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 $S\subset Q$ (and a corresponding localic 'etale groupoid $G$) together with a map of involutive quantales $p:Q\to {\mathcal{L}}^{\bigvee }(S)[=\mathcal{O}(G)]$. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of 'etale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame whose partial units cover $Q$.