  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 $Q$.

Rings and Algebras;Category Theory