ON THE CANONICAL LATTICE EXTENSION OF A DISTRIBUTIVE NEARLATTICE WITH A RESIDUATION OPERATION

Abstract

AS shown by Macneille, any distributive nearlattice A has a canonical distributive lattice extension A” that contains A as a hereditary subalgebra. Here, we consider distributive nearlattices A equipped with a binary residuation operation—that, essentially, models the implication connective of Intuitionistic Propositional Logic without the exchange and contraction rules. These form both a category and a quasivariety of algebras. We show that the canonical distributive lattice extension of a such a ‘distributive residuation nearlattice’ AMaybe enriched with a residuation operation which extends that of A and which behaves well with respect to morphisms. We also show that the lattices of relative congruences of A and A° are isomorphic, so that second order properties such as subdirect irreducibility and simplicity are preserved by the extension.