1 Galois Connections for Operations and Relations

R. Pöschel

Abstract

This paper reports on various Galois connections between operations and relations. Several specifications and generalizations are discussed. AMS Mathematics Subject Classzfication: 08455, 03B10. Key words: Operations, Relations, Clones, Relational algebras.

I

lntroduction

Operational systems and relational systems are basic structures in algebra and logic. Here we focus on one fundamental connection be- tween operations and relations which is based on the invariance rela- tion "an operation f preserves a relation e". This indiices a Galois connection' (Pol - Inv, see 1.3, 2.2) between Sets of operations and sets of relations. Many well-known algebraic concepts fit into this framework (automorphisms, subalgebras, congruences, see e.g. proof of 3.5). , There are numerous results on various Galois connections hetween operations and relations characterizing the Galois closed sets. In this paper we undertake the attempt to collect and systematize these re- sults and to give a kind of survey. Therefore almost no proofs are given and besides new formulations there are no new results. More- over, we were not able to give a complete survey. We mainly report on

P

In [McKMT87, p. 1471 this Galois connection is emphasized as the most basic Galois connection in algebra. In [Wi103] it is mentioned as a general framwork for the development of dyadic algebra.

231 K. Denecke et al. (eds.), Galois Connections and Applications. 231-258. O 2004 KluwerAcademic Publishers. Printed in the Nerherlands.

*

232 R. Pöschel

results which can be found in the following (not so easily accessible) papers: [Pös80a] (some of the results of this report can also be found in [Pös79]), [Pös84], [BörOO] and [BörPS] . For an orientation to which Galois connections are discussed the reader may at first consult the Table 1 in Section 3 and Table 2 in Section 4.

1 Basic notions and notations

1.1. Operations and Relations. For operations f : An + A and relations p C Am (m, n E N+ = {1,2,3,. . . )) on a fixed base set A we introduce the following notation:

o ~ ( ~ ) ( A ) := {f 1 f : An + A) (n-ary operations)

Op(A) := Ur=, o ~ ( ~ ) ( A ) (finitary operations)

:= O ~ ( ' ) ( A ) (transformations, i.e. unary permutations)

Sym(A) := { f E Tr(A) I f bijective) (permutations) Rel (" ) (~) := {p 1 p C Am) (m-ary relations) Rel(A) := ~ e l ( ~ ) (A) (finitary relations) ( P P ) := Rel ( ' ) (~ ) (power set) Eq(A) := {B E ~ e l ( ~ ) ( ~ ) 1 B equivalence) (equivalence

relations) The (A) may be omitted if it is clear from the context. 0-ary functions or relations are not considered (mainly for technical reasons). An m-tuple r E Am sometimes will be regarded as a mapping r :

m + A with m := (1,. . . , m), and its components are given by r =

( ~ ( l ) , . . . , +)).

1.2. Invariance. Operations and relations can be connected by the invariance property: A relation p E ~ e l ( ~ ) ( ~ ) is invariant for an operation f E o ~ ( ~ ) ( A ) (we also say, f preserves p, or f is a polymor- phism of p, notation [Wi103]: f D p) if f [p, . . . , p] C p, i.e., if for all r l , . . . ,T, E p we have f [T',. . . , T,] E p. Here the m-tuple f Irl,. . . , T,]

is defined component-wise by

Operations and Relations 233

Remark: In the universal algebra setting this means that p is (the carrier of) a subalgebra of the algebra (A; f )m (m-th direct power); from the relational point of view f is a (relational) homomorphism from the relational system (A; Q ) ~ = (An; pn) into (A; p). For permutations f E Sym(A) we define: f strongly preserves p if f [p] = p, i.e., if the mapping p + p : r t-, f [T] is bijective.

