Choice-Free de Vries Duality

TL;DR

This paper develops a choice-free duality theory connecting de Vries algebras with a new class of spaces, expanding algebraic and topological tools for region-based spatial reasoning.

Contribution

It establishes a choice-free duality between de Vries algebras and de Vries spaces, extending recent duality results and linking to point-free topology.

Findings

Established a choice-free duality for de Vries algebras and spaces

Connected the duality with the Vietoris functor and regular frames

Provided a new topological semantics for the Symmetric Strict Implication Calculus

Abstract

De Vries Duality generalizes Stone duality between Boolean algebras and Stone spaces to a duality between de Vries algebras (complete Boolean algebras equipped with a subordination relation satisfying some axioms) and compact Hausdorff spaces. This duality allows for an algebraic approach to region-based theories of space that differs from point-free topology. Building on the recent choice-free version of Stone duality developed by Bezhanishvili and Holliday, this paper establishes a choice-free duality between de Vries algebras and a category of de Vries spaces. We also investigate connections with the Vietoris functor on the category of compact Hausdorff spaces and with the category of compact regular frames in point-free topology, and we provide an alternative, choice-free topological semantics for the Symmetric Strict Implication Calculus of Bezhanishvili et al.