Reconstructing étale groupoids from semigroups

Abstract We unify various étale groupoid reconstruction theorems such as the following: • Kumjian and Renault’s from a C*-algebra; Exel’s an ample inverse semigroup; Steinberg’s ring; Choi, Gardella Thiel’s Lp {L^{p}} -algebra. do this by working with certain bumpy semigroups S of functions defined on G . The semigroup structure together diagonal subsemigroup D then yields natural domination relation xmlns:m="http://www.w3.org/1998/Math/MathML">? {\prec} -ultrafilters is isomorphic to original