Modal compact Hausdorff spaces

Abstract

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved. These extend the familiar Isbell and de Vries dualities for compact Hausdorff spaces, as well as the duality between modal spaces and modal algebras. As the first step in the logical treatment of modal compact Hausdorff spaces, a version of Sahlqvist correspondence is given for the positive modal language.

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

“…If X is a Stone space, then an immediate application of Esakia's lemma ( [6,19]) yields that we can restrict the condition to proper clopen subsets of X.…”

“…Definition 6. 6 We call a pair (X, R) a Gleason space if X is an ED-space and R is an irreducible equivalence relation on X.…”

Modal Operators on Compact Regular Frames and de Vries Algebras

“…Also in [7] 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 [7], 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 [19, 20], discuss a Vietoris construction for de Vries algebras, and how it is linked to modal de Vries algebras.…”

“…In [7] 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 [7] 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 [7], MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another.…”

“…Also in [7] 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 [7], MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another.…”