Partition-induced natural dualities for varieties of pseudo- complemented distributive lattices

A natural duality is obtained for each finitely generated variety Bn (n < ω) of distributive p-algebras. The duality for Bn is based on a schizophrenic object: P-1 in Bn is the algebra 2n ⊕ 1 which generates the variety and P-1 is a topological relational structure carrying the discrete topology and a set of algebraic relations. The relations are (i) the graphs of a (3-element) generating set for the endomorphism monoid of P-1 and (ii) a set of subalgebras of P2-2 in one-to-one correspondence with partitions of the integer n. Each of the latter class of relations, regarded as a digraph, is 'nearly' the union of two isomorphic trees. The duality is obtained by the piggyback method of Davey and Werner (which has previously yielded a duality in case n ≤ 2), combined with use of the restriction to finite p-algebras of the duality for bounded distributive lattices, which enables the relations suggested by the general theory to be concretely described. © 1993