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

Abstract

AbstractWe 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.