Reuter, Klaus

Wille, Rudolf

English

Reuter Klaus

Wille Rudolf

University of Szeged

Acta Sci. Math., 51 (1987), 319—327 Complete congruence relations of concept lattices KLAUS REUTER and RUDOLF WILLE To the memory of András Huhn 1. Introduction. Although complete lattices have been a main subject of lattice theory for a long time, complete congruence relations of complete lattices have only rarely been studied. In this paper we describe a general approach to complete congruence relations generalizing ideas introduced in [3]. In our approach we under-stand complete lattices as concept lattices. This enables us to establish a one-to-one correspondence between complete congruence relations and compatible saturated subcontexts of suitable contexts. The use of this correspondence is demonstrated by proving that every distributive complete lattice in which each element is the supremum of v-irreducible elements is isomorphic to the lattice of all complete congruence relations of some complete lattice. The question remains open which complete lattices are isomorphic to such lattices of complete congruence relations. Examples are given that they need not be distributive. 2. Compatible and saturated subcontexts. A subcontext of a context (G, M, I) is understood as a triple (H, N, IC\(HXN)) with H<gG and NQM\ we often write (H, N) instead of (H, N, If](HxN)). Throughout this section, (G, M, 1) will be a context and (H, N) a subcontext of (G, M, I). For g£G and m£M, g' and m stands for {g}' and {m}', respectively. By n(H, N)(A, B):=(AC\H, BC\N) for any concept (A, B) of (G, M, I), we define a map it(H, N) from ©(G, M, I) into where, in general, ^(S1) is the complete lattice of all subsets of a set S. The subcontext (H, N) of (G, M, I) is said to be compatible if the following conditions are satisfied: (la) For all h£H and m£M\h' there exists an n£N\h' with rí m'; (lb) for all n£N and g€G\n' there exists an h£H\n' with / i ' i g ' . The notion of a compatible subcontext is the same as in [3] which follows from Proposition 1. Received January 28, 1987, and in revised form April 17, 1987. 320 К. Reuter and R. Wille P r o p o s i t i o n 1. (H, N) is compatible if and only if n(H, N) is a complete lattice homomorphism from 5B(G, M, I ) onto 93(Я, N, IC\(HXN)). P r o o f . Let (H, N) be compatible. By the basic theorem in [2], it must only be shown that (АПН, BC\N) is a concept of ( i f , N, / П ( # Х Л 0 ) for (A, B)e<B(G, M, I). Let h£H\A. Then there is an m£B with (h, m)$I, i.e. m£M\h'. By (la), there exists an n£N\ti with n'^m'. Hence и£ЯП W and so h$(Br\N)'. It follows that Af\H= =(Bf]Nyr\H and dually that ЯПЛГ=(ЛП#) 'ГШ. Thus, (ADH, BC\N) is a •concept of [H, N, 1ПНХЮ). Conversely, let (АПН, BC\N)£iB(H, N, 1C\(HXN)) for all (A, B)£<B(G, M, I). Now, let h£H and m£M\h'. Since (m'DH, mT\N) is a concept of (H, N, IC\(HxN)) and there exists an n£m"(~)N with ih,n)$I, i.e. n£N\h' and n' m'. Hence ( # , N) satisfies (la). Dually we obtain i(lb). Thus, (H, N) is compatible. Let &(H,N) be the set of all pairs of concepts (A, B) and (C, D) of ( G , M , I ) such that n(H, N)(A, B)—n(H, N)(C, D), i.e. @(H,N) is the kernel of n(H, N). If (H, N) is compatible, Proposition 1 yields that 0 (H, N) is a complete con-gruence relation on » (G, M, I) and that » ( # , N, / П ( # Х Л 0 ) = ® (G, M, I)/0(H, N). Let us recall that a complete congruence relation of a complete lattice L is an equiv-alence relation 0 on L satisfying ( Д xА О ( Д у.