  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.
