Priestley Duality for Strong Proximity Lattices

In 1937 Marshall Stone extended his celebrated representation theorem for Boolean algebras to distributive lattices. In modern terminology, the representing topological spaces are zero-dimensional stably compact, but typically not Hausdorff. In 1970, Hilary Priestley realised that Stone's topology could be enriched to yield order-disconnected compact ordered spaces.In the present paper, we generalise Priestley duality to a representation theorem for strong proximity lattices. For these a “Stone-type” duality was given in 1995 in joint work between Philipp Sünderhauf and the second author, which established a close link between these algebraic structures and the class of all stably compact spaces. The feature which distinguishes the present work from this duality is that the proximity relation of strong proximity lattices is “preserved” in the dual, where it manifests itself as a form of “apartness.” This suggests a link with constructive mathematics which in this paper we can only hint at. Apartness seems particularly attractive in view of potential applications of the theory in areas of semantics where continuous phenomena play a role; there, it is the distinctness between different states which is observable, not equality.The idea of separating states is also taken up in our discussion of possible morphisms for which the representation theorem extends to an equivalence of categories.