Choice-Free de Vries Duality

Abstract

De Vries Duality generalizes Stone's 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.

Topological duality for orthomodular lattices

“…Therefore, as an application of our duality for orthomodular lattices, we develop a topological semantics for quantum logic using orthomodular spaces. For other papers which develop a topological semantics for logical calculi by directly exploiting a duality theorem for the Lindenbaum algebra of the logic, we refer the reader to for instance Bezhanishvili, Grilletti, and Holiday [3] as well as Massas [23].…”