Research Article Yoneda Philosophy in...

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2013 Article ID 758729 11 pageshttpdxdoiorg1011552013758729

Research ArticleYoneda Philosophy in Engineering

Lambrini Seremeti1 and Ioannis Kougias2

1 Faculty of Science and Technology Hellenic Open University 26222 Patras Greece2 Department of Telecommunication Systems and Networks Technological Educational Institute of Messolonghi30200 Messolonghi Greece

Correspondence should be addressed to Ioannis Kougias kougiasteimesgr

Received 25 March 2013 Revised 18 July 2013 Accepted 9 August 2013

Academic Editor Shouming Zhong

Copyright copy 2013 L Seremeti and I Kougias This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Mathematical models such as sets of equations are used in engineering to represent and analyze the behaviour of physical systemsThe conventional notations in formulating engineering models do not clearly provide all the details required in order to fullyunderstand the equations and thus artifacts such as ontologies which are the building blocks of knowledge representationmodelsare used to fulfil this gap Since ontologies are the outcome of an intersubjective agreement among a group of individuals aboutthe same fragment of the objective world their development and use are questions in debate with regard to their competenciesand limitations to univocally conceptualize a domain of interest This is related to the following question ldquoWhat is the criterionfor delimiting the specification of the main identifiable entities in order to consistently build the conceptual framework of thedomain in questionrdquo This query motivates us to view the Yoneda philosophy as a fundamental concern of understanding theconceptualization phase of each ontology engineering methodology In this way we exploit the link between the notion of formalconcepts of formal concept analysis and a concluding remark resulting from the Yoneda embedding lemma of category theory inorder to establish a formal process

1 Introduction

In the computer science context the terms ontology andconcept are defined in many various ways and sometimestheir use is a confused mixture of both terms On the onehand an ontology is defined as a common conceptualizationof a general idea or a domain of interest [1 2] while onthe other hand a concept is defined as a general idea whichis created by removing the uncommon characteristics fromseveral particular ideas [3]Thus it is obvious that ontologiesuse concepts to convey their semantics and concepts aredefinitional structures that are explicitly encoded withinontologies

In practice an ontology consists of classes that areconcepts of a particular domain relations between classesa hierarchy of classes properties of classes that describetheir attributes instances of classes with specific values andaxioms that specify additional constraints over classes So themost crucial task of building an ontology is to consistentlyexpress its classes that is the concepts of a particular domain

An indispensable condition for formally expressing a conceptthat should carry a meaning besides its name that does notcarry any meaning is to determine all its stakeholders thatis other terms interrelated with the concept in questionproperties attached to it and specific values directly orindirectly affecting its noema since in natural language aconcept can be described by the objects belonging to it or byits determining features [4]

In order to make the ontological status of conceptsapparent independently of the language used to expressthem and of the techniques used for embedding them withinontologies we have exploited the link between the notionof formal concepts of formal concept analysis (FCA) anda concluding remark resulting from the Yoneda embeddinglemma of category theory This enables the formalization ofconcepts by considering their semantics since the notion of aconcept in computer science is very subtle and does not relyon mathematical constructors This step intends to eliminateanymisunderstanding that occurs due to ontological seman-tic heterogeneity arising from synonymy (ie ontological

2 International Journal of Engineering Mathematics

concepts for a domain described with different terminolo-gies) and polysemy (ie different meanings assigned to thesame word in different contexts)

The basics and the role of formal concept analysis indefining ontology concepts are discussed in Sections 2 and4 respectively while the basics and a discussion of relatedwork on category theory are presented in Sections 3 and5 respectively In Section 6 the main point of this work isillustrated by means of a simple example In Section 7 otherpossible meeting points of FCA and category theory aimingat direct research into the critical path of formalizing tasksconcerning ontology engineering and facilitating progress ofthe field are discussed The conclusion is drawn in the lastsection

2 Formal Concept Analysis BackgroundFormal Context Formal Concept andConcept Lattice

This section introduces basic terms of formal concept anal-ysis among which are the fundamental notions of a formalcontext formal concept and concept lattice as well as theirbasic properties

Formal concept analysis (FCA) introduced by Wille [5]is a method for analyzing data which describe relationshipsbetween a set of objects and a set of attributes From the inputdata FCA produces a concept lattice which is a collection offormal concepts in the data Formal concepts are clusters ofdata drawn together by having common attributes [6] rep-resenting natural human-like concepts and are hierarchicallyordered by a subconcept-superconcept relation FCA alsoproduces attribute implications which describe particulardependencies valid in the data It integrates the discovery andreasoning with concepts in data the discovery and reasoningwith dependencies in data and the visualization of dataand concepts and dependencies with folding and unfoldingcapabilities which are referred to as the reconstruction of adiagram of concepts and their dependencies in order to bemore simplified but as informative as the original one Formore details see [7 8]

In FCA a cross-table with logical attributes is representedby a triplet ⟨119883 119884 119868⟩ (a so-called context) where the elementsof the set 119883 are called objects and correspond to the rows ofthe table the elements of the set 119884 are called attributes andcorrespond to the columns of the table and 119868 is a binary(incidence) relation between 119883 and 119884 ⟨119909 119910⟩ isin 119868 or 119909119868119910indicates that object 119909 has attribute 119910 or that 119910 holds for 119909while ⟨119909 119910⟩ notin 119868 indicates that object119909 does not have attribute119910 An example of such a table is shown in Table 1

A partially ordered collection of particular clusters ofobjects and attributes is obtained out of a given context cross-table and it is called a concept latticeThe clusters of a conceptlattice called formal concepts are pairs ⟨119860 119861⟩ where 119860 sube 119883

is a set of objects and 119861 sube 119884 is a set of attributes such that119860 is the set of all objects which have all attributes from 119861

and 119861 is the set of all attributes which are common to allobjects from119860 For instance ⟨119909

1 1199092 1199101 1199102⟩ is an example

of a formal concept for the table of Table 1 Under suitable

Table 1 A table defining an incidence relation

1199101

1199102

1199103

1199091

times times times

1199092

times times

1199093

times times

permutations of the rows and the columns of the cross-tablea formal concept ⟨119860 119861⟩ is represented by amaximal rectanglefull of crosses

Moreover formulas describing dependencies which aretrue in the table are obtained which are called attributeimplications An attribute implication is an expression 119860 rArr

119861 with 119860 and 119861 being sets of attributes 119860 rArr 119861 is true inthe context ⟨119883 119884 119868⟩ if each object having all attributes from119860 has all attributes from 119861 as well For instance 119910

3 rArr 119910

2

is true for the context of Table 1 while 1199101 1199102 rArr 119910

3 is not

1199092being a counterexampleIn order to produce the formal concepts of a formal

context ⟨119883 119884 119868⟩ the so-called concept-forming operators areused defined between the powersets of119883 and 119884

uarr 119860 isin 2119883997888rarr 119861 isin 2

119884 119860uarr

= 119910 isin 119884for each 119909 isin 119860 ⟨119909 119910⟩ isin 119868

(1)

where 119860uarr is just the set of all attributes shared by all objectsof 119860 and

darr 119861 isin 2119884997888rarr 119860 isin 2

119883 119861darr

= 119909 isin 119883for each 119910 isin 119861 ⟨119909 119910⟩ isin 119868 (2)

where119861darr is just the set of all objects sharing all attributes from119861

A formal concept now in ⟨119883 119884 119868⟩ is just a pair ⟨119860 119861⟩ of119860 sube 119883 and 119861 sube 119884 such that 119860uarr = 119861 and 119861darr = 119860 119860 iscalled the extent and 119861 the intent of ⟨119860 119861⟩ respectively Theformal concepts are naturally ordered by using a subconcept-superconcept relation based on inclusion relation on objectsand attributes that is for formal concepts ⟨119860

1 1198611⟩ and

⟨1198602 1198612⟩ of the context ⟨119883 119884 119868⟩ ⟨119860

1 1198611⟩ le ⟨119860

2 1198612⟩ if and

only if 1198601sube 1198602(if and only if 119861

2sube 1198611)

By using basic mathematical structures behind FCAit can be proven that the pair ⟨uarr darr⟩ of concept-formingoperators induced by ⟨119883 119884 119868⟩ forms a Galois connectionbetween119883 and 119884 and the formal concepts ⟨119860 119861⟩ are just thefixed points of this Galois connection

More precisely given a pair of sets 119883 and 119884 a pair ⟨119891 119892⟩of operators 119891 2

119883rarr 2119884 and 119892 2

119883rarr 2119884 is a Galois

connection if for 1198601198601 1198602sube 119883 119861 119861

1 1198612sube 119884 it satisfies the

following

1198601sube 1198602997904rArr 119891 (119860

2) sube 119891 (119860

1)

1198611sube 1198612997904rArr 119892 (119861

2) sube 119892 (119861

1)

119860 sube 119892 (119891 (119860))

119861 sube 119891 (119892 (119861))

(3)

