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

Abstract

Abstract 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 ) {\operatorname{\mathcal{S}}(G)} on the semigroup I ⁢ ( X ) {I(X)} of partial bijections of X, with 𝒮 ⁡ ( 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 𝒮 ⁡ ( 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.