A Class of Actions of Inverse Semigroups

Abstract

We generalise the classical Munn representation of an inverse semigroup with the introduction of what we call ordered representations of inverse semigroups. Both the Wagner–Preston Representation and the effective actions of O'Carroll and McAlister are examples of such representations. We show that every ordered representation of an inverse semigroup Sdetermines and is determined by a special kind of cover of S. As applications, we provide a fully categorical account of the theory of idempotent pure congruences, and we show that every inverse semigroup which is a semilattice with respect to the natural partial ordering is an image of a combinatorial inverse semigroup under an L -bijective, prehomomorphism.