Priestley-type dualities for partially ordered structures

We introduce a general framework for generating dualities between categories of partial orders and categories of ordered Stone spaces; we recover in particular the classical Priestley duality for distributive lattices and establish several other dualities for different kinds of partially ordered structures.

💡 Research Summary

The paper “Priestley‑type dualities for partially ordered structures” develops a unified categorical framework that generates dualities between a wide class of partially ordered algebraic structures and a corresponding class of ordered Stone spaces. The authors begin by observing that the classical Priestley duality, which establishes an equivalence between the category of bounded distributive lattices and the category of Priestley spaces (compact, totally order‑disconnected, partially ordered topological spaces), is limited to distributive lattices. Modern applications in domain theory, logic, and computer science, however, involve many other ordered structures such as complete lattices, focal orders, frames, and co‑frames, for which no systematic Priestley‑type duality is known.

To overcome this limitation, the authors introduce the notion of representability for a partially ordered structure (A). Representability requires the existence of an ordered Stone space (X_A) together with a bijective map \