International Journal of Engineering Mathematics 3

For a Galois connection ⟨119891 119892⟩ between the sets119883 and 119884the fixpoint set of ⟨119891 119892⟩ is defined as the set

fix (⟨119891 119892⟩) = ⟨119860 119861⟩ isin 2119883 times 2119884

119891 (119860)= 119861 119892 (119861) = 119860 (4)

Also for a Galois connection ⟨119891 119892⟩ between the sets 119883and 119884 we have

119891 (119860) = 119891 (119892 (119891 (119860))) 119892 (119861) = 119892 (119891 (119892 (119861))) (5)

Now given a set 119883 a mapping 119862 2119883rarr 2119883 such that

for each 1198601198601 1198602sube 119883

1198601sube 119862 (119860)

1198601sube 1198602997904rArr 119862 (119860

1) sube 119862 (119860

2)

119862 (119860) sube 119862 (119862 (119860))

(6)

is called a closure operator on119883 and its fixpoint set is given by

fix (119862) = 119860 sube119883

119862 (119860)= 119860 (7)

In addition for a Galois connection ⟨119891 119892⟩ between thesets119883 and119884 the composition119862

119883 119891∘119892 is the closure operator

on119883 and 119862119884 119892 ∘ 119891 is the closure operator on 119884

Finally since the pair ⟨uarr darr⟩ of concept-forming operatorsinduced by ⟨119883 119884 119868⟩ forms a Galois connection between 119883

and 119884 the compositions uarrdarr and darruarr of the concept-formingoperators are closure operators on119883 and 119884 respectively

Another fundamental notion in FCA is that of the set ofall formal concepts of ⟨119883 119884 119868⟩ denoted byB (119883 119884 119868) the so-called concept lattice where

B (119883 119884 119868) = ⟨119860 119861⟩ isin 2119883times2119884

119860uarr= 119861 119861

darr= 119860 (8)

Then the fixpoints of the composition operator uarrdarr givethe set of all extents of the formal concepts of ⟨119883 119884 119868⟩ that is

Ext (119883 119884 119868) = 119860 isin2119883

⟨119860 119861⟩isin B (119883 119884 119868) for some 119861

= fix (uarrdarr)

(9)

and the fixpoints of the composition operator darruarr give the setof all intents of the formal concepts of ⟨119883 119884 119868⟩ that is

Int (119883 119884 119868) = 119861 isin 2119884

⟨119860 119861⟩isin B (119883 119884 119868) for some 119860

= fix (darruarr) (10)

The extents and intents are just the images under theconcept-forming operators

Ext (119883 119884 119868) = 119861darr

119861sube 119884

Int (119883 119884 119868) = 119860uarr

119860sube 119883

(11)

Hence

B (119883 119884 119868) = ⟨119860119860

uarr⟩

119860isin Ext (119883 119884 119868)

= ⟨119861darr 119861⟩

119861isin Int (119883 119884 119868)

(12)

Now a partial order le can be defined onB(119883 119884 119868) sincefor ⟨119860

1 1198611⟩ ⟨1198602 1198612⟩ isin B(119883 119884 119868) 119860

1sube 1198602if and only if

1198612sube 1198611 Thus one puts ⟨119860

1 1198611⟩ le ⟨119860

2 1198612⟩ meaning that

⟨1198601 1198611⟩ is more specific than ⟨119860

2 1198612⟩ or equivalently that

⟨1198602 1198612⟩ is more general than ⟨119860

1 1198611⟩ Then ⟨B(119883 119884 119868) le⟩

is a complete lattice which means that for any collection offormal concepts 119870 sube B(119883 119884 119868) B(119883 119884 119868) contains boththeir direct generalization (the supremum of119870 noted as or119870)and their direct specialization (the infimum of 119870 noted asand119870) The main theorem of concept lattices states that thesupremum and infimum of a collection of formal concepts⟨119860119895 119861119895⟩ 119895 isin 119869 119869 = 1 2 3 are given by

⋀119895isin119869

⟨119860119895 119861119895⟩ = ⟨⋂

119895isin119869

119860119895 (⋃119895isin119869

119861119895)

darruarr

⟩

⋁119895isin119869

⟨119860119895 119861119895⟩ = ⟨(⋃

119895isin119869

119860119895)

uarrdarr

⋂119895isin119869

119861119895⟩

(13)

3 Category Theory Background CategoriesFunctors Natural Transformations andYoneda Embedding Lemma

In this section we briefly refer to the basic notions ofcategory theory such as categories functors and naturaltransformations that guide to the comprehension of theYoneda embedding lemma

Category theory [9ndash12] which can in general be consid-ered as a mathematical theory of structures and of systemsof structures occupies a central position in contemporarymathematics and theoretical computer science since (a) itfocuses on relationships (categorical morphisms functorsand natural transformations) and not on entities (categor-ical objects and categories) (b) it allows the coexistenceof heterogeneous entities since it provides the ability todefine several categories according to the kind of entitiesto be described which can be related by the definition ofspecial morphisms (categorical functors) (c) it offers a set ofcategorical constructors for creating new categories by usingpredefined ones (d) it provides a means for the combinationof categorical objects (colimits can be used to compose themand limits to decompose them) and for the combinationof categorical functors (natural transformations) and (e)it provides a multilevel study of its categorical notions bydefining three interrelated levels (the level of categories offunctors and of natural transformations)

A category is a mathematical structure consisting ofentities of some kind together with the relationships between


A B C Df g h

h ∘ (g ∘ f) = (h ∘ g) ∘ f

g ∘ f

h ∘ g

Figure 1 The composition operation is associative

A B1B ∘ f = f = f ∘ 1A

Figure 2 The unit of the composition

them that express the structure Formally a category consistsof objects (entities) and morphisms (relationships) togetherwith a law of composition for the morphisms and a uniqueidentitymorphism for each objectMore rigorously speakinga categoryC is anything comprising the following data

(i) A classC0of objects is denoted by 119860 119861 119862

(ii) A class C1of morphisms between objects is denoted

by 119891 119892 ℎ In order to denote a morphism 119891 fromobject 119860 to object 119861 we write 119891 119860 rarr 119861 where 119860 isthe domain of 119891(dom119891 = 119860) and 119861 is the codomainof 119891(cod119891 = 119861) In general there may exist morethan one morphism from 119860 to 119861 The collection of allmorphisms from119860 to 119861 is denoted by HomC(119860 119861) =

C(119860 119861)(iii) A composition operator is denoted by ∘ assigning to

each pair of composable morphisms 119891 119860 rarr 119861 and119892 119861 rarr 119862 a composite arrow ℎ 119860 rarr 119862 ℎ = 119892 ∘ 119891The composition of morphisms is associative that isfor any morphisms 119891 119860 rarr 119861 119892 119861 rarr 119862 andℎ 119862 rarr 119863 ℎ ∘ (119892 ∘ 119891) = (ℎ ∘ 119892) ∘ 119891 as depicted inFigure 1

(iv) For each object 119860 in C0 a unique morphism 1

119860

119860 rarr 119860 which acts as the unit of the compositionthat is for any morphism 119891 119860 rarr 119861 1

119861∘ 119891 = 119891 =

119891 ∘ 1119860 as depicted in Figure 2

A categoryC is called small when the objectsC0and the

morphisms C1are sets and not proper classes A less strict

concept is that of a locally small category when for all objects119860 and 119861 in C

0 the collection of morphisms between them

C(119860 119861) is a set called a Hom-set (in this case we may stillhave ldquotoo manyrdquo pairs of objects)

An important category is 119878119890119905 the category having sets asobjects total functions as morphisms and identity functionsas identity morphisms The composition of total functions119891 119860 rarr 119861 and 119892 119861 rarr 119862 is the total function ℎ 119860 rarr 119862mapping 119886 isin 119860 to (119892 ∘ 119891)(119886) = 119892(119891(119886)) isin 119862

For each category C the dual category Cop is defined asthe category having the same objects as those of C and as

A1

A2

A30

Figure 3 An initial object

A1

A2

A3

1

Figure 4 A terminal object

morphisms the opposites of the morphisms in C that is if119891 119860 rarr 119861 is a morphism in C

1 then 119891op

119861 rarr 119860 is amorphism inC

op1

Another essential feature of category theory is that ofan isomorphism which is mainly used in order to compareobjects An isomorphism is a morphism 119891 119860 rarr 119861 that hasan inverse that is a morphism 119891

minus1 119861 rarr 119860 such that

119891 ∘ 119891minus1

= 1119861and 119891minus1 ∘ 119891 = 1

119860 If 119891 119860 rarr 119861 is an iso-

morphism then its inverse 119891minus1 119861 rarr 119860 is unique By usingthis definition two objects 119860 and 119861 are isomorphic (119860 cong 119861)if there is a pair of inverse morphisms 119891 119860 rarr 119861 and 119891minus1 119861 rarr 119860 between them Two isomorphic objects are said tobe identical up to an isomorphism if every object satisfying aproperty 119901 of an object 119860 is isomorphic to 119860

There are two special objects that are defined in acategory the initial and the terminal objects More formallyan object of a categoryC is called an initial object denoted by0 in Figure 3 if for every object 119860 inC there exists exactly aunique morphism 0 rarr 119860