1.3. The operators Pol - Inv. For F C Op(A), G C Sym(A) and Q Rel(A) we define

PolA Q := { f E Op(A) I every p E Q is invariant for f ) , pol$) Q := O~(") (A) n polA Q , (n E N) EndA Q := n ( A ) f l PolA Q (endomorphisms),

W A U ~ A Q := Sym(A) n PolA Q (weak automorphisms),

AutA Q := { f E W A U ~ A Q I f-' E W A U ~ A Q)

(au tomorphisms),

InvA F := {p E Rel(A) I p is invariant for every f E F) , 1nvLrn) F := ~ e l ( ~ ) ( ~ ) n Inv F (m E N) ,

(1) SubA F := Inv, F (subalgebras),

ConA F := Eq n InvA F (congruence relations),

sInvA G := {p E Rel(A) I p is strongly invariant for every g E G ) .

Note that the notation Pol stands for polymorphisms, not for polyno- mials! The subscript A may be omitted if A is clear from the context. The notion of weak automorphism applies here to relational systems and differs from the weak automorphisms for algebras introduced by A. Goetz ([Goe66]) and investigated in many papers of K. Glazek (e.g. [Gla97]).

1.4. Clones. Recall, a clone is a set of operations (on a fixed base set A) which is closed with respect to composition and contains all projections er E o ~ ( ~ ) given by er(xl, . . . , xn) = xi. The composition f . . . , gn] of f E o ~ ( ~ ) and gl, . . . , gn E o ~ ( ~ ) is the m-ary function defined by

234 R. Pöschel

The least clone containing F C Op(A) (i.e., the clone generated by F) will be denoted by (F ) or Likewise, (H)n(A) and (G)Sym(A) denote the submonoid of Tr(A) generated by H C Tr(A) and the subgroup of Sym(A) generated by G C Sym(A), respectively.

1.5. Relational algebras (clones). We define the following (set- theoretical) operations:

(1) AA (nullary operation: to contain the diagonal (or equality) relation AA := {(U, a) I a E A)),

(2) n (intersection of relations of the Same arity),

(3) X (product for m-ary e and s-ary a :

(4) pr, (projection onto a subset I of coordinates: For m-ary Q and I = (21, ..., it) with 15 il < i 2 - . . < it 5 m we define

(5) T, (permutation of coordinates: For m-ary e and a permutation Q of (1 , . . . , m ) let

(6) U (union of relations of the Same arity),

(7) 7 (complementation: TQ := Am \ Q for m-ary Q)

A set Q C Rel(A) of relations is called a relational algebra, weak Krasner algebra or Krasner algebra, respectively, if Q is closed with respect to (1)-(5), (1)-(6) or (1)-(7), respectively. These closures will be denoted by [QIRA, [QIWKA and [Q]KA, resp. For finite sets A these algebras are also called clones (relational clone, (weak) Krasner clone). For infinite A See 2.4. Note that relational algebras contain relations of arbitrary (finite) ar- ity and are much more general than Tarski's relation algebras ([Tar41]).

Operations and Relations 235

1.6. Remark. Relational algebras can be characterized also via clo- Sure with respect to first order formulas. To each first order formula y (Rl , . . . , R,; xl , . . . , X,) with free variables xl , . . . ,X, and mi-ary re- lation (predicate) symbols R, (i E {I , . . . , q)) one can assign an Oper- ation (called logical operation)

according to

Hereby, for atomic formulas e.g. R,(x, y) we interpret k ei(a, b) as (U, b) E ei (for elements a, b E A). E.g., the formula

(with binary relation symbols) induces the operation relational prod- uct:

for binary relations e2, ~2 E ~ e l ( ~ ) ( ~ ) . Let Q(3, A, . . . ) denote the set of all first order formulas that contain only the indicated quantifier 3, the indicated connectives A, . . . and relation symbols and variables. Let LopA(3, A, . . . ) := {F, I cp E Q(3, A, . . . )) denote the corre- sponding logical operations. Then we have (cf. [PösK79, 2.1.31) for Q C Rel(A):

Q relational .algebra H Q closed w.r.t. LopA(3, A, =) , Q weak Krasner algebra H Q closed w.r.t. LopA(3, A, V, =) ,

Q Krasner algebra H Q closed w.r.t. L0pA(3, A , V , l , = ) .

