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

Document Type : Research Paper

Authors

1 Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.

2 H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.

Abstract

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{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 $U(g)=\{P\in Min(G):g\notin P\}$ ($g\in G$), the Stone space $\mathcal{A}\mathscr{A}$ is a cover of $YG$ with the minimal property of (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.

Highlights

Dedicated to Bernhard Banaschewski on the Occasion of his 90th Birthday, Communicated by Themba Dub

Keywords

References

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