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

TL;DR

This paper explores a topological generalization of the correspondence between partial group actions and inverse semigroup actions, extending known algebraic results to topological settings and introducing new inverse semigroup topologies.

Contribution

It generalizes Exel's correspondence to unital premorphisms and inverse semigroups, including a topological version with a minimal Hausdorff inverse semigroup topology.

Findings

Extended the correspondence to unital premorphisms and inverse semigroups.

Established a topological version with a minimal Hausdorff inverse semigroup topology.

Provided conditions under which the extension of premorphisms is possible.

Abstract

We present some generalizations of the well-known correspondence, found by R. 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 $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $\theta:G\to S$, where $S$ is an inverse monoid, can be extended to a semigroup homomorphism $\theta^*:T\to S$ for any inverse semigroup $T$ with $S(G)\subseteq T\subseteq P^*(G)\times G,$ being $P^*(G)$ the semigroup of non-empty subset 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 homeomorphism between open subsets of a locally compact Hausdorff space $X$.