Connecting generalized Priestley duality to Hofmann-Mislove-Stralka duality

TL;DR

This paper explores the connections between different duality theories in lattice and semilattice structures, unifying several dualities including Priestley, Hofmann-Mislove-Stralka, and Stone dualities.

Contribution

It establishes a unified framework linking Priestley duality, its generalizations, and Hofmann-Mislove-Stralka duality, including the role of morphisms between algebraic frames.

Findings

Unified duality framework for distributive lattices and semilattices

Connections between Stone duality and generalized dualities

Insights into morphisms between algebraic frames

Abstract

We connect Priestley duality for distributive lattices and its generalization to distributive meet-semilattices to Hofmann-Mislove-Stralka duality for semilattices. Among other things, this involves consideration of various morphisms between algebraic frames. We also show how Stone duality for boolean algebras and generalized boolean algebras fits as a particular case of the general picture we develop.