Reconstructing a Totally Disconnected Groupoid from Its Ample Semigroup

Recall that an open subset S of a (not necessarily Hausdorff) étale groupoid G is said to be a slice (sometimes also a G-set , or a bissection) if the domain and range maps are injective on S. The collection of all slices forms an inverse semigroup which has often been studied alongside the groupoid itself. Among other things the natural action of this semigroup on the unit space G highlights the dynamical nature of groupoids. The set S formed by all compact (open) slices is called the ample semigroup of G [6: 2.10]. A compact slice S is somewhat special because its characteristic function 1S is in Cc(G) and hence also in C (G). We therefore get a map ρ : S → C(G), (1.1)