Distributive lattice orderings and Priestley duality

TL;DR

This paper explores the relationship between distributive lattice orderings and Priestley duality, characterizing certain lattices through a functorial construction and duality interactions.

Contribution

It introduces a functor  from bounded distributive lattices to itself and characterizes lattices related via this functor to Priestley duality.

Findings

Characterization of lattices where (K) is isomorphic to L

Analysis of 's interaction with Priestley duality

Conditions for  to produce specific lattice structures

Abstract

The ordering relation of a bounded distributive lattice L is a (distributive) (0, 1)-sublattice of L \times L. This construction gives rise to a functor \Phi from the category of bounded distributive lattices to itself. We examine the interaction of \Phi with Priestley duality and characterise those bounded distributive lattices L such that there is a bounded distributive lattice K such that \Phi(K) is (isomorphic to) L.