The projectable hull of an archimedean l-group with weak unit

The much-studied projectable hull of an ℓ-group G≤pG is an essential extension, so that, in the case that G is archimedean with weak unit, G∈W , we have for the Yosida representation spaces a covering map YG←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 A, the boolean algebra in the power set of the minimal prime spectrum Min(G), generated by the sets U(g)={P∈Min(G):g∉P} (g∈G), the Stone space AA 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 [1]. The present methods are largely topological, involving details of covering maps and Stone duality.