Modal Operators on Compact Regular Frames and de Vries Algebras

Abstract

Abstract. 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.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. Also described is an alternative approach to the duality of MKRFrm and MKHaus obtained by using modal de Vries algebras as an intermediary.

Choice-Free de Vries Duality

“…Let us conclude this section by noting that connections between de Vries duality and the Vietoris functor on compact Hausdorff spaces have already been studied in [3,4]. In particular, the authors define modal de Vries algebras and prove that they are the duals of coalgebras of the Vietoris functor.…”

“…We conclude with the main result of this paper: Theorem 5. 4 The functors Φ and Λ establish a dual equivalence between the categories deV and dVS.…”

Modal compact Hausdorff spaces

“…There are connections between MKR-frames and the construction of Vietoris frames of compact regular frames [5,28]. In fact, MKR-frames are algebras for the Vietoris functor on KRFrm.…”

“…We remark that the dualities involving MKHaus use some version of the axiom of choice. In [5] choice-free equivalences between MKRFrm and both LMDV and UMDV, and hence MDV, are given.…”