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INVARIANT SETS AND NORMAL SUBGROUPOIDS OF UNIVERSAL ÉTALE GROUPOIDS INDUCED BY CONGRUENCES OF INVERSE SEMIGROUPS

Part of: Semigroups Selfadjoint operator algebras Topological and differentiable algebraic systems

Published online by Cambridge University Press:  13 April 2021

FUYUTA KOMURA Open the ORCID record for FUYUTA KOMURA [Opens in a new window]
FUYUTA KOMURA*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
*
e-mail: fuyuta.k@keio.jp
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Abstract

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For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

Keywords

C*-algebrasétale groupoids

MSC classification

Primary: 20M18: Inverse semigroups
Secondary: 22A22: Topological groupoids (including differentiable and Lie groupoids) 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 1 , August 2022 , pp. 99 - 118
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021

Footnotes

Communicated by Aidan Sims

This work was supported by JSPS KAKENHI 20J10088.

References

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