Difference-restriction algebras of partial functions with operators: discrete duality and completion

TL;DR

This paper develops a duality and completion theory for algebras of partial functions with operators, generalizing classical dualities for Boolean algebras and sets, and extends these results to algebras with additional operators.

Contribution

It introduces an adjunction and duality framework for atomic algebras of partial functions with operators, generalizing Boolean algebra-set duality and defining compatible completions.

Findings

Established an adjunction between algebras of partial functions and set quotients.

Defined compatible completion and showed it forms a monad for atomic representable algebras.

Extended duality and completion results to algebras with additional operators.

Abstract

We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators.