Equational Classes of Distributive Pseudo-Complemented Lattices

Abstract

A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.