A topological correspondence between partial actions of groups and inverse semigroup actions

We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of S(G) on the semigroup I(X) of partial bijections of X, with S(G) being an inverse monoid introduced by Exel. We show that any unital premorphism theta : G -> S, where S is an inverse monoid, can be extended to a semigroup homomorphism theta* : T -> S for any inverse semigroup T with S(G) subset of T subset of P* (G) x G, with P* (G) being the semigroup of non-empty subsets of G, and such that E(S) satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on Gamma(X), the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X.