The symbol is used to characterize unique morphismsOn the other hand an object of a category C is called aterminal object denoted by 1 in Figure 4 if for every object119860 inC there exists exactly a unique morphism 119860 rarr 1

In this mathematical theory any part of a category canbe pictured as a directed diagram which is a set of objectsand morphisms A significant operation in category theory isthe pushout which is defined over a diagram A pushout ofa pair of morphisms 119891 119860 rarr 119861 and 119892 119860 rarr 119862 is anobject 119876 together with a pair of morphisms 119902

119891 119862 rarr 119876

and 119902119892 119861 rarr 119876 such that 119902

119891∘ 119892 = 119902

119892∘ 119891 and with the

universal property stating that whenever 1199021015840119891 119862 rarr 119876

1015840 and1199021015840

119892 119861 rarr 119876

1015840 are such that 1199021015840119891∘ 119892 = 119902

1015840

119892∘ 119891 there exists a

unique morphism 119904 119876 rarr 1198761015840 such that 119904 ∘ 119902

119891= 1199021015840

119891and

119904 ∘ 119902119892= 1199021015840

119892 In Figure 5 the pushout is indicated by drawing

a square corner lceil next to it


A

B

Cg

f qf

qg

sq998400g = s ∘ qg

q998400f = s ∘ qf

Q

Q998400

Figure 5 The pushout tuple (119876 119902119891 119902119892) of 119891 and 119892

P998400

A

B

Cf

P

m

g

pf

pg

p998400f = pg ∘ m

p998400g = pf ∘ m

Figure 6 The pullback tuple (119875 119901119891 119901119892) of 119891 and 119892

The pushout is the categorical dual of pullback which isdepicted in Figure 6 which is obtained by reversing arrowsin the graph of pushout

These categorical constructions are characterized as uni-versal constructions describing a class of objects and theaccompanying morphisms that share a common property

Another important notion of category theory is functorswhich are structure-preservingmappings between categoriesGiven two categoriesC andD a functor 119865 C rarr Dmaps

(i) each object 119860 isin C to an object 119865119860 isin D(ii) each morphism 119891 119860 rarr 119861 isin C to a morphism

119865119891 119865119860 rarr 119865119861 isin D(iii) identity morphisms ofC to identity morphisms ofD

that is 119865(1119860) = 1119865119860

(iv) the composition of morphisms 119891 and 119892 in C to thecomposition of their mappings in D that is with 119891and 119892 composable inC 119865(119892 ∘ 119891) = 119865(119892) ∘ 119865(119891)

For a given categoryC the identity functor 1C maps eachobject and each arrow inC to itselfMoreover functors can becomposed Given two functors119865 C rarr D and119866 D rarr Etheir composition 119866 ∘ 119865 C rarr E is a functor mappingan object 119860 isin C

0to the object 119866(119865119860) isin E

0and mapping a

morphism 119891 isin C1to the morphism 119866(119865119891) isin E

1

Let us now consider two locally small categories C andD For every pair of objects119860 119861 isin C

0 a functor 119865 C rarr D

defines a function119865119860119861

C(119860 119861) rarr D(119865119860 119865119861) in 119878119890119905Thenthe functor 119865 C rarr D is

(i) faithful if the function 119865119860119861

is injective

(ii) full if the function 119865119860119861

is surjective

(iii) fully faithful if the function 119865119860119861

is bijective

A functor 119865 C rarr D is an embedding if it is fullfaithful and injective on objects A functor from C to D iscalled a covariant functor while a functor 119865 Cop

rarr Dis called a contravariant functor from C to D if it reversesarrow that is it is a covariant functor 119865 Cop

rarr D fromthe opposite categoryCop toD

In order to comprehend the Yoneda embedding themore elaborate categorical notions of representable functorsare needed The representable functors map a locally smallcategoryC to the category 119878119890119905 They refer either to incomingmorphisms to a fixed object in C (generalized elements) orto outgoing morphisms from a fixed object inC (generalizedproperties)

Given a category C the covariant representable functorC(119860 minus) C rarr 119878119890119905 which is depicted in Figure 7 canbe defined for each fixed object 119860 isin C

0 and it takes each

object 119861 isin C0to the set C(119860 119861) isin 119878119890119905

0of morphisms

from119860 to 119861 (C(119860 119861) is an object of 119878119890119905) and each morphism119891 119861 rarr 119862 isin C

1to the function

C (119860 minus) (119891) = C (119860 119891)

= 119891 ∘ minus C (119860 119861) 997888rarr C (119860 119862) isin 1198781198901199051

(14)

This last function takes a morphism 119892 119860 rarr 119861 isin C1

(an element of the set C(119860 119861) in 1198781198901199050) to the morphism

C(119860 119891)(119892) = 119891 ∘ 119892 isin C1(an element of the set C(119860 119862)

in 1198781198901199050) The covariant representable functor refers to the

outgoing morphisms from a fixed object 119860 in C and to theircompositions with other morphisms inC

Given a categoryC the contravariant representable func-tor C(minus 119861) Cop

rarr 119878119890119905 which is depicted in Figure 8is defined for each fixed object 119861 isin C

0 taking each object

119860 isin C0to the set C(119860 119861) isin 119878119890119905

0of morphisms from 119860 to

119861 (C(119860 119861) is an object of 119878119890119905) and each morphism 119891 119860 rarr

119862 isin C1to the function

C (minus 119861) (119891) = C (119891 119861)

= minus ∘ 119891 C (119862 119861) 997888rarr C (119860 119861) isin 1198781198901199051

(15)

which takes a morphism 119892 119862 rarr 119861 isin C1(an element of

the setC(119862 119861) in 1198781198901199050) to themorphismC(119891 119861)(119892) = 119892∘119891 isin

C1(an element of the setC(119860 119861) in 119878119890119905

0) The contravariant

representable functor refers to the incoming morphisms toa fixed object 119861 in C and their compositions with othermorphisms inC

Moreover given a category C the Hom-set bifunctorC(minus minus) Cop

times C rarr 119878119890119905 which is depicted in Figure 9 isdefined for each pair of fixed objects 119860 isin C

0and 119861 isin C

0

and it takes each such pair of fixed objects 119860 isin C0and




forallg B

A

C

f

Fixed

119966

119966(A ndash)(B) = 119966(A B)

119966(A ndash)(f) = 119966(A f) = f ∘ ndash

119966(A ndash)(C) = 119966(A C)

119966(A ndash)

119966(A ndash)

119966(A ndash)

119966(A ndash)

f ∘ g

Set

Figure 7 The covariant representable functorC(119860 minus)

forallg forallgB B

A A

C Cf f

Fixed Fixed

119966

g ∘ f g ∘ f

119966op119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)(A) = 119966(A B)

119966(ndash B)(f) = 119966(f B) = ndash ∘ f

119966(ndash B)(C) = 119966(C B)

Set

Figure 8 The contravariant representable functorC(minus 119861)

119861 (C(119860 119861) is an object of 119878119890119905) and each pair of morphisms119891 1198601015840rarr 119860 isin C

1and 119892 1198611015840 rarr 119861 isin C

1to the function

C (minus minus) (119891 119892)

= C (119891 119892)

= 119892 ∘ minus ∘ 119891 C (119860 1198611015840) 997888rarr C (119860

1015840 119861) isin 119878119890119905

1

(16)

This last function takes a morphism ℎ 119860 rarr 119861 isin C1(an

element of the set C(119860 119861) in 1198781198901199050) to the morphism

C(119891 119892)(ℎ) = 119892 ∘ ℎ ∘ 119891 isin C1(an element of the set C(1198601015840 1198611015840)

in 1198781198901199050)The contravariant representable functor refers to the

incoming morphisms to a fixed object 119860 in C and to theoutgoing morphisms from a fixed point 119861 inC

Another important notion of category theory is that ofnatural transformations which relate two functors Giventwo functors 119865 C rarr D and 119866 C rarr D from C toD a natural transformation 120578 119865 rArr 119866 which is depicted inFigure 10 is a map ℎ C

0rarr D

1 assigning to each object

119860 isin C0a morphism 120578

119860 119865119860 rarr 119866119860 isin D

1 called the

component of 120578 at119860 such that for every morphism 119891 119860 rarr

119861 isin C1 it holds that 120578

119861∘ 119865119891 = 119866119891 ∘ 120578

119860 The components 120578

119860

are morphisms inD1and can thus be composed

Given two categories C and D a functor categorydenoted byDC is defined having as objects the functors fromC toD and having asmorphisms the natural transformationsbetween them In this functor category HomDC(119865 119866) =

Nat(119865 119866) is the set of natural transformations from functor 119865to functor 119866 (119865 and 119866 are objects of DC) The contravariantrepresentable functor for DC is then denoted by Nat(minus 119866)

and sends a functor 119865 isin DC to the set Nat(119865 119866) isin 119878119890119905

of natural transformations between 119865 and 119866 and a naturaltransformation 120578 119865 rArr 119867 to the function

Nat (minus 119866) (120578) = Nat (120578 119866)

