Modal Operators on Compact Regular Frames and de Vries Algebras

Abstract

In Bezhanishvili et al. (2012) we introduced the category MKHaus of modal compact Hausdorff spaces, and showed these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in Bezhanishvili et al. (2012) we introduced the categories MKRFrm and MDV of modal compact regular frames, and modal de Vries algebras as algebraic counterparts to modal compact Hausdorff spaces, much as modal algebras are algebraic counterparts to modal spaces. In Bezhanishvili et al. (2012), MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another. Here we provide a direct, choice-free proof of the equivalence of MKRFrm and MDV. We also detail connections between modal compact regular frames and the Vietoris construction for frames (Johnstone 1982, 1985), discuss a Vietoris construction for de Vries algebras, and how it is linked to modal de Vries algebras. Also described is an alternative approach to the duality of MKRFrm and MKHaus obtained by using modal de Vries algebras as an intermediary.