2 Galois connections and Galois closures

2.1. Galois connections. We use here the canonical definition of a Galois connection between sets A und B, namely as a pair of mappings

4

236 R. Pöschel

cp : p ( A ) + p ( B ) , $J : p(B) + p ( A ) induced by a binary relation I C A x B v i a

for X C A, Y C B. It is well-known that such pairs (V,$) of map- pings can be characterized by the following properties: Both cp and $J are anti-monotone (i.e. inclusion-reversing) and their compositions cp$J and $Jcp are closure operators (i.e., extensive, monotone, idempotent). Therefore elements X E p ( A ) and Y E p(B) satisfying X = $J(cp(X)) and Y = cp($J(Y)) are called Galois closed w.r.t. (cp, $J). The sets of Galois closed elements are

and they form dually isomorphic complete lattices (w.r.t. inclusion).

2.2. With the above definitions in 2.1, the (strong) invariance relation D (cf. 1.2) gives rise to the following Galois connections:

Pol - Inv between Rel(A) und Op(A) , End - Inv between Rel(A) und Tr(A) ,

wAut - Inv between Rel(A) und Sym(A) , Aut - sInv between Rel(A) und Sym(A) .

Whenever a Galois connection is given there arises the question of how to characterize the corresponding closure operators and the Galois closed sets. This is well-known for these listed Galois connections (cf. e.g. [PösK79]) and we summarize the results for finite A in the next theorem (for notation See 1.4, 1.5 and 1.6).

Theorem 2.3. Let A be finite und let F C Op(A), H C Tr(A), G C Sym(A), Q C Rel(A). Then

( F ) o ~ ( A ) = Po1 Inv F , . [QI RA = Inv Pol Q , = End Inv F , [Q]WKA = Inv End Q ,

( G ) s ~ ~ ( A ) = Aut Inv G [QIKA = I ~ v Aut Q .

O~erations and Relations 237

In order to include results for infinite sets A we still need some more notions and notations.

+ 2.4. Relational clones and local closures. For b = (cl , . . . , C,) E Bm and a mapping f E AB let

For a family (ei)iEr of relations ei E ~ e l ( ~ ' ) ( ~ ) (I being an arbitrary index set) we introduce the general superposition with respect to given b E Bm, bi E Bmi (i E I) for some set B as follows:

S(i,p)(ei)it~ := {f [q E Am I f E AB and V i E I : f[bil E eil. The strong superposition w.r.t. ä E Am, äi E Ami is given by

~ s ( ä , a ~ ) ( e i ) i , ~ := {f [ä] I f E Tr(A) surjective and Vi E I : f [Gi] E e i l .

Let Q C Rel(A). The least set of relations which contains Q and the trivial (unary) relations 0, A, and is closed w.r.t. general superposition, will be denoted by [Q](s). Note that arbitrary intersections are a particular case of general su-

+ +

perposition: niEI ei = S(bb.,(ei)iEI for b = bi E Am (mi = m, i E I ) . Further we introduce locai Llosure operators which are in fact a kind of interpolation operation (and can also be characterized topologically). Let F C Op(A), G C Sym(A), Q C Rel(A), m E W. Then

m-LocF := {f E o ~ ( ~ ) ( A ) I Val, ..., Zn E Am39 E F :

f [ ä l , - . . , ä n ] = g [ ~ i , - . - , & ] ) , 00

Loc F := n m-Loc F , m=l

Loc,G := {f E Sym(A) I Vn E N+VÜE An3g E G : f[q = g[q), m-LOCQ:= { @ E Rel(A) I Vrl, . . . , rm E e3a E Q :

{ r l , . . . , r m ) C a C Q ) , M

LOC F := n m-LOC F . m=l

Let [Q]Rc := LOC[Q](s) and [Q]WKC := l-LOCIQ](S). We call Q C Rel(A) a relational clone (weak Krasner clone, resp.) if Q = [QIRc

