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

TL;DR

The paper develops a framework to extract Morita-invariant, combinatorial data from inverse semigroups by constructing labelled graphs from coherent idempotent data. It proves that, for combinatorial inverse semigroups with finite intervals, the inverse semigroup of a labelled graph is Morita equivalent to the original, and extends this to inverse hulls of Markov shifts. A crucial contribution is the embedding of a labelled-graph inverse semigroup $S_{\mathcal{C}}$ into $S$ and the demonstration that $S$ is Morita equivalent to $S_{\mathcal{C}}$, with the data encoded in a coherent set $\mathcal{C}$ and the core-based CD$(S)$ construction. For Markov shifts, the authors show that the associated labelled graph is a complete invariant of Morita equivalence among inverse hulls, reducing the problem to finite semilattice conditions and core structure. Together, these results provide a pathway for geometric/classification-style invariants of large classes of inverse semigroup $C^*$-algebras via labelled graphs.

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.

Theorems & Definitions (75)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• proof
• Definition 3.1
• Remark 3.2
• Proposition 3.3