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The HSP-classes of Archimedean \(l\)-groups with weak unit. (English) Zbl 1262.06009

Summary: Let W denote the class of abstract algebras of the title (with homomorphisms preserving unities). The familiar H, S, and P from universal algebra are here meant in W. Let \({\mathbb Z}\) and \({\mathbb R}\) denote the integers and the reals, with unity 1, qua W-objects. Let V denote a non-void finite set of positive integers. Let \({\mathcal G}\subseteq {\mathrm W}\) be non-void and not \(\{\{0\}\}\). We show
(1)
\({\mathrm{HSP}}\,{\mathcal G}={\mathrm{HSP}}({\mathrm{HS}}\,{\mathcal G}\cap {\mathrm S}\,{\mathbb R})\), and
(2)
\({\mathrm W}\neq{\mathcal G}={\mathrm{HSP}}\,{\mathcal G}\;\text{ if and only if }\;\exists {\mathrm V}({\mathcal G}={\mathrm{HSP}}\{1\;v{\mathbb Z}\mid v\in {\mathrm V}\})\).
Our proofs are, for the most part, simple calculations. There is no real use of methods of universal algebra (e.g., free objects), and only one restricted use of representation theory (Yosida). Note that (1) implies the basic fact the \({\mathrm{HSP}}\,{\mathbb R}={\mathrm W}\) (which can be proved in several ways). Note that (2) contrasts W with \({\mathcal C}=\) Archimedean \(l\)-groups, and \({\mathcal C}=\) abelian \(l\)-groups, where \({\mathrm{HSP}}\,{\mathbb Z}={\mathcal C}\) in each case.

MSC:

06F15 Ordered groups
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References:

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