Algebra Universalis, 14 (1982) 235-243 0002-5240/82/002235-09501.50 + 0.20/0 (~) 1982 Birkhiiuser Verlag, Basel Equational classes of distributive double p-algebras ALASDAIR UROUHART An algebra (A, A, V, --1, --, 0, 1) is a double p-algebra if (A, A, V, 0, 1) is a bounded lattice, -1 is a pseudocomplementation operator and - a dual pseudocomplementation operator. In this paper, we give two results on equational classes of distributive double p-algebras. First, we show that the number of equational classes of such algebras is uncountable; secondly, we exhibit a finitely based equational class which is not determined by its finite members. These results are in contrast with those of K. B. Lee [8], who showed that the lattice of equational classes of distributive p- algebras (distributive pseudocomplemented lattices) is a chain of type ~o+ 1; furthermore, all such equational classes are determined by their finite members. An essential tool in proving these results is Priestley's duality as it applies to distributive double p-algebras; we summarize these results in the first section. For general lattice theory, we refer to Griitzer [3]. For graph theory, we refer to Harary [4]. Earlier results on distributive double p-algebras are in [1], [2], [5], [6], [71, [12], [13], [14]. 1. Priestley's duality In this section we describe briefly Priestley's duality (see Priestley [9], [10], [lI]) as it applies to distributive double p-algebras. For a fuller discussion, the reader is referred to Davey [2]. Let A be a distributive lattice, P(A) the family of prime filters on A. For a an element of A define: e(a) = {x e P(A): a ~ x}. Topologize P(A) by taking as a sub-base for a topology T the family {P(a):a A}UIP(A)-P(a):aeA}. Ordering P(A) by inclusion, we find S(A)= (P(A), T, _c), the dual space of A, forms a compact space which is totally Presented by G. Gr~itzer.Received January 17, 1979. Accepted for publication in final form March 11, 198l. 235