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FINITELY PRESENTED INVERSE SEMIGROUPS WITH FINITELY MANY IDEMPOTENTS IN EACH
$\mathcal D$-CLASS AND NON-HAUSDORFF UNIVERSAL GROUPOIDS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 116 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 13 December 2023, pp. 384-398
- Print publication:
- June 2024
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- Article
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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.