*

238 R. Pöschel

(Q = [QIWKC, resp.). It should be noted that the definition of rela- tional clone here differs from that in other Papers (e.g. [Pös80a]) where sets Q with Q = [Q](s) were also called relational clones. One easily checks that 1-LOCQ is the closure of Q with respect to arbitrary unions. F'urther, let [QIKc be the closure of Q with respect to [.IWKC (weak Krasner clone), complementation (1) (cf. 1.5) and strong superposi- tion (symbolically we express this by (KC) = (WKC)&(i)&(sS)). Sets Q with Q = [QIKC are called Krasner clones. A weak Krasner clone will be called a Pre-Krasner clone if it is closed with respect to strong superposition and contains the inequaltity re- lation U = VA := {(X, Y) E A2 1 X # Y ) . The corresponding closure is denoted by [QIPKC (thus we have symbolically (PKC) = (WKC)&(v)&(sS)). Remark: In [BörOO] it is shown that (Pre-)Krasner clones are just the (Pre-)Krasner algebras (cf. 1.5) which are closed w.r.t. strong super- position and arbitrary (i.e. also infinite) intersections and unions. If A is countable then one can even drop the closure w.r.t. strong super- position.

With these notions we can state the following theorem which ex- tends 2.3 to infinite sets A.

Theorem 2.5. Let F C Op(A), H C Tr(A), G C Sym(A), Q C Rel(A). Then

Loc(F)o,(A) = Pol Inv F , [Q]Rc = Inv Pol Q , L o c ( H ) n ( ~ ) = End Inv F , [Q]WKC = InvEndQ,

L O C , ( G ) ~ ( ~ ) = wAut Inv G , [Q]PKC = Inv wAut Q , Loc,(G)s~,,,(A) = Aut sInv G , [Q] KC = sInv Aut Q .

Remark: We have sInv G = Inv G whenever G 5 Sym(A) is a permu- tation group. Therefore e.g. sInv Aut Q = Inv Aut Q. In the next sections we report on further Galois connections which appear in connection with operations and relations. There are several (mutually connected) possibilities for specialization and generalization of those Galois connections considered in the previous section, e.g.:

Operations and Relations 239

(A) Restricting the operations and/or relations under consideration but keeping the invariance relation "Q is invariant for f" .

(B) Generalizing the operations (e.g. to partial operations or multi- operations) and/or relations with "canonically" modified invari- ance relation.

(C) Considering "natural" closure operators on operations and/or relations and trying to characterize the closed sets as Galois closures of a suitable Galois connection.

3 Galois connections with restricted operations and relations

We start with the approach (A) which was already described in [Pös80a, 15.11.

3.1. The Galois connections EPol-R~nv. Let E C Op(A) and R C Rel(A). Then EPol -RInv defined by

E ~ o l ~ : = E ~ P O ~ Q for Q C R ,

R ~ n v ~ := R ~ I ~ V F for F C E

is a Galois connection between R and E. For motivation, E and R may be regarded as those sets of operations and relations which are of particular interest or especially important in some context. For arbitrary E and R, there is no general proceedure for how to characterize the Galois closed sets of operations or relations. However, if we assume that E is a locally closed clone and R is a relational clone then (by 2.5) there exist F. C Op(A) and Qo C Rel(A) such that

E = PolQo and R = Inv F. .

Then we have the following characterization for the Galois closures.

Theorem 3.2. Let E , R, Fo, Qo be as above. Then

240 R. Pöschel

Forfinite A the 1ocal.operator.s can be omitted and [.IRC can be replaced by [.IRA. For E = Op(A) or R = Rel(A) one can choose Ro = 0 or F, = 0.

Proof. The result immediately follows from Theorem 2.5 and E n PolQ = PolQonPo1Q = Pol(QouQ), R n I n v F = I n v F 0 n I n v F = Inv(Fo U F). 0

Remark 3.3. Let G := Pol R = Pol Inv F0 (then R = Inv G). Every pair (E, R) as above gives rise to the induced closure operator

CIS(E,G) (H) := E ~ ~ l RInv H

on the principle ideal $E =$LaE :=. { H I H E CA and H C E) in the lattice CA of all clones in Op(A). In case of finite A we have

c~s(E,G)(H) = E A (G V H) (see Fig. 1)

(A, V denoting meet and join in the lattice CA) according to Theo- rem 3.2 (where the local operators can be omitted since A in finite); for infinite A one has to restrict to the lattice of locally closed clones.

