Inverse semigroups associated with labelled spaces and their tight spectra

TL;DR

This paper links labelled spaces with inverse semigroups and characterizes their tight spectra, extending the understanding of boundary path spaces similar to directed graphs, with implications for C*-algebras.

Contribution

It introduces a method to associate inverse semigroups with labelled spaces and describes their tight spectra, inspired by Exel's work and boundary path space concepts.

Findings

Established a correspondence between labelled spaces and inverse semigroups.

Characterized the tight spectrum in a manner similar to boundary path spaces.

Provided insights relevant to the study of C*-algebras associated with labelled spaces.

Abstract

The notion of a labelled space was introduced by Bates and Pask in generalizing certain classes of C*-algebras. Motivated by Exel's work on inverse semigroups and combinatorial C*-algebras, we associate each weakly left resolving labelled space with an inverse semigroup, and characterize the tight spectrum of the latter in a way that is reminiscent of the description of the boundary path space of a directed graph.