The study of algebras of partial functions is an active area of research that investigates collections of partial functions and their interrelationships from an algebraic perspective. The partial functions are treated as abstract elements that may be combined algebraically using various natural operations. Many different selections of operations have been considered, each leading to a different class/category of abstract algebras. In this talk, we will consider algebras of partial functions for a foundational signature consisting o  two operations, both binary: the standard set-theoretic relative complement operation and a domain restriction operation. We will exhibit an adjunction between the category whose objects are the atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction and whose morphisms are the complete homomorphisms, and a category of set quotients. 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. If time allows, we will then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely  additive and compatibility preserving operators.

This is based on joint work with Brett McLean.