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The projectable hull of an Archimedean \(\ell\)-group with weak unit. (English) Zbl 1423.06057

Summary: 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, “\(G\in \mathbf W\)”, we have for the Yosida representation spaces a “covering map” \(YG \leftarrow YpG\). We have earlier [C. M. Kimber et al., Rend. Circ. Mat. Palermo (2) 52, No. 3, 453–480 (2003; Zbl 1072.06013)] 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 \(\mathcal{A}\), the boolean algebra in the power set of the minimal prime spectrum \(\operatorname{Min}(G)\), generated by the sets \(U(g)=\{P\in \operatorname{Min}(G):g\notin P\}\) (\(g\in G\)), the Stone space \(\mathcal{SA}\) is a cover of \(YG\) with the minimal property of (1); this extends the result from [R. N. Ball et al., Forum Math. 30, No. 2, 513–526 (2018; Zbl 1528.06024)] for the strong unit case. Then, applying (2) gives the pre-existing description of \(pG\), which includes the strong unit description of [Ball et al., loc. cit.]. The present methods are largely topological, involving details of covering maps and Stone duality.

MSC:

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
46A40 Ordered topological linear spaces, vector lattices
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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