De Vries Powers and Proximity Specker Algebras

Abstract

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