DETERMINED CONGRUENCE CLASSES IN BROUWERIAN SEMILATTICES

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1 East-WestJ.ofMathematics:Vol. 2,No1(2000) pp DETERMINED CONGRUENCE CLASSES IN BROUWERIAN SEMILATTICES Ivan Chajda and Josef Zedník Dept. of Algebra and Geometry Palacký University Olomouc, Tomkova 40, Olomouc, Czech Republic Dept. of Mathematics, FT VUT Zlín, T.G.M. 275, Zlín, Czech Republic Abstract Let Θ be a congruence of an algebra A =(A, F )anda, b A. We say that the class [a] Θ determines [b] Θ if for each Φ Con A, [b] Φ =[b] Θ whenever [a] Φ =[a] Θ. We characterize those classes [a] Θ of a Brouwerian semilattice which determine [b] Θ for b a. An algebra A =(A, F ) is called regular if for any two congruences Θ, Φ Con A it holds: if [a] Θ =[a] Φ for some a A then Θ = Φ. Varieties of regular algebras were characterized by B. Csákány, G. Gr atzer and R. Wille independently by various Mal cev conditions in For several comments and some generalizations of the concept of regularity, see e.g. [2]. The aforementioned regularity condition can be expressed in the following form: if Θ Con A and [a] Θ =[a] Φ for some Φ Con A and a A, then [b] Θ =[b] Φ for each b A. Key words and phrases: Brouwerian semilattice, regularity of congruences, congruence class, relative pseudocomplement (1991) AMS Mathematics Subject Classification: 08A30, 06D20, 06A12 49

2 50 Determined congruence classes in Brouwerian semilattices Hence, it properly says that every class of Θ Con A determines each other class of Θ. It can happen that an algebra A is not regular but some of Θ Con A has still a class [b] Θ determined by [a] Θ for some a, b of A. For example, the non-modular lattice N 5 is not regular but for non-trivial congruences Θ 1, Θ 2, Θ 3 Con N 5, (see Fig.1), the class [a] Θi determines [c] Θi for i =1, 2, 3 but e.g. [c] Θ3 does not determine [a] Θ3 since, for the identical congruence ω Con N 5,also[c] Θ3 = {c} =[c] ω but [a] Θ3 = {a, b} {a} =[a] ω. This motivates us to define precisely: Definition Let A =(A, F ) be an algebra, let Θ Con A and a, b A. We say that the class [a] Θ determines [b] Θ if for every Φ Con A, if[a] Θ =[a] Φ then [b] Θ =[b] Φ. Of course, if A is regular then every class of each Θ Con A determines every other class of Θ. Some other interesting results were published earlier: Proposition 1 (Corollary 2.1 in [3]). Let L =(L;,, 0, 1) be a complemented lattice. Then for each Θ Con L and every a L; the class [a] Θ determines [0] Θ. An analogous result for ortholattices L =(L;,,, 0, 1) is published in [1]. Proposition 2 (Corollary 3.2 in [6]). Let P be a p algebra or a distributive p semilattice. Then for each Θ Con P and every a of P, the class [a] Θ determines [0] Θ. Recall that by a p-algebra is meant a bounded lattice L =(L;,,, 0, 1) with a unary operation (the so-called pseudocomplementation) such that a b = 0iffb a for each a, b L.

3 I. Chajda and J. Zedník 51 For semilattices with pseudocomplementation, the so-called pseudocomplemented semilattices, the assertion of Proposition 2 does not hold in general, see e.g. the Example in [3]. However, we are successful in a particular case: Proposition 3 (Theorem 3.1 in [3]). Let P= (P ;,, 0) be a pseudocomplemented semilattice and Θ Con P, a A. If a [a] Θ then [a] Θ determines [0] Θ. The aim of this note is to prove a similar result for the so called Brouwerian semilattices. These algebras were introduced by W.C. Nemitz [5] (by the name implication semilattices). We can adopt the definition from [4]: Definition Let (S; ) be a semilattice, let x, y S. An element x y is called a relative pseudocomplement of x with respect to y if z x y holds iff x z y. If for every x, y S there exists x y, the binary operation is called a relative pseudocoplementation and the algebra S =(S;, ) is called a Brouwerian semilattice. Of course, every Brouwerian semilattice S has the greatest element 1 (since x x =1foreachx S). One can easily check that for x, y S, therelative pseudocomplement x y is the greatest element of S satisfying x x y = x y. (P) The following statement was proved in [4]: Lemma 1 Let S =(S;, ) be a Brouwerian semilattice. For x, y, z S we have: (i) x y implies x y =1; (ii) 1 x = x; (iii) y x y; (iv) x (x y) y; (v) x y implies y z x z. For our purposes, we prove the following Lemma 2 Let S= (S;, ) be a Brouwerian semilattice and a, c, x, z S. (a) If a x then (c a x a) a x = x; (b) If Θ Con S and (z a) a [z] Θ then (c a) a [z] Θ for each c [z] Θ.

