

(33)


119891



and1198922 isin 1198662

(1198702 ∘ (120573 120593)⊢[

(119905)) (1198922)

= ⋀

1198982isin1198722

(((120573 120593)⊢[

(119905)) (1198982) 997888rarr1199031198682 (1198922 1198982))

= ⋀

1198982isin1198722

(120593119900119901⊢

( ⋁

120573(1198981)=1198982

119905 (1198981))997888rarr119903 1198682 (1198922 1198982))


(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(dagger)

= ⋀

1198982isin1198722

(( ⋁

120573(1198981)=1198982

120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982)))

= ⋀

1198982isin1198722

⋀

120573(1198981)=1198982

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 1198982))

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 1198682 (1198922 120573 (1198981)))

(dagger)

= ⋀

1198981isin1198721

(120593119900119901⊢

∘ 119905 (1198981) 997888rarr119903 120593119900119901⊢

∘ 120593119900119901∘ 1198682 (1198922 120573 (1198981)))

(dagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901∘ 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (1198922 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 120593119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(119905 (1198981) 997888rarr119903 1198681 (1198921 1198981))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 (119905)) (1198921))

= ((120572 120593)⊢[

∘ 1198701 (119905)) (1198922)

(34)


thatK1

119891






K1

119891


120572

997888rarr 1198662 1198721

120573

997888rarr 1198722


120593119900119901



lowast

119867119866119863997888997888997888rarr


119891

997888rarr K2) =

K1

((120572120593)⊢[

((120573120593)larr)119900119901)

997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr K2


11987111986611

11987111986611

11987111986622

11987111986622119871

11987211

11987111987211

11987111987222

11987111987222

1198671 1198672 1198701 1198702

(120572 120593)⊢

(120572 120593)⊢(120573 120593)larr

(120573 120593)larr

(35)


1and

1198981 isin 1198721

((120573 120593)larr∘ 1198672 ∘ (120572 120593)

⊢[(119904)) (1198981)

= 120593119900119901∘ (1198672 ((120572 120593)

⊢[(119904))) ∘ 120573 (1198981)

= 120593119900119901( ⋀

1198922isin1198662

(((120572 120593)⊢[

(119904)) (1198922) 997888rarr119897 1198682 (1198922 120573 (1198981))))

= 120593119900119901( ⋀

1198922isin1198662

(120593119900119901⊢

( ⋁

120572(1198921)=1198922

119904 (1198921))

997888rarr119897 1198682 (1198922 120573 (1198981))))

(dagger)

= 120593119900119901( ⋀

1198922isin1198662

( ⋀

120572(1198921)=1198922

(120593119900119901⊢

∘ 119904 (1198921)

997888rarr119897 1198682 (1198922 120573 (1198981)) )))

= 120593119900119901( ⋀

1198921isin1198661

(120593119900119901⊢

∘ 119904 (1198921)997888rarr119897 1198682 (120572 (1198921) 120573 (1198981))))

(dagger)

= ⋀

1198921isin1198661

(120593119900119901∘ 120593

119900119901⊢∘ 119904 (1198921) 997888rarr119897 120593

119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(dagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 120593119900119901∘ 1198682 (120572 (1198921) 120573 (1198981)))

(daggerdagger)

= ⋀

1198921isin1198661

(119904 (1198921) 997888rarr119897 1198681 (1198921 1198981)) = (1198671 (119904)) (1198981)

(36)


thatK1

119891




and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and1198922 isin 1198662

((120572 120593)⊢[

∘ 1198701 ∘ (120573 120593)larr(119905)) (1198922)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

(1198701 ∘ (120573 120593)larr(119905)) (1198921))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(((120573 120593)larr(119905)) (1198981)

997888rarr119903 1198681 (1198921 1198981) )))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981)

997888rarr119903 1198681 (1198921 1198981))))

(dagger)

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (120572 (1198921) 120573 (1198981)))))

= 120593119900119901⊢

( ⋁

120572(1198921)=1198922

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903 120593

119900119901

∘1198682 (1198922 120573 (1198981)))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198981isin1198721