Fig. 1: The closure cIs(E,~)(H) = E A (G V H ) in $E

Operations and Relations 241

Let E = A~~ and let G be the clone of all term operations of an algebra A = (A; (f&) from a class IC of algebras. The C l ~ ( ~ , ~ ) - c l ~ ~ e d sets H A~~ (monoids in case k = 1) can be equipped with the algebraic structure inherited from A. They were investigated in [JakMPOS] as so-called operation IC-algebras (transformation IC-algebras for k = 1) and generalize the concept of near-rings.

In particular cases the Galois closures may be characterized more intrinsically. Note, e.g., that the Galois connections End - Inv and wAut - Inv coincide with EPol - Inv for E = Tr(A) and E = Sym(A), respectively (and trivial R = Rel(A)). As a further example we mention the following characterization theo- rem where the arity of operations or relations will be restricted (E = o ~ ( ~ ) (A) or R = ~ e l ( ~ ) (A)) .

Theorem 3.4. Let m E N+, F C Op(A), Q Rel(A). Then

pol ~nv(") F = m-Loc(F)o,,(~) , Inv pol(") Q = m - L o c [ Q ] ~ ~ = m-LOc[Q](s) .

Many other specializations of the Galois connection Pol - Inv were investigated, in particular, special Galois closures were characterized for various purposes. We collect some of these specializations in Ta- ble l below where we briefly mention which Galois closures have been characterized and give some references. The first two columns show how to choose the sets E and R in order to apply Theorem 3.2 (how- ever, although our approach via relational clones can be applied to every particular case there are often other characterizations which are more practicable; some details can be found e.g. in [Pös80a] and in the references of Table 1). In order to read Table 1 correctly we mention that sometimes the reader Sees a set F of operations where a set of relations should be expected. Here we use the convention that an n-ary function f : An + A also may be treated as (n + 1)-ary relation, namely the graph of f which we denote by

Thus, e.g., the so-called bicentralizer Pol Pol F of F means Pol(Po1 F')' (with F' := {f' 1 f E F) for a set F of operations). Note that e.g.

4

242 R. Pöschel

an automorphism f of an algebra A = (A; F) may be characterized in different ways:

Orice the Galois closures are characterized the way is Open for applica- tions. As an example we take the so-called concrete characterization of related structures like the subalgebra lattice SubA, the automor- phism group Aut A or the congruence lattice ConA of an algebra A = (A; F ) . The problem may be posed as follows: The concrete sirnultaneous characterization problern: Given a set L C y ( A ) of subsets of A, a set C C Eq(A) of equivalence relations on A and a set G C Sym(A) of permutations on A. Does there exist an algebra A such that G is the automorphism group of A, L is the subalgebra lattice of A, und C is the congruence lattice of A ? Note that we do require exact equality (e.g. C = ConA) and not equality up to isomorphism (the latter is the abstract characteriza- tion problem which sometimes is easier to answer and gives satisfac- tory globally valid results; here are some references for the abstract characterization problem: [Bir46], [BirF48], [GräS63], [Sch63], [Sch64], [GräL67], [J6n74], [SchT79]). The answer to the concrete characterization problem in principle is already provided with Theorem 3.2, but we shall make it more explicit:

Theorem 3.5 (cf. [Pös80a, (7.2)]). For a given permutation group G < Sym(A) und given L C !J3(A), C C Eq(A), there exists an algebra A = (A; F) such that G = Aut A, L = Sub A und C = Con A if und only if

where Q := G' U L U C. The algebra A can be chosen with at most m-ary operations if und only if [Q]Rc is replaced by m-LOC[QIRc.

Proof. Note that f E Au tA W f' E Sym(A)' n Inv F , U E SubA W U E I ~ v ( ' ) F = S u b F and 0 E ConA W 0 E Eq(A) n InvF .

Thus the theorem immediately follows from Theorem 3.2. (For more details we refer to [Pös80a]).

