A Class of Actions of Inverse Semigroups Original Research Article

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 OʹCarroll and McAlister are examples of such representations. We show that every ordered representation of an inverse semigroupSdetermines and is determined by a special kind of cover ofS. 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 image-bijective, prehomomorphism.