De Vries powers and proximity Specker algebras
Abstract
By de Vries duality [9], the category KHaus of compact Hausdorff spaces is dually equivalent to the category DeV of de Vries algebras. In [5] an alternate duality for 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 DeV is equivalent to the category PBSp of proximity Baer-Specker algebras. The equivalence is obtained by passing through 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.