(120593119900119901∘ 119905 ∘ 120573 (1198981) 997888rarr119903120593

119900119901

∘1198682 (1198922 120573 (1198981))))

(daggerdagger)

= 120593119900119901⊢

( ⋀

1198982isin1198722

(120593119900119901∘ 119905 (1198982) 997888rarr119903 120593

119900119901∘ 1198682 (1198922 1198982)))

(daggerdagger)

= ⋀

1198982isin1198722

(120593119900119901⊢

∘ 120593119900119901∘ 119905 (1198982)997888rarr119903 120593

119900119901⊢∘ 120593

119900119901∘ 1198682 (1198922 1198982))

(daggerdagger)

= ⋀

1198982isin1198722

(119905 (1198982) 997888rarr119903 1198682 (1198922 1198982)) = (1198702 (119905)) (1198922)

(37)


119891


525 From S-FC119863119872119877

119886to S-FC119866119882




119898 We





whose morphisms K1

119891


maps 1198711198661 120572

997888rarr 1198711198662 1198711198721

120573



119886i

119867i119863119886119866119898

997888997888997888997888997888rarr

S-FC119866119882


i119863119886119866119898

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








119871-FC119863119872119877

119886i

119867i119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882






119886i120572 (resp119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr



119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where










the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













the functor 119871-FC119863119872119877

119886idagger

119867idagger119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882





119886119900119890120572(resp

119871-FC119863119872119877


119871-FC119863119872119877


119891

997888rarr


997888rarr 1198711198662 (resp 1198711198721

120573

997888rarr


order-reflecting)


119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882


119863119886119866119898(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 119871 1)(1205721205731119871)

997888997888997888997888997888997888rarr (1198711198662 119871

1198722 119871 2) where








the functor 119871-FC119863119872119877

119886119900119890dagger

119867119900119890dagger

119863119886119866119898

997888997888997888997888997888rarr S-FC119866119882




53 S-FC119866119882 and S-FC119866119882


119898 In this




119898 on the other

531 From S-FC119866119882

119898to S-FC119862



119898and S-FC119862

119898 respectively





119898(resp S-FC119862

119898) with the


119891



120573

997888rarr 1198722 (resp 1198722

120573119900119901




(1) S-FC119866119882

119898lowast

119867119866119898119862119898997888997888997888997888997888rarr S-FC119862


119867119866119898119862119898(K1

119891

997888rarr K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

119898lowast

119867119862119898119866119898997888997888997888997888997888rarr S-FC119866119882


119867119862119898119866119898(K1

119891

997888rarr K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


and119867119862119898119866119898 ∘ 119867119866119898119862119898 =

1S-FC119866119882119898lowast


minus1(1198982)) = 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 1198682(120572(1198921) 1198982)


119900119901(1198982)) = 1198682(120572(1198921) 1198982) =

1198682(120572(1198921) (120573119900119901)minus1(1198981))







119891


the map1198721

120573

997888rarr 1198722 (resp1198722

120573119900119901




(1) S-FC119866119882

lowast

119867119866119862997888997888997888rarr S-FC119862


119891

997888rarr

K2) = K1

(120572(120573minus1)119900119901120593)

997888997888997888997888997888997888997888997888997888rarr K2

(2) S-FC119862

lowast

119867119862119866997888997888997888rarr S-FC119866119882


119891

997888rarr

K2) = K1

(120572(120573119900119901)minus1120593)

997888997888997888997888997888997888997888997888997888rarr K2


lowast


minus1(1198982)) = 120593

119900119901∘ 1198682(120572(1198921) 120573 ∘ 120573

minus1(1198982)) = 120593

119900119901∘



119900119901(1198982)) =

120593119900119901∘ 1198682(120572(1198921) 1198982) = 1198682(120572(1198921) (120573

119900119901)minus1(1198981))








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863997888997888997888997888rarr


119891

997888rarr K2) =

K1

(120572rarr

119871((120573119900119901)rarr

119871)119900119901)




