2022/07/22 by Pedro V. Silva, Silva, Pedro V., Benjamin Steinberg +1
Computer Science · Mathematics · #20M05 #20M18 #22A22 #Advanced Algebra and Logic #Advanced Operator Algebra Research #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2207.11206
openalex publication_date 2022/07/22 · openalex created_date 2022/07/27 · openalex updated_date 2026/07/28
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.