%0 Journal Article
%T The projectable hull of an archimedean $ell$-group with weak unit
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Hager, Anthony W.
%A McGovern, Warren Wm.
%D 2017
%\ 07/01/2017
%V 7
%N Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
%P 165-179
%! The projectable hull of an archimedean $ell$-group with weak unit
%K Archimedean $l$-group
%K vector lattice
%K Yosida representation
%K minimal prime spectrum
%K principal polar
%K projectable
%K principal projection property
%R
%X The much-studied projectable hull of an $ell$-group $Gleq pG$ is an essential extension, so that, in the case that $G$ isĀ 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.
%U http://cgasa.sbu.ac.ir/article_46629_aaedb6eda82247753a33798657cb5075.pdf