Finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids

TL;DR

This paper constructs countably many finitely presented inverse semigroups with finitely many idempotents per $\,\mathcal D$-class and non-Hausdorff universal groupoids, highlighting limitations of current $C^*$-algebraic techniques.

Contribution

It introduces new examples of inverse semigroups with non-Hausdorff universal groupoids that are finitely presented and have finitely many idempotents per $\,\mathcal D$-class, expanding understanding of their structure.

Findings

Constructed countably many non-isomorphic examples.

Showed these examples cannot be direct limits of semigroups with Hausdorff groupoids.

Highlighted current limitations in $C^*$-algebraic methods for these cases.

Abstract

The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupoids. We construct here countably many non-isomorphic finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids. At this time there is not a clear $C^*$-algebraic technique to prove these inverse semigroups have stably finite complex algebras.