Difference-restriction algebras with operators

TL;DR

This paper establishes a duality and adjunction between difference-restriction algebras, a class of partial function algebras, and Hausdorff étale spaces, generalizing classical dualities and extending to algebras with operators.

Contribution

It introduces the concept of difference-restriction algebras, constructs an adjunction with Hausdorff étale spaces, and extends duality results to algebras with additional operators.

Findings

Established an adjunction between difference-restriction algebras and Hausdorff étale spaces.

Defined the finitary compatible completion and showed it forms a monad.

Extended duality to algebras with arbitrary compatibility-preserving 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 \'etale 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 \'etale 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.