Remarks on Hyperspaces for Priestley Spaces

TL;DR

This paper unifies various hyperspace constructions for Priestley spaces, crucial for coalgebraic semantics in positive modal logic, using category theory, topology, and duality techniques.

Contribution

It offers a unified framework for hyperspaces of Priestley spaces, simplifying and generalizing previous approaches in the context of positive modal logic.

Findings

Provides a unifying approach to hyperspaces of Priestley spaces.

Connects category theory, topology, and duality in this context.

Facilitates applications in coalgebraic semantics for positive modal logic.

Abstract

The Vietoris space of a Stone space plays an important role in the coalgebraic approach to modal logic. When generalizing this to positive modal logic, there is a variety of relevant hyperspace constructions based on various topologies on a Priestley space and mechanisms to topologize the hyperspace of closed sets. A number of authors considered hyperspaces of Priestley spaces and their application to the coalgebraic approach to positive modal logic. A mixture of techniques from category theory, pointfree topology, and Priestley duality have been employed. Our aim is to provide a unifying approach to this area of research relying only on a basic familiarity with Priestley duality and related free constructions of distributive lattices.