1198911

9994909994731198912

K2 such that


119871 ((120573

119900119901

1)rarr

119871)119900119901) =

(1205722rarr

119871 ((120573

119900119901

2)rarr

119871)119900119901) and therefore 1205721

rarr

119871= 1205722

rarr

119871 (120573119900119901

1)rarr

119871=

(120573119900119901

2)rarr


(1205721rarr

119871(120594

⊤119871

1198921)) (1198922) = ⋁

1205721(1198921015840

1)=1198922

120594⊤119871

1198921(119892

1015840

1)

= ⊤119871 1198922 = 1205721 (1198921)


⊤119871

1205721(1198921)(1198922)

(41)


1205721(1198921)= 1205721

rarr

119871(120594

⊤119871

1198921) =

1205722rarr

119871(120594

⊤119871

1198921) = 120594

⊤119871



1= 120573

119900119901

2



(1) 119871-FC119863119872119877

i

119867i119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867i119871119863119871119862

(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(2) 119871-FC119863119872119877

119903119891119901

119867119903119891119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119903119891119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where



(3) 119871-FC119863119872119877

119900119903119901

119867119900119903119901

119871119863119871119862

997888997888997888997888rarr 119871-FC119862 given by 119867119900119903119901

119871119863119871119862(K1

119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where




119891

997888rarr

K2) = (1198711198661 119871

1198721 1)(120572120573)

997888997888997888rarr (1198711198662 119871

1198722 2) where






997888rarr P(119866)P(119872)120573119900119901

997888997888rarr P(119872) is then


997888997888997888rarr

119867119871119863119871119862(K) Define now P(119866)120572

997888rarr P(119866) = P(119866)

1P(119866)

997888997888997888997888rarr

P(119866) P(119872)120573119900119901

997888997888rarr P(119872) = P(119872)0

997888rarr P(119872) and get

an 119871-FC119862-morphism119867119871119863119871119862(K)(120572120573)

997888997888997888rarr 119867119871119863119871119862(K) However120573119900119901∘ 119867 ∘ 120572(119866) = 120573

119900119901∘ 119867(119866) = 120573

119900119901(119872) = 0 =119872 = 119867(119866) and

therefore |K|(120572120573)



lowast

119867119871119862119871119863997888997888997888997888rarr 119871-FC119863119872119877 119871-FC119863119872119877

119867119871119863119871119862997888997888997888997888rarr

119871-FC119862


lowastis the (non-




119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891


the maps 1198711198661 120572

997888rarr 1198711198662 1198711198722

120573119900119901



119871-FC119862

lowast

119867lowast

119871119862119871119863

997888997888997888997888rarr 119871-FC119863119872119877

lowast 119871-FC119863119872119877

lowast

119867lowast

119871119863119871119862

997888997888997888997888rarr 119871-FC119862

lowastof the



119871119862119871119863(resp 119867lowast

119871119863119871119862) is not a left


119871119863119871119862(resp119867lowast

119871119862119871119863)



119871119863119871119862 Then




is an 119871-FC119862

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119863119871119862∘ 119867

lowast

119871119862119871119863(K) =

(119871119866 119871




119872

997888rarr




119871119862119871119863 Then every

object K of 119871-FC119863119872119877



an 119871-FC119863119872119877

lowast-morphism K

120578=(120572120573)

997888997888997888997888997888rarr (119867lowast

119871119862119871119863∘ 119867

lowast

119871119863119871119862(K) =

(119871119866 119871



contradiction

55 119871-FC119862 versus 119871-FC119863119872119877








with morphisms K1

119891


120572

997888rarr 1198662

1198722

120573119900119901



lowast

119867119871119862119871119863119886997888997888997888997888997888rarr

119871-FC119863119872119877


119891

997888rarr

K2) = K1

(120572rarr

119871(120573119900119901)larr

119871)

997888997888997888997888997888997888997888997888997888rarr K2










119886(cf













119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119868


1119871

119892))(119898) and

1198682(119892119898) = (120573(1205941119871

119898))(119892) where 119871119866

120572

999448999471120573












Acknowledgments


119886following

References










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Lattice-Valued Topological Systems as a...

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