Operations and Relations 243

In Table 1 we also mention several further results for the concrete characterization problem.

Tab. 1: Several closures of the Galois connections EPol - R ~ n v ( E C OP(A), R C Rel(A))

Galois closure References Pol Inv F [BodKKRgga],

[BodKKRGSb], [Gei68], [BakP75], [Rom76], [Rom77a], [Rom77b] ,[PösK79], cf. 2.3, 2.5

Inv Pol Q [Gei68], [BodKKR69a], [BodKKR69b], [Sza78], [PösK79], [Pös79], [Pös80a], cf. 2.3, 2.5

Generalization to infinitary relations or operations Pol Invm F [Ros72], [KraP76],

[Poi8l] Invm Pol Q [Ros79] Pol" Invm F [Kra76b], [KraP76], Invm Polm Q [Poi81]

arity restrictions Pol lndrn) F [Gei68], [BakP75],

[Pös80a] lndm) Pol Q [Ros78]

Pol Sub F [Sch82, Thm. 1.61, [Pös8Oa]

Sub Pol Q see below Inv Q [Sza78], [Pös8Oa]

to be continued at nezt page

244 R. Pöschel

Inv Aut Q [Kra38], [BodKKRGga],

wAut Inv F [BodKKRGgb], Aut sInv F [Gou72a], [Pös80a],

[BörOO]

Table 1, continued from previous Page

implicit operations) I End P o l 0 see below

Galois closure References [Kra38], [Kra5O],

Inv End End Inv F ) [Kra76a], [Kra86], [Gou68], [BodKKRGga], [BodKKR69b], [Pös80a], [BörOO]

E

Tr(A)

S Y ~ ( A ) I ~ e l ( ~ ) (A) I Aut 1nv(") F [Wie691 restriction to (graphs o f ) operations only

concrete characterization problems

Aut PolQ [J6n68] (cf. [J6n72, (2.4.3)]), [Kra50], [ArmS64], [Sza75], [Bre76]

R Re1 ( A)

OP(A)

concrete characterization of A u t d for algebras A with at most m-ary operations

, ~ u t pol(") Q [Pio68], [Gou72a],

L [J6n72, (2.4.1)] concrete characterization of Sub A

OP(A)'

Sub Pol Q [BirF48] (cf. [J6n72, (3.6.4)]), [Gou68], [Gou72b]

to be wntinued at next Page

Pol Pol F [Sza78, Thm. 131, [Faj77], [Dan771 (for [Al = 3), (also Kuz- necov, cf. [Va176])

Operations and Relations 245

Op( A)' U Sub(A)

Table 1, continued from previous jage

Aut A & Sub A [Sto72], [Gou72b], cf. 3.5

Aut A & Con A [Wer74](conjecture) cf. [Pös80b], (for simple A [Sch64]), cf. 3.5

End A &Sub A [SauS77b] (cf. [J6n74])

Galois closure References for unary algebras: [J6n72, (3.6.7)], [JohS67]

concrete characterization of Con A Con Pol Q [Arm701 (partial

solution), [J6n72, (4.4.1)], [QuaW7l], [Wer74], [Dra74]

Pol Con F for p-rings (A; F) [Isk72]

wncrete characterization of End d End Pol Q [Lam68], [GräL68],

[SauS77a], [Sto69], [Sto75], [Jei72], [Sza78, Thm. 151

concrete characterization of

E

OP(A)

OP(A)

I I Cf. 3.5 I

R

Tr(A)'

op(A)'USub(A) U Eq(A)

I Generalization to partial functions and rnultifunctions See 4.1 1

Aut A & SubA & Con A

[Sza78], [Pös80a],

4 Further Galois connections

Now let us consider further Galois connections from the points of view (B) and (C) mentioned a t the end of Section 2. Concerning (B) we mention here some generalizations of the opera- tions under consideration while the other side of the Galois connection

246 R. Pöschel

- the relations Rel(A) - remains unchanged. The invariance relation which induces the Galois connection is a more or less straightforward generalization and we refer to the references.

