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- Title
FINITELY PRESENTED INVERSE SEMIGROUPS WITH FINITELY MANY IDEMPOTENTS IN EACH $\mathcal D$ -CLASS AND NON-HAUSDORFF UNIVERSAL GROUPOIDS.
- Authors
SILVA, PEDRO V.; STEINBERG, BENJAMIN
- 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 nonisomorphic 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.
- Subjects
IDEMPOTENTS; GROUPOIDS; SEMIGROUP algebras; ALGEBRA
- Publication
Journal of the Australian Mathematical Society, 2024, Vol 116, Issue 3, p384
- ISSN
1446-7887
- Publication type
Article
- DOI
10.1017/S1446788723000198