Dualities for Boolean Contact Algebras

The well-known de Vries duality, established by H. de Vries in 1962, states that the category of compact Hausdorff spaces is dually equivalent to that of de Vries algebras. The notion of Boolean contact algebra (BCA) was developed independently in the context of region-based theory of space. Düntsch and Winter established a representation theorem for BCAs, showing that every BCA is isomorphic to a dense subalgebra of the regular closed sets of a T_1 weakly regular space. It appears that BCAs are a direct generalization of de Vries algebras, and that the representation theorem for complete BCAs generalizes de Vries duality for objects. During a conference, Vakarelov raised the question of dualizing morphisms. We answer this question using concepts similar to those of modal logic's neighborhood semantics.