Difference-restriction algebras with operators

Abstract

We exhibit an adjunction between a category of abstract algebras of partial functions that we call difference-restriction algebras and a category of Hausdorff étale spaces. Difference-restriction algebras are those algebras isomorphic to a collection of partial functions closed under relative complement and domain restriction. Our adjunction generalises the adjunction between the category of generalised Boolean algebras and the category of Hausdorff spaces. We define the finitary compatible completion of a difference-restriction algebra and show that the monad induced by our adjunction yields the finitary compatible completion of any difference-restriction algebra. As a corollary, the adjunction restricts to a duality between the finitarily compatibly complete difference-restriction algebras and the locally compact zero-dimensional Hausdorff étale spaces, generalising the duality between generalised Boolean algebras and locally compact zero-dimensional Hausdorff spaces. We then extend these adjunction, duality, and completion results to difference-restriction algebras equipped with arbitrary additional compatibility preserving operators.