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 풮 ⁡ (G) {\operatorname{\mathcal{S}}(G)} on the semigroup I ⁢ (X) {I(X)} of partial bijections of X, with 풮 ⁡ (G) {\operatorname{\mathcal{S}}(G)} being an inverse monoid introduced by Exel. We show that any unital premorphism θ : G → S {\theta:G\to S} , where S is an inverse monoid, can be extended to a semigroup homomorphism θ * : T → S {\theta^{*}:T\to S} for any inverse semigroup T with 풮 ⁡ (G) ⊆ T ⊆ P * ⁢ (G) × G {\operatorname{\mathcal{S}}(G)\subseteq T\subseteq P^{*}(G)\times G} , with P * ⁢ (G) {P^{*}(G)} being the semigroup of non-empty subsets of G, and such that E ⁢ (S) {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 Γ ⁢ (X) {\Gamma(X)} , the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X. [ABSTRACT FROM AUTHOR]