4.1. Galois connections wi th generalized operations.

partial functions (i.e. mappings f : B + A with B C An): [Ros83] ([FleR78]), [HadR92] (cf. also [Fre66]), [Rom81], [BörHP91], [SusK94],

multifunctions (i.e. mappings f : An + ()7(A)): [BörOl], [DreP],

heterogeneous functions (i.e. mappings f : Ai, X Ai, + Ai, for a family (Ai)iEr of base sets, 20, i l , . . . , in E I): [Pös73].

Another generalization can be found in [PösROO] where cofunctions (i.e. mappings f : A + (1,. . . , n) X A) and corelations (i.e. subsets e (1, . . . , n)") are considered instead of functions and relations. A Galois connection which generalizes many features of functions, cofunctions, partial functions and multifunctions was investigated in [RößOO] .

Finally we demonstrate (C) point of view.

4.2. Closures of relational systems as Galois closures. In Table 2 below we show how several closures of systems of rela- tions can be characterized by Galois connections. These closures can be classified by logical operatioris (cf. 1.6) of particular sets of first order formulas (cases (1)-(8)) or other operators on relations (cases

(9)-(15)). However the investigations were not really motivated by this classifica- tion. In cases (1)-(8) the algebraic structure of the relational systems under consideration was of most important interest and gave such a hierarchy of relational systems. In cases (9)-(15) also the closed sets of operations were the starting point of interest. We mainly report on results from [BörPS](cases (1)-(8)) and [Pös84] (cases (9)-(14)) and refer to these papers for further details. Some of the cases are already discussed in this paper and were mentioned in Table 1; however they fit into the scheme used here and therefore they are listed once more. Table 2 is organized as follows. The first column shows the closure by

Operations and Relations 247

indicating under which operations the relational system is to be closed. The next column gives a notation for the closure (if it was introduced somewhere). The last two columns indicate the Galois connection and briefly describe the Galois closed sets (the last column hereby shows what is the operational counterpart for the Galois connection under consideration). For simplicity we assume that the base set A is finite for the cases (1)-(8). The generalization to arbitrary base sets A for ( I) , (2), (3) is given in (9), (14), (15). In cases (6)- (9) one has essentially to consider n-complete relational systems and locally closed opera- tional system in order to get the results also for infinite A (for details See [BörPS]). We shall provide here explicitly only those notions in Table 2 which concern the relational systems (for the operational part See the above mentioned papers) . A Boolean system, briefly BS, is a set Q C_ Rel(A) of relations which is closed with respect to the Boolean operations (union U, intersection n, complement 1, cf. also 1.5), contains as constants the sets 0 and Am (m E N+) and is closed with respect to the following operation W, : Rel(")(A) + Rel(")(A) for all mappings s : + (n, m E N'):

A Boolean system with identity, briefly BSI, is a Boolean system which contains the diagonal relation AA (cf. 1.5). A Boolean system with projections, briefly BSP, is a Boolean system which is additionally closed under the projections prl (cf. 1.5).

Tab. 2: Relational closures 'as Galois closures

for finite base set A:

T ~ ~ ~ A T (cf. 2.3 and Tab. 1)

(1) LopA(3,A,v, T,=) [QIKA Krasner aige- I I bra (cf. 1.5, group of per- mutations

248 R. Pöschel Operations and Relations 249

Closure Closure I Galois connection Galois connection closed rela- ( closed opera-

I sInv - smEnd

Table

(2)

(3)

(4)

(5)

(6)

I tional system I tional system

Table 2, continued from previous Page

involuted mo- noid of bitotal multifunc-

to be continued on next Page

Inv -End (cf. 2.3 and Tab. 1)

noid of unary multifunctions

2, continued from

LopA(3,A,V,=)

LopA (3, A, =)

L o ~ ~ ( A 7 =)

L o ~ ~ ( A )

LopA(3, A, V, 1 )

weak Kras- ner algebra (cf. 1.5, 1.6)

I I LopA(A, V, 1 )

