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

The much-studied projectable hull of an \ell-group G\leq pG is an essential extension, so that, in the case that G is  archimedean with weak unit, ``G\in {\bf W}", we have for the Yosida representation spaces a ``covering map" YG \leftarrow YpG. We have earlier \cite{hkm۲} shown that (۱) this cover has a characteristic minimality property, and that (۲) 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)=\{P\in Min(G):g\notin P\} (g\in G), the Stone space \mathcal{A}\mathscr{A} is a cover of YG with the minimal property of (۱); this extends the result from \cite{bmmp} for the strong unit case. Then, applying (۲) 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.

Keywords:

Archimedean l-group , vector lattice , Yosida representation , minimal prime spectrum , principal polar , projectable , principal projection property