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

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