= minus ∘ 120578 Nat (119867 119866) 997888rarr Nat (119865 119866) (17)

The function Nat(120578 119866) sends a natural transformation 120579 119867 rArr 119866 to the natural transformation 120579 ∘ 120578 119865 rArr 119866 (which isamorphism inDC and an element of the set Nat(119865 119866) isin 119878119890119905)

An isomorphism can be defined in a functor categoryDC Given two functors 119865 119866 C rarr D (objects of DC)if a natural transformation 120578 119865 rarr 119866 (morphism in DC)has an inverse natural transformation 120578minus1 119866 rarr 119865 such thatfor every object 119860 in 119862 it is (120578 ∘ 120578minus1)

119860= (1119865)119860

= 1119865119860

and (120578minus1∘ 120578)119860= (1119866)119860= 1119866119860 then 120578 is called a natural

isomorphism and 119865 and 119866 are said to be naturally isomor-phic that is 119865 cong 119866

Given an object 119860 in a category C there is a functor 119910

C rarr 119878119890119905Cop

from C to the category 119878119890119905Copof functors from

Cop to 119878119890119905 (called the category of presheaves) that sends 119860 tothe functor C(minus 119860) isin 119878119890119905

Cop This functor 119910 is the Yoneda

embedding The Yoneda embedding also sends a morphism119891 119860 rarr 119861 isin C

1to the natural transformation 119910

119891

C(minus 119860) rarr C(minus 119861) between functors C(minus 119860) and C(minus 119861)having a component (119910

119891)119883 C(119883 119891) = 119891∘minus at119883 isin 119862

0 which

sends a morphism 119892 119883 rarr 119860 isin C(119883 119860) to the morphism119891 ∘ 119892 119883 rarr 119861 isin C(119883 119861)

The Yoneda lemma is an abstract result on functors ofthe type morphisms into a fixed object which allows the


A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)

119966(ndash ndash)(f g) = 119966(f g) = g ∘ h ∘ f

119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set

Figure 9 The Hom-set bifunctorC(minus minus)

119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G

Figure 10 The definition of the natural transformation

embedding of any category into a category of functors definedon that category It states that every functor 119865 Cop

rarr 119878119890119905 isin

119878119890119905Cop

is naturally isomorphic to the functor Nat(119910minus 119865) thatis for every 119860 isin C

0 Nat(119910119860 119865) cong 119865119860 and this isomorphism

is natural both in119860 and 119865 One instance of Yonedarsquos lemma isthe Yoneda embedding which informally states that in orderto determine an object of a category it suffices to determineall its generated elements or all its generated properties

4 The Interplay between FCA andOntological Concepts

An ontology is the formal expression of relations betweenrelevant notions in order to consistently describe a domain ofinterest and in this perspective it must clearly conceptualizethese notions At the conceptualization level it is necessaryto formally describe the relevant concepts which are to beincluded in the ontology

FCA on the other hand is a mathematical theory whichis based on the philosophical statement that a concept isconstituted by two parts its extension consisting of all objectsbelonging to the concept and its intension comprising allattributes shared by those objects [13] In FCA a formalconcept is produced from a formal context such as ⟨119883 119884 119868⟩of Section 2 by applying the concept-forming operators to itsintension or to its extension that is a formal concept is a pair⟨119860 119861⟩ where 119860 sube 119883 is a set of objects and 119861 sube 119884 is a set ofattributes such that 119860 is the set of all objects which have all

attributes from 119861 and 119861 is the set of all attributes which arecommon to all objects from 119860

Based on the assumption that an isolated concept hasno meaning since meaning is a product of the relationshipsamong concepts in a specific context an extension of FCAhas been designed the so-called relational concept analysis(RCA) [14] in order to deal with multirelational data [15]RCA provides a generalization of FCA in the case of many-valued contexts A many-valued context is defined as acontext (119866119872119882 119868) consisting of three sets namely119866 whoseelements are called formal objects 119872 with many-valuedattributes and 119882 whose elements are values and a ternaryrelation 119868 between them with the conditions (119892119898 119908) isin 119868

and (119892119898 V) isin 119868 rArr 119908 = V with (119892119898 119908) being read as ldquotheattribute119898 has the value 119908 for the object 119892rdquo Thus relationalcontext schemas are useful for attributes that have not justonly one but several values for an object

Moreover a triadic approach to FCA has been designedthe so-called triadic concept analysis (TCA) [16] as anotherFCA extension which is based on a formalization of thetriadic relation connecting objects attributes and conditionswhere a concept is described by its extension intention andmodus respectively [17] In this framework instead of thedyadic relation between objects and attributes used in FCAthe triadic approach is based on a triadic relation sayingthat the object 119892 has the attribute 119898 under the condition119887 Similar to the dyadic form of FCA a triadic conceptis produced from a triadic context which is defined as aquadruple ⟨119883 119884 119885 119868⟩ where 119883 119884 and 119885 are sets of objectsattributes and conditions respectively and 119868 is a ternaryrelation between them by applying derivation operators toits intension extension or modusThus triadic concepts canbe considered for example as those formal concepts thathave the same extent but differ according to their intent andmodus

The interplay between FCA and ontologies has beenstudied to a considerable extent [18] It turns out that thereare basically two ways concerning how FCA and ontologicalconcepts can be combined The most obvious way is toidentify the ontology concepts with the formal concepts ofFCA However in many applications the ontology conceptscorrespond to the FCA attributes [19 20] Since the onto-logical status of concepts has an internal structure within


the framework of FCA this is sometimes generated bytertiary relations withinmultivalued contexts [21] or by usingdistributional formal contexts [22]

The crucial point in specifically describing a conceptwithin an ontology is to find out all the attributes that areshared by this concept Thus from an FCA-based pointof view for formally describing a concept its intentionaldescription is required In the case where a concept isdescribed by a set of attributes under specific conditions thenotion of triadic contexts is used for its representation [23]Moreover each concept within an ontology denotes a specificrole emerging from adding specific values to its intentionalstructure that is its attributes In this perspective ontologyconcepts are considered as the relational structures that theRCA process provides [24 25]

5 The Interplay between Category Theory andOntological Concepts

Since ontology can be regarded as a categorization of thatwhich exists issues involving the ontological status of con-cepts and the notion of a category are important According toAristotle a class or collection of things all of one type wheretypicality is determined by a concept is a category Later ona category as a purely mathematical notion was defined ashaving an internal structure given by relationships amongmembers Thus the exact match between the terms categoryand concept has emerged because a concept can be defined ashaving internal structure due to their explanatory capabilities[26]

Category theory has been established as an appropriatemathematical basis for the formalization and study of ontolo-gies [27 28] since it focuses on relationships (categoricalmorphisms) between entities (categorical objects) in such away that an entity can be defined according to its interactionwith other entities [29] There have been various categorytheoretic approaches to ontology engineering which involvestasks such as development mapping andmerging of ontolo-gies In particular in the case of ontology mapping andmerging the category 119874119899119905 of ontologies is defined and thecategorical constructions of pullback and pushout are usedrespectively [30ndash32]

Formally speaking ontology is a theory over logic Butmany ontologies are described in different languages eachone with its own logic Thus in order to integrate ontologiesmany approaches using the theory of institutions [33] haveemerged such as in [34 35] for mapping between differentlogics to deal with heterogeneous formalization Moreoverthe constructions of category theory have been combinedwith information flow theory [36] which describes howinformation about one thing can convey information aboutsomething else for integrating distributed ontologies [37 38]Finally in [39] a rigorous foundation for integrating heteroge-neous information is provided by unifying information flowtheory institutions formal concept analysis and categorytheory

By using a category theoretic language ontologies canbe represented as categories with finite limits and finite

A1

A2 A3

A4

A5 A6

Figure 11 A simple example of an ontology

coproducts where classes (concepts) are represented bynodes and functional relationships by arrows of a directedgraph [40] and if an ontology has a representation by acategory then one may apply homological and homotopicalalgebra constructions to study the ontology [41] Anotherview of the category theoretic perspective of ontologies andtheir underlying classes (concepts) is given in [42] where anontology is regarded as a mathematical category of theoriesand a concept is represented by the ideas of theories andmodels that are borrowed from FCA Moreover in [43] anontology can be described by formal concepts and thus it canbe viewed as a category

6 From the Yoneda Embedding toFormal Concepts

Our main point is the fact that from an ontological point ofview the formal determination of a concept is guided by theinterconnection existing between a concluding remark of theYoneda embedding lemma of category theory and the intentsof the formal concepts of FCA In order to illustrate this pointan example is used in this section relying on the assumptionthat attributes can also be considered as concepts accordingto the German standard DIN 2330 stating that attributes ldquoareunits of thought which are gained by abstraction and hencethey are also conceptsrdquo [19 44]

Following this statement we can represent binary rela-tions between entities within an ontology by using a suitableformal context For this purpose we consider the entities ofthe ontology both as objects and as attributes of the contextA binary relation present in the context signifies that therespective entities are related As an example the ontologyof Figure 11 where the concepts 119860

