On an order based construction of a groupoid from an inverse semigroup

Abstract

We present a construction, which assigns two groupoids, $\Gugamma$ and $\Gmgamma$, to an inverse semigroup $\Gamma$. By definition, $\Gmgamma$ is a subgroupoid (even a reduction) of $\Gugamma$. The construction unifies known constructions for groupoids. More precisely, the groupoid $\Gugamma$ is shown to be isomorphic to the universal groupoid of $\Gamma$ introduced by Paterson. For $\Gamma$ arising from graphs resp. tilings, the groupoid $\Gmgamma$ is the graph groupoid introduced by Kumjian et al. resp. the tiling groupoid introduced by Kellendonk. We obtain a characterisation of open invariant sets in $\Gmgamma^{(0)}$ in terms of certain order ideals of $\Gammanull$ for a large class of $\Gamma$ (including those arising from graphs and from tilings). If $\Gmgamma$ is essentially principal this gives a characterization of the ideal structure of $\Cred(\Gmgamma)$ by a theory of Renault. In particular, we then obtain necessary and sufficient conditions on $\Gamma$ for simplicity of $\Cred(\Gmgamma)$. Our approach relies on a detailed analysis of the order structure of $\Gamma$.