1 for arbitrary base set A:

previous Page

[&]wKA

[&]RA

[QIssp

[&Issi (7)

I

monoid of unary func- tions

sInv - spmEnd

(KC) = (WKC) & (7) &(SC)

LopA(A, V, -, =) BSI (cf. 4.2)

[QIss

sInv - Aut (cf. 2.5 and Tab. 1)

(cf. 2.4) group of per-

Inv - wAut (cf. 2.5 and Tab. 1)

down-closed involuted monoid of pp-multifunc- tions (partial

Inv - Pol (cf. 2.3 and Tab. 1)

(10) 1 (PKC) = [QIPKC Pre-Krasner locally closed (WKC) & (U) & (SC) 1 1 clone (cf. 2.4) monoid of per-

relational alge- bra (cf. 1.5, 1.6)

BS (cf. 4.2)

clone of fini- tary functions

down-closed involuted mo-

tive unary jective unary functions funct ions

(1 I )

Inv - pPol

weak system with identity

(WKC)&(sS)

Inv - inb-End

down-closed clone of fini- tary partial functions

Inv H for 10- cally invertible monoids H

I mutations

Inv - sur-End

locally closed locally invert- ible monoid o f unary functions

Inv - mPol

Inv H for sets H of surjec-

t o be continued on next Page

weak system of relations

locally closed monoid of sur-

down-closed clone of finitary multi- functions

sInv - sEnd (sbmEnd, resp.)

BSP (cf. 4.2)

Special mo- noid of unary functions (cf. [BörPS, 7.91)

(down-closed

250 R. Pöschel

There are many interdependences between the operations on relations mentioned in this Paper. E.g., with intersections and the operations W, (cf. 4.2) one can express the operations (3) and (5) from 1.5. Of- ten it depends on the cardinality of the base set (finite, countable, un- countable) which operations on relations are needed for the characteri- zation of the Galois closed sets (detailed investigations can be found in [BörOO]) and how they are interrelated (e.g. (KC) e (RC)&(l) holds for finite and countable but not for uncountable A, [DroKMPOl]).

Closure

Finally we mention that there is an interesting interpretation of the operations of relational algebras (and Krasner algebras) as operations of the Peircean algebraic logic (PAL) proposed also by R.W. Burch in [Bur911 and modelled mathematically in power context families, see [Po1021 and [ArnOl], for more details we refer to [HerPOS]. PAL is closely related to the existential graphs that Peirce developed in the late 1890s, and to the conceptual graphs (cf. e.g. [Sow92]).

Operations and Relations 251

Galois connection References closed rela- tional system

Table 2, continued frona previous page [Arm701 M. Armbrust. On set-theoretic characterization of congru-

ence lattices. Z. Math. Annalen, 16: 417-419, 1970.

closed opera- tional system

(13)

(14)

(15)

[Arms641 M. Armbrust and J . Schmidt. Zum Cayleyschen Darstel- lungssatz. Math. Annalen, 154: 70-73, 1964.

[ArnOl] M. Arnold. Einführung in die kontextuelle Relationenlogik. Diplomarbeit, Technische Universität Darmstadt, 2001.

(WKC) & (Y)

(WKC) =

(RC)&(l-LOC)

(RC) =

(s)&(Loc)

[BakP75] K.A. Baker and A.F. Pixley. Polynomial interpolation and the chinese remainder theorem for algebraic systems. Math. Z., 143: 165-174, 1975.

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[Q]WKC

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Trans. Amer. Math. Soc., 64: 299-316, 1948.

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Inv - inj-End

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Inv H for monoids H of injective funct ions

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locally closed monoid of injective unary functions

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*

Inv - End (cf. 2.5 and Tab. 1)

weak Krasner clone (cf. 2.4)

locally closed monoid of unary func- tions

Inv - Pol (cf. 2.5 and Tab. 1)

relational clone (cf. 2.4)

locally closed clone of fini- tary funct ions

252 R. Pöschel

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Author's address: Reinhard Pöschel Institut für Algebra TU Dresden D - 01062 Dresden GERMANY e-mail: poeschel@math.tu-dresden.de