Restricted Priestley dualities and discriminator varieties

TL;DR

This paper introduces an abstract definition of restricted Priestley dualities for algebraic varieties with distributive lattice reducts, characterizes finitely generated discriminator subvarieties, and applies these results to Cornish and Ockham algebras.

Contribution

It provides the first abstract definition of restricted Priestley dualities and characterizes discriminator subvarieties using duality theory, with applications to Cornish and Ockham algebras.

Findings

A new abstract definition of restricted Priestley duality is established.

Characterization of finitely generated discriminator subvarieties is achieved.

Conditions for finite Cornish algebras to share a discriminator term are provided.

Abstract

Anyone who has ever worked with a variety~$\boldsymbol{\mathscr{A}}$ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one---but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category $\boldsymbol{\mathscr{X}}$ of such a variety, we give a characterisation, in terms of $\boldsymbol{\mathscr{X}}$, of finitely generated discriminator subvarieties of~$\boldsymbol{\mathscr{A}}$. As a first application of our characterisation, we give a new proof of Sankappanavar's characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly's characterisation of quasi-primal Ockham algebras.