-) and ( V x j ) ® ( V У,) if Xj @У,-j£J j£J JiJ JiJ for all jdJ. The question arises how to reconstruct the compatible subcontext (H , N) from the complete congruence relation 0 ( H , N). By the following definition, a complete congruence relation 0 of S(G, M, I) is naturally transformed into a subcontext of (G, M, I ) : G(0) := {g€G|yg := (g", g') is the smallest element of a ©-class}, M(0) := {m£M\pm := (m', m") is the greatest element of a ©-class}. To obtain (H, N) from &(H, N) via this definition, (H, N) has to be saturated, i.e. (H, N) must satisfy the following conditions: (2a) For g<EG and XQH, g'=X' implies (2b) for meM and YQN, m'=Y' implies m£N. P r o p o s i t i o n 2. Let (H, N) be compatible. Then (H, N) is saturated if and •only if H=G(Q(H, N)) and N=M(&(H, N)). P r o o f . First we assume that (# , N) is saturated. Let h£H and (A, B)€93(G, M, I). Then h"C\H=AC\H implies he A a n d so h"QA; hence heG(Q(H, N)). Now, let g £ G ( 0 ( # , N)). Since yg is the smallest element of a <=>(#, A^-class, it follows that yg=((Hr\g'T, (Hilg")') and therefore g'=(HOg")'. Hence g € # by (2a). Complete congruence relations of concept lattices 321 This proves that H=G(®(H, N)) and dually N=M(&(H, N)). Let us assume these equalities for the opposite direction of the proof. Now we use that for a com-plete congruence relation 0 of a complete lattice L the set of the smallest elements of the (9-classes is closed under suprema and the set of the greatest elements of the 0-classes is closed under infima. Let s'—^' for g£G and XQH=G(®(H, N)). Then, by the basic theorem in [2], yg= V yx and so g£G(Q(H, Nj)=H. In x £ X this way we obtain (2a) and dually (2b). Thus, (H, N) is saturated. The next proposition clarifies the nature of the complete congruence relation &(H, N). In the formulation we use the notation [z] 0 for the equivalence class of 0 represented by z. P ropos i t ion 3. Let & be a complete congruence relation of 8(G, M, I). Then (G(0), M(0)) is a compatible and saturated subcontext of (G, M, 1) sat-isfying 0 = 0(G(0) , M(0)) if and only if {[y/i] & \ li£G(&)} is a supremum-dense and {[[m]0 \n£M(&)} is infimum-dense in 5B(G, M, I)/0. Proof . Assume that (G(0), M(0)) is a compatible and saturated sub-context of ( G , M , I ) satisfying 0 = 0 ( G ( 0 ) , M(0)). By Proposition 1, {(/TT)G(0), h'f)M(0)) | A€G(0)} is supremum-dense in ®(G(0), M(0), /n (G(0)XM(0))) . Since 0 is the kernel of n(G(0), M(0)), it follows that {[yh]0 | h£G(0)} is supremum-dense in 2?(G, M, I)/0 and dually that {[firi]0 | n£M(0)} is infimum-dense in 23(G, M, I)/0. Let us assume these properties for the opposite direction of the proof. First we show that (G(0), M(0)) is compatible. Let h£G(0) and m£M\h'. Then [yh]0^[¡im]0. Since {[nn]0 \ n£M(0)} is infi-mum-dense in S(G, M, /), there exists an n£M(0)\h' with /xrzS îm, i.e. n ' 2 m ' . This proves (la) and dually (lb). From the fact that the smallest and greatest elements of the ©-¡classes are closed under suprema and infima, respectively, it follows that (G(0) ,M(0)) is saturated. Now, let (A, 2?)€®(G, M, / ) and let (A_,B_) and (A~,B~) be the smallest and greatest concept in the 0-class containing (A, B). Then yg^(A, B) for g£G(0) implies yg^(A_,B^). Therefore Af)G(0)= =A-DG(0) and dually BDM(0)=B~ f W ( 0 ) . Hence (A, B)0 (C, D) implies v4 f lG(0)=CnG(0) andf inM (0 )=I>nM (0 ) , i.e. (A, B) 0 (G(0) , M(0)) (C, D). Thus, we have 0 g 0 ( G ( 0 ) , M(0)). The equality follows from (A_,B_)= =y(Af)G(0)) and ( .4 - ,5 - )= / i ( .BfW(0) ) . For subcontexts (H1, Nj) and (H2, N2) of ( G , M , I ) we define (H1, N±) ^ S ^ i . ^ ^ g ^ and N ^ N 2 . The set of all compatible and saturated sub-contexts of (G, M, / ) together with this order relation is denoted by <S(G, M, /). For the complete lattice of all complete congruence relations of a complete lattice L we use the notation <f (L). From Propositions 1, 2, and 3 we obtain the following theorem: 322 К. Reuter and R. Wille Theorem 4. Let (G, M, I) be a context such that, for all complete congruence relations 0 o/®(G, M, I), {[уй]0 | h€G(0)} issupremum-dense and {[рп]0 | n£M(@)} is infimum-dense in SB(G, M, 1)10. Then ®<-*(G(0), M(0)) describes an anti-isomorphism from C(»(G, M, I)) onto <5(G, M, I). For the study of C(L) it is interesting to find suitable contexts (G, M, I) with Lsi5B(G, M, I) satisfying the assumption of Theorem 4. Obviously, (L, L, Ш) will do, but it would be better to find smaller contexts. The following lemma serves us with one method recognizing such contexts. Another method is given by Lemma 7. Lemma 5. Let {yg | g£G}U{(M', M)} be an order ideal and let {pm | mZM) U U{(G, G')} be an order filter of S(G, M, I). Then {{yh]0 \h<iG(0)} is supremum-dense and {[/ги]0 | n£M(0)} is infimum-dense in 5B(G, M, I)/0 for each complete congruence relation 0 of S (G, M, I). Proof . For a complete congruence relation 0 of 2?(G, M, I) let (A, B) e be the smallest concept in the 0-class containing the concept (A, B). It can be easily seen that (A, B)>->(A, B)e describes a V-preserving map from S (G, M, I ) into itself. From ( A , B ) = V yg we obtain (A, B)e= \J (yg)e and so [(A, B)]0 = в(.л дел = V [(yg)e]0- since (yg)e=yh for all g€G with ( y g ) e ^ ( M ' , M ) by assump-tion, the first assertion follows (and dually the second). 3. Closed subcontexts. After establishing the correspondence between complete congruence relations and compatible saturated subcontexts, the question arises how to construct compatible and saturated subcontexts. In general, this question seems difficult to answer. But there is a method which can be successfully applied in special cases. This method is based on the relations / and / of a context (G, M, / ) which have been introduced in [3] as follows (g£G, ш€М): g/m:-t>(g,m)$I and m' is maximal in «CA/\g'}, g/m : o ( g , m)$I and g' is maximal in {h'\h^G\m'}. It has been useful to fill in the arrows in the cross-table describing the given con-text. An example is shown in Figure 1. A subcontext (H, N) of (G, M, I) is called (arrow-) closed if h/m implies m£N for h£H and m£M and if g/n implies g£H for g£G and n£N. For example ({1,4}, {a, d}) is a closed subcontext of the context described in Figure 1. A context (G, M, 1) is called doubly founded if for all ( g , m ) £ G x M \ I there exists h£G and n£M with g/n, n' m' and h/m, h'^g' (cf. [4]). Lemma 6. A compatible subcontext of a context (G, M, I) for which gi=g'& implies gx=g2 for gi,g2£G and m[—m'2 implies m1—m2 for n\,m£M, is closed. Conversely, a closed subcontext of a doubly founded context is compatible. Complete congruence relations of concept lattices 323 a b c d e 1 X* X X X X 2 jr • X X* X IF 3 X X* X JR 4 X * * *r Figure I Lemma 6 is an immediate consequence of the definitions. Let us recall that a context (G, M, I) is said to be reduced if g'=X' implies g£X for g£G, XQG and if m'=Y' implies Y for m£M, YQM. Observe that each subcontext of a reduced context is saturated. A complete lattice L is called doubly founded if, for every pair x<y in L, there exists a minimal element s£L with s^y and s ^ x and a maximal element /€L with x=t and y^t. Such minimal and maximal elements are just the V-irreducible and A-irreducible elements of L, respectively, and every element of L is the supremum of V-irreducible elements and the infimum of A-irreducible elements of L. If J(L) denotes the set of all V-irreducible elements of L and M{L) the set of all A-irreducible elements of L, then L ^ ® (/(L), Af(L), s ) by the basic theorem in [2], and (/(L), M(L), s ) is a reduced context. For a doubly founded lattice L, (/(L), M(L), s ) is a doubly founded context; but the con-cept lattice of a doubly founded context need not be doubly founded (take (N, N, S)) . Lemma 7. Let L :=S(G, M , / ) be doubly founded and let 0 be a com-plete congruence relation of L. Then {[yh]0 \ h£G(0)} is supremum-dense and {\jiri]0 | n£M(0)} is infimum-dense in L/0. Proof . Suppose there is a concept (A,B) with [ V y ( ^ n G ( 0 ) ) ] 0 < [ ( ^ , B)]0. Let ( I , B) and (C, B) be the greatest element in [(A, B)]0 and [ V y ( ^ n G ( 0 ) ) ] 0 , respectively. Because of ( C , D ) < ( A , B ) , there exists a minimal concept (E, F) in L with (E, F)S(A, B) and (£, D). Since (E, F) is V-irreducible in L, there is a g£G with yg=(E, F). Moreover, (E, F) must be the smallest element of [(£, F)]0 and so geA(~}G(Q). This contradicts y g $ V y ( ^ n G ( 0 ) ) . Thus, the first assertion is proved and dually the second. Lemmas 6 and 7 together with Theorem 4 yield the following theorem: . T h e o r e m 8. For a doubly founded complete lattice L, C(L) is antiisomorphic to the complete lattice of all closed subcontexts of (/(L), M(L), S). Notice that the supremum and infimum of closed subcontexts (Hk, Nk) with kZK are just given by ( U Hk, U Nk) and ( f | Hk, f l Nk). *6K *£K kOC k€K 324 К. Reuter and R. Wille Coro l l a ry . For a doubly founded complete lattice L, (£(/.) is completely dis-tributive. 4. Lattices of complete congruence relations. It is a challenging problem to de-termine the class of all complete lattices which are isomorphic to some <£(L). Up to now, no complete lattice is known which does not belong to this class. As a positive result we prove that every distributive complete lattice with enough v-irreducible elements is isomorphic to some (E(L). T h e o r e m 9. Let D be a distributive complete lattice in which each element is a supremum of v-irreducible elements. Then there exists a complete lattice L with D^H(L). Proof . Let J(D) be the set of all v-irreducible elements of D (notice that 0$/(D)). The following construction of a context (G, M, / ) was stimulated by an (unpublished) idea of E. T. Schmidt: G: = J(D) X {1, 2, 3}, M := J(D) X {4,6}\JD X {5}, and / : = (J(D) X {1}) X(J(D) X {4}) U (/(£>) X {2}) X(Z) X {5}) U (J(D) X {3})x(/(/)) X {6}) U U{((SX , i), (S2, j))|Sl, S2£J(D), Sl * s2, (i, J ) € { ( 2 , 4), (3, 4), (1, 6), (2, 6)}}U U{((s, i), (x, 5)) |s£J(D), x€L>, s x, i£{l, 3}}. For a concept (A, B) of the context (/(£>),!), = ) we define g(A, B) := (¿X{1, 2, 3}, l x { 4 , e j U ^ X ^ } ) where A:=J(D)\A. We shall show that q is an antiisomorphism from »(/(£>), A ==) onto <2(G, M, I) which leads to DssG(®(G, M, /)) using Theo-rem 4 and the fact that D s=® (/(£>), Z), s ) . Figure 2 visualizes the foregoing definitions. • . . ; J(P)x{4} , Z?x{5} J(P)*{6} D <7 (D) x{ 1} - X • • . < 1 + JW) , A JW)*{ 2} 1 . < t • t . * B• i ' / ' - ' J(0) x(3} 1 - - - . J * 4= . 1 . < •>< J f. 1 1 Figure 2' • • .!-•'/ Complete congruence relations of concept lattices 325 Let (A, B) be a concept of (J(D), D, S ) . First we show that the subcontext (A,B) of (J(D), D, . satisfies the conditions (la) and (2b). Let he A and m€D\h'. Then yh^(A, B) and yh^/xm. Since yh is v-irreducible and since S( / (D) , D, s ) is distributive, it follows that yh^(A, B)vfim. Hence there exists an n£B\h' with n'3m ; this proves (la). Let m^D and YQB with m'=Y As AQ Y' we get mdB and so (2b). Now we shall verify the conditions (lb), (2a), (la), and (2b) for the subcontext g(A, B) of (G, M, I). Define (H, N):= g(A, B). For geG and m£M we write g1 and in1 instead of g' and m', respectively, to avoid confusion; the prime symbol is used in this proof with respect to the context (J(D), D, S ) . Because of G\n'QH for all n£N, (H, N) satisfies (lb). (2a) fol-lows from the fact that g[^g[ implies gx=g2 for all g!,g2(iG. For h£H we have M\hr<^NUDx{5}. Therefore (la) holds because (A, B) satisfies (la) in (7(7>), D, S ) as shown above. Since tn[ ZDm'2 for m l 5 m 2 £M and '«I^OD , 5) implies mi,m2e_Dx{5}, (H, N) satisfies (2b) because this condition holds for (A, B) in (7(D), D, Thus, (H, N) is a compatible and saturated subcontext of (G, M, I). Now we shall show that a compatible and saturated subcontext of (G, M, / ) equals q(A, B) for some (A, 5)£S(7(D), D, s ) . It can be easily seen that g/M fo r al l (g , m)eGxM\T a n d g / m f o r al