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ON AN ORDER-BASED CONSTRUCTION OF A TOPOLOGICAL GROUPOID FROM AN INVERSE SEMIGROUP

Published online by Cambridge University Press:  28 July 2008

Daniel H. Lenz
Daniel H. Lenz
Affiliation:
Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany (dlenz@mathematik.tu-chemnitz.de)
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Abstract

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We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

Keywords

Primary 22A2220M18Secondary 18B3552C23groupoidsinverse semigroupsgraphstilings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 51 , Issue 2 , June 2008 , pp. 387 - 406
Copyright
Copyright © Edinburgh Mathematical Society 2008