119894 119894 = 1 2 6 are

hierarchically structured is represented by the formal contextof Figure 12 by using Lattice Miner [45] which is a formalconcept analysis software tool for the construction of conceptlattices

In Figure 12 the concept lattice and the formal conceptsassociated with the formal context of Figure 11 are depictedIt is a representation of a formal context called cross-tableThe context has six attributes 119886

119894 119894 = 1 2 6 An ldquo119909rdquo in a


Figure 12 The formal context for the ontology of Figure 11

A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6

Figure 13 The formal concepts and the concept lattice for theontology of Figure 11

particular location or cell in the cross-table indicates that anobject has an attribute so for example the object ldquo119860

2rdquo has

the attribute ldquo1198861rdquo Conversely a blank location indicates that

the object does not have that attributeThe concept lattice of Figure 13 contains five concepts

themost general being ⟨1198601 119860

6⟩The bottom concept is

⟨ 1198861 119886

6⟩ because it is the concept of objects that have

all attributes The remaining concepts are marked with theirintent and their extent

The diagram shows that there is an ordering over theconcepts which is a specialization ordering denoted by le[46]

It is interesting to notice here that the formal conceptsrepresent the incoming relations to an ontology entity thatis the intents of each formal concept correspond to theincoming arrows to each respective ontological concept Forexample the formal concept ⟨119886

1 1198602 1198603⟩ represents the

incoming relations (arrows) to ontological entity (concept)1198601 Now if we consider a categorical perspective of the

ontology where the ontology entities are the objects ofa category and the relations between the entities are the

Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to

Belongs to Belongs to

Belongs to

Figure 14 A segment of an ontology conceptualizing Europeanplaces

morphisms (arrows) then the intent of a formal concept rep-resents the incoming arrows to an entity (object) But from acategory theoretic point of view this remark is equivalent tothe remark concluded from the Yoneda embedding lemmawhich states that an object is completely determined by theincoming arrows to this object

This statement yields from the Yoneda lemma whichis interpreted as follows an object 119883 of a category 119862 isdetermined by the network of relationships that the object119883 has with all the other objects in 119862 [47] Thus a conceptcan be clearly specified only by identifying the set of allits attributes and a set of attributes that are common fora concept determines completely a concept This reciprocalcondition can guide the building of the ontological statusof concepts completely and consistently since it eliminatespolysemy and synonymy

In amore illustrative example such as that of the ontologyof Figure 14 following the same process and taking intoaccount its category theoretical representation where con-cepts are objects of a category and their attributesrelationsare morphisms we observe the alignment between the intentof the formal concept ⟨119887119890119897119900119899119892119904 119879119900 119866119903119890119890119888119890 119865119903119886119899119888119890⟩ andthe incoming arrows to the ontological entity ldquoEuroperdquo thatis the Yoneda philosophy in unequivocally determiningconcepts in ontology engineering

7 Equivalent Points of FCA andCategory Theory

In this section we refer to the possiblemeeting points of FCAand category theory aiming at direct research into the criticalpath of formalizing tasks concerning ontology engineeringand facilitating progress of the field

Due to the fact that we focus on the abstract notion ofconcepts as well as their formal representation and not onstrict mathematical objects it is incongruous to talk about


equality between different points of mathematical theoriesrather than equivalence or isomorphism between themThus the main question to be answered is not if two conceptsare equivalent but if they are isomorphic that is if theyare interchangeable in the sense that they have the samerelationships with all concepts within a context In thatsense it will be interesting to investigate the correspondencebetween the basic notions and constructions of FCA andcategory theory such as between (a) the supremum of anycollection of formal concepts of a complete lattice and theinitial object of a category (b) the infimum of any collectionof formal concepts of a complete lattice and the terminalobject of a category and (c) the application of the concept-forming operators to the intention or to the extension of aformal context and the fact that from an object of a categoryone can define a specific functor that encodes the essence ofthese objects as well as between other notions

8 Conclusion

A formal view on ontologies or on ontological concepts cancontribute a lot to the direction of achieving effective inter-operability since the coexistence of multiple heterogeneousontologies within or among ontology-based applications isan objective reality Thus by carefully studying FCA andcategory theory as the mathematical theories that provideframeworks for structuring analyzing and visualizing datato make them more understandable and for combining andmaintaining heterogeneous collections of data to enable themto interoperate respectively we investigate some equivalentnotions of both theories This can lead despite the abstractbut quite accurate techniques of these theories to create aunified framework for the ontological representation as wellas their management

Acknowledgments

The authors are indebted to the editor and the anonymousreviewers for their valuable suggestions and comments thatgreatly contributed to improving the paper

References

[1] T R Gruber ldquoToward principles for the design of ontologiesused for knowledge sharingrdquo in Formal Analysis in Con-ceptual Analysis and Knowledge Representation Kluwer Aca-demicPlenum New York NY USA 1993

[2] W N Borst Construction of Engineering Ontologies Centre forTelematica and Information Technology University of TweentyEnsschede The Netherlands 1997

[3] J Goguen ldquoWhat is a conceptrdquo in Proceedings of the 13thConference on Conceptual Structures F Dan and M MungierEds vol 3596 of Lecture Notes in Artificial Intelligence pp 52ndash77 Springer 2005

[4] L Lambrechts Formal Concept Analysis Vrije UniverstiteitBrussels Belgium 2012

[5] R Wille ldquoRestructuring lattice theory an approach based onhierarchies of conceptsrdquo in Ordered Sets I Rival Ed pp 445ndash470 Reidel Dordrecht The Netherlands 1982

[6] R Belohlavek and V Sklenar ldquoFormal concept analysis overattributes with levels of granularityrdquo in Proceedings of theInternational Conference on Communicational Intelligence forModelling Control and Automation amp Inter Conf on IntelligentAgents Web Technologies and Internet Commerce ViennaAustria November 2005

[7] R Belohlavek Introduction to Formal Concept Analysis UPOlomouc 2008

[8] B Ganter and R Will Formal Concept Analysis MathematicalFoundations Springer Berlin Germany 1999

[9] SAwodeyCategoryTheory OxfordUniversity PressNewYorkNY USA 2010

[10] B Pierce Basic Category Theory for Computer Scientists TheMIT Press Cambridge Mass USA 1991

[11] W Lawvere and S Schanuel Conceptual Mathematics A FirstIntroduction to Categories Cambridge University Press NewYork NY USA 1997

[12] J Fiadeiro Categories for Software Engineering Springer NewYork NY USA 2004

[13] M Obitko V Snasel and J Smid ldquoOntology design with formalconcept analysisrdquo in Proceedings of the International Conferenceon Concept Lattices and Their Applications (CLA rsquo04) V Snaseland R Belohlavek Eds pp 111ndash119 2004

[14] U E PrissRelational concept analysis semantic structures in dic-tionaries and lexical databases [Dissertation] TH-Darmstadt1996

[15] M Huchard M R Hacene C Roume and P ValtchevldquoRelational concept discovery in structured datasetsrdquo Annals ofMathematics and Artificial Intelligence vol 49 no 1ndash4 pp 39ndash76 2007

[16] F Lehmann and R Will ldquoA triadic approach to formal conceptanalysisrdquo in Conceptual Structures Applications Implementa-tion and Theory vol 954 of Lecture Notes in Computer Sciencepp 32ndash43 Springer Berlin Germany 1995

[17] J Konecny and P Osicka ldquoGeneral approach to triadic conceptanalysisrdquo in Proceedings of the 7th International Conferenceon Concept Lattices and Their Applications (CLA rsquo10) Mkryszkiewicz and S A Obiedkov Eds pp 116ndash126 SevillaSpain 2010

[18] P Hitzler ldquoWhatrsquos happening in semantic webmdashand what FCAcould have to do with itrdquo in Formal Concept Analysis PValtchev and R Jaschke Eds vol 6628 of Lecture Notes inComputer Science pp 18ndash23 Springer Berlin Germany 2011

[19] P Cimiano A Hotho G Stumme and J Tane ldquoConceptualknowledge processing with formal concept analysis and ontolo-giesrdquo in Proceedings of the 2nd International Conference onFormal Concept Analysis (ICFCA rsquo04) pp 189ndash207 February2004

[20] R Bendaoud A Napoli and Y Toussaint ldquoFormal con-cept analysis a unified framework for building and refiningontologiesrdquo in Knowledge Engineering Practice and PatternsA Gangemi and J Euzenat Eds vol 5268 of Lecture Notes inComputer Science pp 156ndash171 Springer Berlin Germany 2008

[21] J Nanda T W Simpson S R T Kumara and S B Shooter ldquoAmethodology for product family ontology development usingformal concept analysis and web ontology languagerdquo Journal ofComputing and Information Science in Engineering vol 6 no 2pp 103ndash113 2006

[22] D Jurkevicius and O Vasilecas ldquoFormal concept analysis forconcept collecting and their analysisrdquo Scientific Papers Univer-sity of Latvia Computer Science and Information Technologiesvol 751 pp 22ndash39 2009


