The projectable hull of an archimedean $ell$-group with weak unit

The much-studied projectable hull of an $ell$-group $Gleq pG$ is an essential extension, so that, in the case that $G$ is archimedean with weak unit, ``$Gin {bf W}$", we have for the Yosida representation spaces a ``covering map" $YG leftarrow YpG$. We have earlier cite{hkm2} shown that (1) this cover has a characteristic minimality property, and that (2) knowing $YpG$, one can write down $pG$. We now show directly that for $mathscr{A}$, the boolean algebra in the power set of the minimal prime spectrum $Min(G)$, generated by the sets $U(g)={Pin Min(G):gnotin P}$ ($gin G$), the Stone space $mathcal{A}mathscr{A}$ is a cover of $YG$ with the minimal property of (1); this extends the result from cite{bmmp} for the strong unit case. Then, applying (2) gives the pre-existing description of $pG$, which includes the strong unit description of cite{bmmp}. The present methods are largely topological, involving details of covering maps and Stone duality