Resumo:
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].