Difference–restriction algebras of partial functions with operators: Discrete duality and completion

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