[23] R Jaschke A Hotho C Schmitz B Ganter and G StummeldquoDiscovering shared conceptualizations in folksonomiesrdquo Jour-nal of Web Semantics vol 6 no 1 pp 38ndash53 2008

[24] M Rouane-Hacene M Huchard A Napoli and P ValtchevldquoUsing formal concept analysis for discovering knowledgepatternsrdquo in Proceedings of the 7th International Conference onConcept Lattices andTheir Applications (CLA rsquo10) Sevilla SpainOctober 2010

[25] X Dolques M Huchard C Nebut and H Saada ldquoFormal andrelational concept analysis approaches in software engineeringan overview and an application to learn model transformationpatterns in examplesrdquo in Proceedings of the 1st ICESE VirtualWorkshop on Software Engineering and Artificial IntelligenceDoha Qatar 2011

[26] M J Healy ldquoCategoryTheory as a Mathematics for formalizingontologiesrdquo in Theory and Applications of Ontology ComputerApplications R Poli M Healy and A Kameas Eds pp 487ndash510 Springer New York NY USA 2010

[27] M Krotzsch P Hitzler M Ehrig and Y Sure ldquoCategory theoryin ontology research concrete gain from an abstract approachrdquoTech Rep AIFB University of Karlsruhe 2005

[28] R Brown J F Glazebrook and I C Baianu ldquoA conceptual con-struction of complexity levels theory in spacetime categoricalontology non-abelian algebraic topology many-valued logicsand dynamic systemsrdquoAxiomathes vol 17 no 3-4 pp 409ndash4932007

[29] I Cafezeiro and E H Haueusler ldquoSemantic interoperabilityvia Category Theoryrdquo in Proceedings of the 26th InternationalConference on Conceptual Modeling J Grundy S Hartman AH F Laender L Maciaszek and J F Roddick Eds vol 83 pp197ndash202 2007

[30] A Zimmermann M Krotzsch J Euzenat and P Hitzler ldquoFor-malizing ontology alignment and its operations with categorytheoryrdquo inProceedings of the International Conference on FormalOntology in Information Systems (FOIS rsquo06) B Bennett and CFellbaum Eds pp 277ndash288 IOS Press Baltimore Md USA2006

[31] PHitzlerM KrotzschM Ehrig andY Sure ldquoWhat is ontologymergingA category-theoretical perspective using pushoutsrdquo inProceedings of the 1st International Workshop on Contexts andOntologiesTheory Practice andApplications pp 104ndash107 AAAIPress July 2005

[32] L Hu and J Wang ldquoGeo-ontology integration based on cate-gory theoryrdquo in Proceedings of the International Conference onComputer Design and Applications (ICCDA rsquo10) pp V15ndashV18Qinhuangdao China June 2010

[33] J A Goguen and R M Burstall ldquoInstitutions abstract modeltheory for specification and programmingrdquo Journal of the ACMvol 39 no 1 pp 95ndash146 1992

[34] TMossakowski CMaeder andK Luttich ldquoTheheterogeneoustool setrdquo in Tools and Algorithms for the Construction andAnalysis of Systems O Grumberg and M Huth Eds vol 4424of Lecture Notes in Computer Science pp 519ndash522 SpringerHeidelberg Germany 2007

[35] O Kutz D Lucke and T Mossakowski ldquoHeterogeneouslystructured ontologies integration connection and refinementrdquoin Proceedings of the Knowledge Representation Ontology Work-shop (KR rsquo08) vol 90 of Conferences in Research and Practice inInformation Technology Sydney Australia 2008

[36] J Barwise and J Seligman Information Flow The Logic ofDistributed Systems vol 44 of Cambridge Tracts in Theoretical

Computer Science Cambridge University Press New York NYUSA 1997

[37] R Kent ldquoThe IFF foundation for ontological knowledge orga-nizationrdquo Cataloging amp Classification Quarterly vol 37 no 1-22003

[38] Y Kalfoglou and M Schorlemmer ldquoThe information flowapproach to ontology-based semantic alignmentrdquo inTheory andApplications of Ontology Computer Applications chapter 4 pp101ndash114 Springer New York NY USA 2010

[39] J Goguen Information Integration in Institutions L Moss Ed2004

[40] M Johnson and R Rosebrugh ldquoOntology engineering univer-sal algebra and category theoryrdquo in Theory and Applicationsof Ontology Computer Applications R Poli M Healy and AKameas Eds pp 565ndash576 Springer New York NY USA 2010

[41] N Glazunov ldquoSemantic web and category theory modelsrdquo inProceedings of the International Seminar ldquoiscrete Mathematicsand its Applicationsrdquo pp 309ndash311 MGU Moscow Russia 2007

[42] M JHealy andT P Caudell ldquoOntologies andworlds in categorytheory implications for neural systemsrdquoAxiomathes vol 16 no1-2 pp 165ndash214 2006

[43] P Wendorff ldquoA formal approach to the assessment andimprovement of terminological models used in informationsystems engineeringrdquo in Proceedings of the 8th Eiropean Engi-neering Conference and 9th ACM SIGSOFT Symposium on theFoundations of Software Engineering (ESEC rsquo01 and FSE rsquo01) VGruhn Ed pp 83ndash87 Vienna Australia September 2001

[44] Deutsches Institut fur Normung ldquoBegriffe undBenennungenmdashAllgemeine Grundsatzerdquo DIN 2330 1993

[45] 2012 httplattice-minersoftpediacom[46] T Tilley Concept Analysis and Formal Methods 2000[47] B Mazur ldquoWhen is one thing equal to other thingrdquo in Proof

and Other Dilemmas Mathematics and Philosophy B Gold andR Simons Eds 2007

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


concepts for a domain described with different terminolo-gies) and polysemy (ie different meanings assigned to thesame word in different contexts)

The basics and the role of formal concept analysis indefining ontology concepts are discussed in Sections 2 and4 respectively while the basics and a discussion of relatedwork on category theory are presented in Sections 3 and5 respectively In Section 6 the main point of this work isillustrated by means of a simple example In Section 7 otherpossible meeting points of FCA and category theory aimingat direct research into the critical path of formalizing tasksconcerning ontology engineering and facilitating progress ofthe field are discussed The conclusion is drawn in the lastsection

2 Formal Concept Analysis BackgroundFormal Context Formal Concept andConcept Lattice

This section introduces basic terms of formal concept anal-ysis among which are the fundamental notions of a formalcontext formal concept and concept lattice as well as theirbasic properties

Formal concept analysis (FCA) introduced by Wille [5]is a method for analyzing data which describe relationshipsbetween a set of objects and a set of attributes From the inputdata FCA produces a concept lattice which is a collection offormal concepts in the data Formal concepts are clusters ofdata drawn together by having common attributes [6] rep-resenting natural human-like concepts and are hierarchicallyordered by a subconcept-superconcept relation FCA alsoproduces attribute implications which describe particulardependencies valid in the data It integrates the discovery andreasoning with concepts in data the discovery and reasoningwith dependencies in data and the visualization of dataand concepts and dependencies with folding and unfoldingcapabilities which are referred to as the reconstruction of adiagram of concepts and their dependencies in order to bemore simplified but as informative as the original one Formore details see [7 8]

In FCA a cross-table with logical attributes is representedby a triplet ⟨119883 119884 119868⟩ (a so-called context) where the elementsof the set 119883 are called objects and correspond to the rows ofthe table the elements of the set 119884 are called attributes andcorrespond to the columns of the table and 119868 is a binary(incidence) relation between 119883 and 119884 ⟨119909 119910⟩ isin 119868 or 119909119868119910indicates that object 119909 has attribute 119910 or that 119910 holds for 119909while ⟨119909 119910⟩ notin 119868 indicates that object119909 does not have attribute119910 An example of such a table is shown in Table 1

A partially ordered collection of particular clusters ofobjects and attributes is obtained out of a given context cross-table and it is called a concept latticeThe clusters of a conceptlattice called formal concepts are pairs ⟨119860 119861⟩ where 119860 sube 119883

is a set of objects and 119861 sube 119884 is a set of attributes such that119860 is the set of all objects which have all attributes from 119861

and 119861 is the set of all attributes which are common to allobjects from119860 For instance ⟨119909

1 1199092 1199101 1199102⟩ is an example

of a formal concept for the table of Table 1 Under suitable

Table 1 A table defining an incidence relation

1199101

1199102

1199103

1199091

times times times

1199092

times times

1199093

times times

permutations of the rows and the columns of the cross-tablea formal concept ⟨119860 119861⟩ is represented by amaximal rectanglefull of crosses

Moreover formulas describing dependencies which aretrue in the table are obtained which are called attributeimplications An attribute implication is an expression 119860 rArr

119861 with 119860 and 119861 being sets of attributes 119860 rArr 119861 is true inthe context ⟨119883 119884 119868⟩ if each object having all attributes from119860 has all attributes from 119861 as well For instance 119910

3 rArr 119910

2

is true for the context of Table 1 while 1199101 1199102 rArr 119910

3 is not

1199092being a counterexampleIn order to produce the formal concepts of a formal

context ⟨119883 119884 119868⟩ the so-called concept-forming operators areused defined between the powersets of119883 and 119884

uarr 119860 isin 2119883997888rarr 119861 isin 2

119884 119860uarr

= 119910 isin 119884for each 119909 isin 119860 ⟨119909 119910⟩ isin 119868

(1)

where 119860uarr is just the set of all attributes shared by all objectsof 119860 and

darr 119861 isin 2119884997888rarr 119860 isin 2

119883 119861darr

= 119909 isin 119883for each 119910 isin 119861 ⟨119909 119910⟩ isin 119868 (2)

where119861darr is just the set of all objects sharing all attributes from119861

A formal concept now in ⟨119883 119884 119868⟩ is just a pair ⟨119860 119861⟩ of119860 sube 119883 and 119861 sube 119884 such that 119860uarr = 119861 and 119861darr = 119860 119860 iscalled the extent and 119861 the intent of ⟨119860 119861⟩ respectively Theformal concepts are naturally ordered by using a subconcept-superconcept relation based on inclusion relation on objectsand attributes that is for formal concepts ⟨119860

1 1198611⟩ and

⟨1198602 1198612⟩ of the context ⟨119883 119884 119868⟩ ⟨119860

1 1198611⟩ le ⟨119860

2 1198612⟩ if and

only if 1198601sube 1198602(if and only if 119861

2sube 1198611)

By using basic mathematical structures behind FCAit can be proven that the pair ⟨uarr darr⟩ of concept-formingoperators induced by ⟨119883 119884 119868⟩ forms a Galois connectionbetween119883 and 119884 and the formal concepts ⟨119860 119861⟩ are just thefixed points of this Galois connection

More precisely given a pair of sets 119883 and 119884 a pair ⟨119891 119892⟩of operators 119891 2

119883rarr 2119884 and 119892 2

119883rarr 2119884 is a Galois

connection if for 1198601198601 1198602sube 119883 119861 119861

1 1198612sube 119884 it satisfies the

following

1198601sube 1198602997904rArr 119891 (119860

2) sube 119891 (119860

1)

1198611sube 1198612997904rArr 119892 (119861

2) sube 119892 (119861

1)

119860 sube 119892 (119891 (119860))

119861 sube 119891 (119892 (119861))

(3)



fix (⟨119891 119892⟩) = ⟨119860 119861⟩ isin 2119883 times 2119884

119891 (119860)= 119861 119892 (119861) = 119860 (4)


119891 (119860) = 119891 (119892 (119891 (119860))) 119892 (119861) = 119892 (119891 (119892 (119861))) (5)


for each 1198601198601 1198602sube 119883

1198601sube 119862 (119860)

1198601sube 1198602997904rArr 119862 (119860

1) sube 119862 (119860

2)

119862 (119860) sube 119862 (119862 (119860))

(6)


fix (119862) = 119860 sube119883

119862 (119860)= 119860 (7)







B (119883 119884 119868) = ⟨119860 119861⟩ isin 2119883times2119884

119860uarr= 119861 119861

darr= 119860 (8)


Ext (119883 119884 119868) = 119860 isin2119883

⟨119860 119861⟩isin B (119883 119884 119868) for some 119861

= fix (uarrdarr)

(9)


Int (119883 119884 119868) = 119861 isin 2119884

⟨119860 119861⟩isin B (119883 119884 119868) for some 119860



Ext (119883 119884 119868) = 119861darr

119861sube 119884

Int (119883 119884 119868) = 119860uarr

119860sube 119883

(11)

Hence

B (119883 119884 119868) = ⟨119860119860

uarr⟩

119860isin Ext (119883 119884 119868)

= ⟨119861darr 119861⟩

119861isin Int (119883 119884 119868)

(12)


1 1198611⟩ ⟨1198602 1198612⟩ isin B(119883 119884 119868) 119860



1 1198611⟩ le ⟨119860





1 1198611⟩ Then ⟨B(119883 119884 119868) le⟩


⋀119895isin119869

⟨119860119895 119861119895⟩ = ⟨⋂

119895isin119869

119860119895 (⋃119895isin119869

119861119895)

darruarr

⟩

⋁119895isin119869

⟨119860119895 119861119895⟩ = ⟨(⋃

119895isin119869

119860119895)

uarrdarr

⋂119895isin119869

119861119895⟩

(13)






A B C Df g h

h ∘ (g ∘ f) = (h ∘ g) ∘ f

g ∘ f

h ∘ g


A B1B ∘ f = f = f ∘ 1A









119860


119861∘ 119891 = 119891 =









A1

A2

A30


A1

A2

A3

1



1 then 119891op


op1



119891 ∘ 119891minus1

= 1119861and 119891minus1 ∘ 119891 = 1

119860 If 119891 119860 rarr 119861 is an iso-





119891 119862 rarr 119876


119891∘ 119892 = 119902

119892∘ 119891 and with the


1015840 and1199021015840

119892 119861 rarr 119876

1015840 are such that 1199021015840119891∘ 119892 = 119902

1015840



119891= 1199021015840

119891and

119904 ∘ 119902119892= 1199021015840




A

B

Cg

f qf

qg

sq998400g = s ∘ qg

q998400f = s ∘ qf

Q

Q998400


P998400

A

B

Cf

P

m

g

pf

pg

p998400f = pg ∘ m

p998400g = pf ∘ m







that is 119865(1119860) = 1119865119860




0and mapping a


1






is injective


is surjective


is bijective






0 and it takes each


0of morphisms


1to the function

C (119860 minus) (119891) = C (119860 119891)


(14)













C (minus 119861) (119891) = C (119891 119861)


(15)








0and 119861 isin C

0





forallg B

A

C

f

Fixed

119966

119966(A ndash)(B) = 119966(A B)


119966(A ndash)(C) = 119966(A C)

119966(A ndash)

119966(A ndash)

119966(A ndash)

119966(A ndash)

f ∘ g

Set


forallg forallgB B

A A

C Cf f

Fixed Fixed

119966

g ∘ f g ∘ f

119966op119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)(A) = 119966(A B)


119966(ndash B)(C) = 119966(C B)

Set



1and 119892 1198611015840 rarr 119861 isin C

1to the function

C (minus minus) (119891 119892)

= C (119891 119892)

= 119892 ∘ minus ∘ 119891 C (119860 1198611015840) 997888rarr C (119860

1015840 119861) isin 119878119890119905

1

(16)







0rarr D



119860 119865119860 rarr 119866119860 isin D

1 called the



119861∘ 119865119891 = 119866119891 ∘ 120578


119860






Nat (minus 119866) (120578) = Nat (120578 119866)




119860= (1119865)119860

= 1119865119860




C rarr 119878119890119905Cop






119891


119891)119883 C(119883 119891) = 119891∘minus at119883 isin 119862

0 which




A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)


119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set


119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G



rarr 119878119890119905 isin

119878119890119905Cop




















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











fix (⟨119891 119892⟩) = ⟨119860 119861⟩ isin 2119883 times 2119884

119891 (119860)= 119861 119892 (119861) = 119860 (4)


119891 (119860) = 119891 (119892 (119891 (119860))) 119892 (119861) = 119892 (119891 (119892 (119861))) (5)


for each 1198601198601 1198602sube 119883

1198601sube 119862 (119860)

1198601sube 1198602997904rArr 119862 (119860

1) sube 119862 (119860

2)

119862 (119860) sube 119862 (119862 (119860))

(6)


fix (119862) = 119860 sube119883

119862 (119860)= 119860 (7)







B (119883 119884 119868) = ⟨119860 119861⟩ isin 2119883times2119884

119860uarr= 119861 119861

darr= 119860 (8)


Ext (119883 119884 119868) = 119860 isin2119883

⟨119860 119861⟩isin B (119883 119884 119868) for some 119861

= fix (uarrdarr)

(9)


Int (119883 119884 119868) = 119861 isin 2119884

⟨119860 119861⟩isin B (119883 119884 119868) for some 119860



Ext (119883 119884 119868) = 119861darr

119861sube 119884

Int (119883 119884 119868) = 119860uarr

119860sube 119883

(11)

Hence

B (119883 119884 119868) = ⟨119860119860

uarr⟩

119860isin Ext (119883 119884 119868)

= ⟨119861darr 119861⟩

119861isin Int (119883 119884 119868)

(12)


1 1198611⟩ ⟨1198602 1198612⟩ isin B(119883 119884 119868) 119860



1 1198611⟩ le ⟨119860





1 1198611⟩ Then ⟨B(119883 119884 119868) le⟩


⋀119895isin119869

⟨119860119895 119861119895⟩ = ⟨⋂

119895isin119869

119860119895 (⋃119895isin119869

119861119895)

darruarr

⟩

⋁119895isin119869

⟨119860119895 119861119895⟩ = ⟨(⋃

119895isin119869

119860119895)

uarrdarr

⋂119895isin119869

119861119895⟩

(13)






