Congruence-semimodular and congruence-distributive pseudocomplemented semilattices

Abstract

Investigations into the structure of the congruence lattices of pseudocomplemented semilattices (PCS's) were initiated in [10]. In this paper a we characterize the class of congruence-semim odular PCS's (i.e. PCS's with semimodular lattice of congruences) and the class of congruence-distr ibutive PCS's (i.e. with distributive congruence lattices). We give two characterizations of each class; one of these is a Dedekind-Birkhof f-type characterization which says that the exclusion in a certain sense of a single PCS P6 determines the class of congruence-semim odular PCS's, and the exclusion of the two PCS's P6 and P5 (these are defined in the sequel) determines the class of congruence-distr ibutive PCS's. The other characterization shows that each of these classes is strictly elementary and gives explicitly the defining axiom for each class as a universal positive sentence (in the language of PCS's). This paper is a continuation of [10] and borrows the notation and the results from it. For other information see the standard references [6] and [7]. w Basic definitions and lemmas Recall that a pseudocomplement ed semilattice (PCS) is an algebra (S; A, *, 0) where (S; A, 0) is a A-semilattice with zero and * is the pseudocomplement ation and that the class of all PCS's is an equational class. Let S denote an arbitrary PCS and B(S) and N(S) denote respectively the set of closed (i,e. a**=a) elements and that of non-closed (i.e. a < a**) elements of S. It is well-known [5] that B(S) is both a Boolean algebra and a PCS-subalgebra of S. Con S denotes the congruence lattice of S with As and V~ (or simply A and V) as its least and greatest elements respectively; and the kernel of the homomorphism **:S ~ S,