On locally compact semitopological graph inverse semigroups

TL;DR

This paper studies the structure of locally compact semitopological graph inverse semigroups, showing that for strongly connected finite graphs, these semigroups are either compact or discrete, extending previous results on polycyclic monoids.

Contribution

It generalizes the dichotomy of compactness or discreteness to a broader class of semigroups derived from finite, strongly connected graphs.

Findings

Semitopological graph inverse semigroups are either compact or discrete for finite, strongly connected graphs.

The result extends previous work on polycyclic monoids to graph inverse semigroups.

Provides a classification of the topological structure of these semigroups.

Abstract

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for locally compact semitopological polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_{\lambda}$, respectively.