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

Abstract

A natural duality is obtained for each finitely generated variety B n ( n < ω) of distributive p-algebras. The duality for B n is based on a schizophrenic object: P −1 in B n is the algebra 2 n ⊕ 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 P 2 −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.