Labelled graphs as Morita equivalence invariants for a class of inverse semigroups

TL;DR

This paper demonstrates that labelled graphs can serve as invariants for Morita equivalence in a class of inverse semigroups, linking combinatorial structures to algebraic equivalence classes.

Contribution

It introduces a method to construct labelled graphs from inverse semigroups and shows these graphs determine Morita equivalence classes, especially for inverse hulls of Markov shifts.

Findings

Labelled graphs are Morita invariants for certain inverse semigroups.

The construction applies to inverse semigroups with specific idempotent classes.

For inverse hulls of Markov shifts, the graph determines the Morita class.

Abstract

We investigate the use of labelled graphs as a Morita equivalence invariant for inverse semigroups. We construct a labelled graph from a combinatorial inverse semigroup $S$ with $0$ admitting a special set of idempotent $\mathcal{D}$-class representatives and show that $S$ is Morita equivalent to a labelled graph inverse semigroup. For the inverse hull $S$ of a Markov shift, we show that the labelled graph determines the Morita equivalence class of $S$ among all other inverse hulls of Markov shifts.