The many groupoids of a stably Gelfand quantale

We study the projections of an arbitrary stably Gelfand quantale Q and show that each projection determines a pseudogroup S⊂Q (and a corresponding localic étale groupoid G) together with a map of involutive quantales p:Q→L∨(S)[=O(G)]. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of étale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame Q whose partial units cover Q.