Semigroup actions on posets and preimage quasi-orders

Structures consisting of a semigroup of (partial) functions on a set X, a?poset of subsets of X, and a preimage operation linking the two, arise commonly throughout mathematics. The poset may be equipped with one or more set operations, up to Boolean algebra structure. Such structures are finitely axiomatized here in terms of order-preserving semigroup actions on posets. This generalises Schein??s axiomatization of semigroups of partial functions equipped with the first projection quasi-order.