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-semimodular PCS's (i.e. PCS's with semimodular lattice of congruences) and the class of congruence-distributive PCS's (i.e. with distributive congruence lattices). We give two characterizations of each class; one of these is a Dedekind-Birkhoff-type characterization which says that the exclusion in a certain sense of a single PCS P6 determines the class of congruence-semimodular PCS's, and the exclusion of the two PCS's P6 and P5 (these are defined in the sequel) determines the class of congruence-distributive 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].