De Vries Algebras and Compact Regular Frames

Abstract

By the de Vries theorem, the category DeV of de Vries algebras is dually equivalent to the category KHaus of compact Hausdorff spaces. By the Isbell theorem, the category KRFrm of compact regular frames is dually equivalent to KHaus. The proofs of both theorems employ the axiom of choice. It is a consequence of the de Vries and Isbell theorems that DeV is equivalent to KRFrm. We give a direct proof of this result, which is choice-free. In the absence of the axiom of countable dependent choice (CDC), the category KCRFrm of compact completely regular frames is a proper subcategory of KRFrm. We introduce the category cDeV of completely regular de Vries algebras, which in the absence of (CDC) is a proper subcategory of DeV, and show that cDeV is equivalent to KCRFrm. Finally, we show how the restriction of the equivalence of DeV and KRFrm works in the zero-dimensional and extremally disconnected cases.