A B C Df g h

h ∘ (g ∘ f) = (h ∘ g) ∘ f

g ∘ f

h ∘ g


A B1B ∘ f = f = f ∘ 1A









119860


119861∘ 119891 = 119891 =









A1

A2

A30


A1

A2

A3

1



1 then 119891op


op1



119891 ∘ 119891minus1

= 1119861and 119891minus1 ∘ 119891 = 1

119860 If 119891 119860 rarr 119861 is an iso-





119891 119862 rarr 119876


119891∘ 119892 = 119902

119892∘ 119891 and with the


1015840 and1199021015840

119892 119861 rarr 119876

1015840 are such that 1199021015840119891∘ 119892 = 119902

1015840



119891= 1199021015840

119891and

119904 ∘ 119902119892= 1199021015840




A

B

Cg

f qf

qg

sq998400g = s ∘ qg

q998400f = s ∘ qf

Q

Q998400


P998400

A

B

Cf

P

m

g

pf

pg

p998400f = pg ∘ m

p998400g = pf ∘ m







that is 119865(1119860) = 1119865119860




0and mapping a


1






is injective


is surjective


is bijective






0 and it takes each


0of morphisms


1to the function

C (119860 minus) (119891) = C (119860 119891)


(14)













C (minus 119861) (119891) = C (119891 119861)


(15)








0and 119861 isin C

0





forallg B

A

C

f

Fixed

119966

119966(A ndash)(B) = 119966(A B)


119966(A ndash)(C) = 119966(A C)

119966(A ndash)

119966(A ndash)

119966(A ndash)

119966(A ndash)

f ∘ g

Set


forallg forallgB B

A A

C Cf f

Fixed Fixed

119966

g ∘ f g ∘ f

119966op119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)(A) = 119966(A B)


119966(ndash B)(C) = 119966(C B)

Set



1and 119892 1198611015840 rarr 119861 isin C

1to the function

C (minus minus) (119891 119892)

= C (119891 119892)

= 119892 ∘ minus ∘ 119891 C (119860 1198611015840) 997888rarr C (119860

1015840 119861) isin 119878119890119905

1

(16)







0rarr D



119860 119865119860 rarr 119866119860 isin D

1 called the



119861∘ 119865119891 = 119866119891 ∘ 120578


119860






Nat (minus 119866) (120578) = Nat (120578 119866)




119860= (1119865)119860

= 1119865119860




C rarr 119878119890119905Cop






119891


119891)119883 C(119883 119891) = 119891∘minus at119883 isin 119862

0 which




A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)


119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set


119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G



rarr 119878119890119905 isin

119878119890119905Cop




















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A B C Df g h

h ∘ (g ∘ f) = (h ∘ g) ∘ f

g ∘ f

h ∘ g


A B1B ∘ f = f = f ∘ 1A









119860


119861∘ 119891 = 119891 =









A1

A2

A30


A1

A2

A3

1



1 then 119891op


op1



119891 ∘ 119891minus1

= 1119861and 119891minus1 ∘ 119891 = 1

119860 If 119891 119860 rarr 119861 is an iso-





119891 119862 rarr 119876


119891∘ 119892 = 119902

119892∘ 119891 and with the


1015840 and1199021015840

119892 119861 rarr 119876

1015840 are such that 1199021015840119891∘ 119892 = 119902

1015840



119891= 1199021015840

119891and

119904 ∘ 119902119892= 1199021015840




A

B

Cg

f qf

qg

sq998400g = s ∘ qg

q998400f = s ∘ qf

Q

Q998400


P998400

A

B

Cf

P

m

g

pf

pg

p998400f = pg ∘ m

p998400g = pf ∘ m







that is 119865(1119860) = 1119865119860




0and mapping a


1






is injective


is surjective


is bijective






0 and it takes each


0of morphisms


1to the function

C (119860 minus) (119891) = C (119860 119891)


(14)













C (minus 119861) (119891) = C (119891 119861)


(15)








0and 119861 isin C

0





forallg B

A

C

f

Fixed

119966

119966(A ndash)(B) = 119966(A B)


119966(A ndash)(C) = 119966(A C)

119966(A ndash)

119966(A ndash)

119966(A ndash)

119966(A ndash)

f ∘ g

Set


forallg forallgB B

A A

C Cf f

Fixed Fixed

119966

g ∘ f g ∘ f

119966op119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)(A) = 119966(A B)


119966(ndash B)(C) = 119966(C B)

Set



1and 119892 1198611015840 rarr 119861 isin C

1to the function

C (minus minus) (119891 119892)

= C (119891 119892)

= 119892 ∘ minus ∘ 119891 C (119860 1198611015840) 997888rarr C (119860

1015840 119861) isin 119878119890119905

1

(16)







0rarr D



119860 119865119860 rarr 119866119860 isin D

1 called the



119861∘ 119865119891 = 119866119891 ∘ 120578


119860






Nat (minus 119866) (120578) = Nat (120578 119866)




119860= (1119865)119860

= 1119865119860




C rarr 119878119890119905Cop






119891


119891)119883 C(119883 119891) = 119891∘minus at119883 isin 119862

0 which




A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)


119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set


119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G



rarr 119878119890119905 isin

119878119890119905Cop




















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

B

Cg

f qf

qg

sq998400g = s ∘ qg

q998400f = s ∘ qf

Q

Q998400


P998400

A

B

Cf

P

m

g

pf

pg

p998400f = pg ∘ m

p998400g = pf ∘ m







that is 119865(1119860) = 1119865119860




0and mapping a


1






is injective


is surjective


is bijective






0 and it takes each


0of morphisms


1to the function

C (119860 minus) (119891) = C (119860 119891)


(14)













C (minus 119861) (119891) = C (119891 119861)


(15)








0and 119861 isin C

0





forallg B

A

C

f

Fixed

119966

119966(A ndash)(B) = 119966(A B)


119966(A ndash)(C) = 119966(A C)

119966(A ndash)

119966(A ndash)

119966(A ndash)

119966(A ndash)

f ∘ g

Set


forallg forallgB B

A A

C Cf f

Fixed Fixed

119966

g ∘ f g ∘ f

119966op119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)(A) = 119966(A B)


119966(ndash B)(C) = 119966(C B)

Set



1and 119892 1198611015840 rarr 119861 isin C

1to the function

C (minus minus) (119891 119892)

= C (119891 119892)

= 119892 ∘ minus ∘ 119891 C (119860 1198611015840) 997888rarr C (119860

1015840 119861) isin 119878119890119905

1

(16)







0rarr D



119860 119865119860 rarr 119866119860 isin D

1 called the



119861∘ 119865119891 = 119866119891 ∘ 120578


119860






Nat (minus 119866) (120578) = Nat (120578 119866)




119860= (1119865)119860

= 1119865119860




C rarr 119878119890119905Cop






119891


119891)119883 C(119883 119891) = 119891∘minus at119883 isin 119862

0 which




A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)


119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set


119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G



rarr 119878119890119905 isin

119878119890119905Cop




















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










forallg B

A

C

f

Fixed

119966

119966(A ndash)(B) = 119966(A B)


119966(A ndash)(C) = 119966(A C)

119966(A ndash)

119966(A ndash)

119966(A ndash)

119966(A ndash)

f ∘ g

Set


forallg forallgB B

A A

C Cf f

Fixed Fixed

119966

g ∘ f g ∘ f

119966op119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)

119966(ndash B)(A) = 119966(A B)


119966(ndash B)(C) = 119966(C B)

Set



1and 119892 1198611015840 rarr 119861 isin C

1to the function

C (minus minus) (119891 119892)

= C (119891 119892)

= 119892 ∘ minus ∘ 119891 C (119860 1198611015840) 997888rarr C (119860

1015840 119861) isin 119878119890119905

1

(16)







0rarr D



119860 119865119860 rarr 119866119860 isin D

1 called the



119861∘ 119865119891 = 119866119891 ∘ 120578


119860






Nat (minus 119866) (120578) = Nat (120578 119866)




119860= (1119865)119860

= 1119865119860




C rarr 119878119890119905Cop






119891


119891)119883 C(119883 119891) = 119891∘minus at119883 isin 119862

0 which




A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)


119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set


119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G



rarr 119878119890119905 isin

119878119890119905Cop




















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A AB B

f f

h h

g g

A998400 A998400B998400 B998400

119966

g ∘ h ∘ f f ∘ h ∘ g

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966(ndash ndash)

119966op

119966(ndash ndash)(A B) = 119966(A B)


119966(ndash ndash)(A998400 B998400) = 119966(A998400 B998400)

Set


119966 119967

F

F

F

G

G

G

f

A

B FB

GB

Gf

FA120578A

120578B

Ff

GA

F G



rarr 119878119890119905 isin

119878119890119905Cop




















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















A1

A2 A3

A4

A5 A6






119894 119894 = 1 2 6 are






A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











A1 A2 A3 A4 A5 A6

A2A3

a1A4

a3A5 A6

a4

a1 a3 a4a2 a5 a6



2rdquo has





⟨ 1198861 119886





1 1198602 1198603⟩ represents the



Europe

GreeceFrance

Athens

Kifisia Glyfada

Belongs to

Belongs to


Belongs to










8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











8 Conclusion


Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Yoneda Philosophy in...

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