l (g, m ) 6 G x ( / ( D ) X { 4 , 6 } ) \ I . By Lemma 6, a compatible subcontext of (G, M, / ) must be of the form (CX{1, 2, 3}, Cx{4, 6}U5X{5}) with CQJ(D) and BQD; in addition, s^x has to be hold for all sdC:=J(D)\C and x£B. It remains to show that (C, B) is a concept of (7(D), D, M) if (CX{1, 2, 3}, CX {4, 6}U5X {5}) is a compatible and saturated subcontext of (G, M, I). Suppose there is an s£C with s£B'. Because of s'^D, we can choose an x£D\s'. By (la), there exists a 0'» i)£Cx{4, 6}U5x{5}\(i", l) r with (y, iY^(x, 5)1. This implies y£B which contradicts s£B'. Thus, C=B' is shown. Let x£D with CQx'. For each g£G\(x, 5)' we have g £ C x { 1,2,3} and (x, 5 ) £ M \ g J . Hence, by (la), there exists an ag£(CX{4,6}U5x{5}) \g f with (xgY^(x, 5)'. It follows that ag€Bx{5} and (x, 5)' = (a(G\(x, 5)1))1. Now (2b) yields x£B and therefore B=C'. . Since (AI, B])^(A2, B2)oq(A1, B1)^G(A2, B2), it is shown that D is anti-isomorphic to <5(G, M , / ) . We apply Lemma 5 to" see. that (G, M, I) satisfies the assumption of Theorem 4. Obviously, {yg | g£G} U {(M', M)} is an order ideal of 5B(G, M, I). Let NM^(A,B) for M^M and (A, B)€&(G, M, 7)\{(G, G')}. Then 5=i?X{5} and so n(A§, 5)=(A, B). Hence {/¿m \ m£M} U{(G, G')} is an order filter of ®(G, M, J). Finally, Theorem 4 yields D ^ K ( » ( G , M, I)). The assumptions of Theorem 9 are fulfilled by distributive dually continuous lattices [1; p. 69] and, in particular, completely distributive complete lattices [1; p. 58]. Since the construction in the proof yields a finite context for a finite lattice D, the assumption of distributivity is unavoidable for this kind of construction. Nevertheless, 326 К. Reuter and R. Wille there are non-distributive lattices <E(L) for certain infinite complete lattices L where G(L) might even be finite. This we show by two examples. Example 1. Let Z be the set of all integers and let E and O be the set of all •even and odd integers, respectively. We define a context (G, M, I) as follows: G : = Z X {1,2,3}, M:= Zx{4,5 ,6} , / : = {((*> 0. (y,j))\x, yeZ, (i, № {(1, 4), (2, 5), (3, 6)}}. Now, we consider the following subcontexts: JVj):= (ZX{1}U EX{2}UOX {3}, Zx{4}UOX {5}U E x {6}), (H2, N2):= (OX{1}UZx{2}U Ex{3}, Ex{4}UZx{5}UOx{6}) , (.H3,N3):= (EX{1}U0X{2}UZX{3}, 0X{4}UEX{5}UZX{6}). It can be easily checked that (//,, Nt) is a compatible and saturated subcontext of (G,M,I) for i = l , 2, 3; furthermore, the subcontexts (0, 0), (H^ N,), (H2, Ns), (H3, N3), and (G,M) form a sublattice of <S(G,M,I) isomorphic to M3. This shows that K(»(G, M, /)) is not distributive. Example 2. Let Ln be the complete lattice described by Figure 3. Figure 3 The lattice Z has only two non-trivial complete congruence relations. It fol-lows that « ( L J s J ^ ) " © ! . For n s 2 , £(L„) is not distributive. Let us remark that £(£.„) is antiisomorphic to the face lattice of an «-cube. The diagram of (£(¿2) is shown in Figure 4. Figure 4 Complete congruence relations of concept lattices 327 References [1] G. GIERZ, K . H . HOFMANN, K . KEIMEL, J. D . LAWSON, M . MISLOVE, D . S. SCOTT, A compendium of continuous lattices. Springer (Berlin—Heidelberg—New York, 1980). [2] R. WILLE, Restructuring lattice theory: an approach based on hierarchies of concepts, in: Ordered sets (ed. I. Rival), Reidel (Dordrecht—Boston, 1982); pp. 445—470. [3] R. WILLE, Subdirect decomposition of concept lattices, Algebra Universalis, 17 (1983), 275— 287. [4] R. WILLE, Tensorial decomposition of concept lattices, Order, 2 (1985), 81—95. TECHNISCHE HOCHSCHULE DARMSTADT FACHBEREICH MATHEMATIK, ARBEITSGRUPPE ALLGEMEINE ALGEBRA SCHLOSSGARTENSTR. 7 6100 DARMSTADT, BRD 3 

Complete congruence relations of concept lattices

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Complete congruence relations of concept lattices

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