Topological Representation of Precontact Algebras

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

“…By [13], there is a 1-1 correspondence between quasi-modal operators on a Boolean algebra B and closed relations on the Stone space of B, and by [16], the same is true for precontact relations on B. 1 From this it follows that the same characterization also holds for subordinations.…”

“…With a different (more geometric) motivation in mind, Düntsch and Vakarelov [18] (see also Dimov and Vakarelov [16]) introduced the concept of a precontact relation, generalizing that of a contact relation.…”

“…In [16,17] it is shown that the 1-1 correspondence between precontact relations on B and closed relations on the Stone space of B extends naturally to include appropriate isomorphisms, and in [13] it is shown that the 1-1 correspondence between quasi-modal operators and closed relations extends to a full duality of appropriate categories. For our purposes, a more general notion of morphism is required.…”

“…In this section we will discuss how this results in a more general duality. In Remarks 2.14, 2.17, and 2.23 we compare our approach to that of [13] and [16,17].…”

“…Among many examples of subordinations, one of the key examples is that of de Vries' compingent relations. As follows from [13,16] (see also [8,17]), subordinations on a Boolean algebra B correspond to closed relations on the Stone space of B. We extend this correspondence to a full categorical duality, which subsumes the duality of [13] (see Remark 2.23).…”