De Vries powers and proximity Specker algebras

TL;DR

This paper establishes a direct, choice-free algebraic equivalence between de Vries algebras and proximity Baer-Specker algebras, simplifying the duality for compact Hausdorff spaces.

Contribution

It provides a choice-free algebraic proof of the equivalence between de Vries algebras and proximity Baer-Specker algebras, improving upon previous duality approaches.

Findings

Provides a choice-free algebraic proof of the duality.

Offers an alternative description of de Vries powers.

Simplifies the categorical equivalence for compact Hausdorff spaces.

Abstract

By de Vries duality [9], the category ${\sf KHaus}$ of compact Hausdorff spaces is dually equivalent to the category ${\sf DeV}$ of de Vries algebras. In [5] an alternate duality for ${\sf KHaus}$ was developed, where de Vries algebras were replaced by proximity Baer-Specker algebras. The functor associating with each compact Hausdorff space a proximity Baer-Specker algebra was described by generalizing the notion of a boolean power of a totally ordered domain to that of a de Vries power. It follows that ${\sf DeV}$ is equivalent to the category ${\sf PBSp}$ of proximity Baer-Specker algebras. The equivalence is obtained by passing through ${\sf KHaus}$, and hence is not choice-free. In this paper we give a direct algebraic proof of this equivalence, which is choice-free. To do so, we give an alternate choice-free description of de Vries powers of a totally ordered domain.