4 52 Determined congruence classes in Brouwerian semilattices Proof. (a) By Lemma 1(iv), we have (x a) a x for each x S. Since c a x a x a, (v)oflemma1gives(c a x a) a (x a) a x, whence (a) is evident. (b) Suppose c [z] Θ.Then c, z Θ, i.e. also (c a) a, (z a) a Θ. Since (z a) a [z] Θ, it implies (c a) a [z] Θ. Now,weareabletoproveourresult: Theorem Let S= (S;, ) be a Brouwerian semilattice, let a, b S, a b and Θ Con S. Suppose that there exists a least element a 0 of [a] Θ.Then [a] Θ = {x S; c x = a 0 and (c a 0 x a 0 ) a 0 [b] Θ for some c [b] Θ } Proof. Denote by C =[b] Θ and if and only if (z a 0 ) a 0 [b] Θ for some z [b] Θ. M = {x S; c x = a 0 and (c a 0 x a 0 ) a 0 C for some c C}. (1) Let x M. Then there exists an element c C such that c x = a 0 and (c a 0 x a 0 ) a 0 C. By Lemma 2(a) we have a 0,x = c x, (c a 0 x a 0 ) a 0 x Θ, thus x [a 0 ] Θ =[a] Θ. Hence M [a] Θ. (2) Suppose x [a] Θ. Then x, a 0 Θ, whence 1,x a 0 = a 0 a 0,x a 0 Θ (by (i) of Lemma 1). Let c 0 C. Then c 0,c 0 x a 0 = c 0 1,c 0 x a 0 Θ, i.e. c = c 0 x a 0 C and c a 0,c a 0 x a 0 = c a 0 1,c a 0 x a 0 Θ, thus also (c a 0 ) a 0, (c a 0 x a 0 ) a 0 Θ. Since c C, bythe assumption and Lemma 2(b) we conclude (c a 0 ) a 0 C, hence also (c a 0 x a 0 ) a 0 C. (V) By the assumption, we have a 0 a b, i.e. a 0 b = a 0.Sincec 0 [b] Θ, we have c 0,b Θthusalso c 0 a 0,a 0 = c 0 a 0,b a 0 Θ, i.e c 0 a 0 [a 0 ] Θ = [a] Θ. However, a 0 is a least element of [a] Θ thus a 0 a 0 c 0. The converse inequality is trivial, so a 0 = a 0 c 0. Hence we obtain c x =(c 0 x a 0 ) x = c 0 (x a 0 x) =c 0 x a 0 = c 0 a 0 = a 0

5 I. Chajda and J. Zedník 53 by (P), since x [a] Θ implies a 0 x. Together with (V), we obtain x M proving [a] Θ M. We have shown [a] Θ = M. Conversely, let [a] Θ = M. Then a 0 [a] Θ and hence there exists c C such that (c a 0 a 0 a 0 ) a 0 C. Applying (i) of Lemma 1, we have (c a 0 ) a 0 =(c a 0 1) a 0 =(c a 0 a 0 a 0 ) a 0 C. ByLemma2(b),wehave(z a 0 ) a 0 [b] Θ = C for each z C. Corollary Let S= (S;, ) be a Brouwerian semilattice, a, b S, a b and Θ Con S. Let there exists a least element a 0 of [a] Θ. Then [b] Θ determines [a] Θ if and only if (b a 0 ) a 0 [b] Θ. Proof. If the assumptions are satisfied and Φ Con S with [b] Θ =[b] Φ, then by the Theorem, also [a] Θ =[a] Φ by the prescription of the classes [a] Θ and [a] Φ by means of [b] Θ and [b] Φ. The last assertion follows from the Theorem and Lemma 2(b). Example Let S= (S,, ) be a Brouwerian semilattice visualized in Fig. 2 and Θ Con S be given by the drawn partition. 1 b a d a 0 0 Fig. 2 Then a 0 b and a 0 is a least element of [a 0 ] Θ =[a] Θ.Moreover, (b a 0 ) a 0 = a a 0 = b [b] Θ thus [b] Θ determines [a] Θ. On the contrary, [d] Θ does not determine [a] Θ since

6 54 Determined congruence classes in Brouwerian semilattices [d] Θ =[d] ω for the identical congruence ω Sbut [a] Θ = {a 0,a} {a} =[a] ω. Remark Let us note that if S =(S;, ) is a finite Brouwerian semilattice then for each Θ Con S and every a S, theclass[a] Θ has a least element a 0,namelya 0 = {x; x [a] Θ }. The same holds for an arbitrary S under the condition that the class [a] Θ is finite. Hence, for finite [a] Θ, the assumption of the existence of a least element of [a] Θ can be dropped out of the Theorem and its Corollary. References [1] I. Chajda, A note on congruence kernels in ortholattices, Math. Bohem., to appear. [2] I. Chajda, Locally regular varieties, Acta Sci. Math., (Szeged), 64 (1998), [3] I. Chajda and G. Eigenthaler, A remark on congruence kernels in complemented lattices and pseudocomplemented semilattices, Contributions to General Algebra No. 11, Proc. of the Olomouc conf. and the Summer School 1998, Verlag J. Heyn, Klagenfurt (1999), [4] P. Köhler, Brouwerian semilattices: the lattice of total subalgebras, Universal Algebra and Applications 9, Banach Center Publ. (1982), [5] W. C. Nemitz, Implicative semilattices, Trans. Amer. Math. Soc. 117(1965), [6] J. C. Varlet, Regularity in p-algebras and in p-semilattices, Universal Algebra and Applications 9, Banach Center Publ. (1982),