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Banaschewski, Bernhard; Hager, Anthony
The $HSP$-Classes of Archimedean $l$-groups with Weak Unit. Annales de la faculté des sciences de Toulouse Mathématiques, Sér. 6, 19 no. S1 (2010), p. 13-24
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$W$ denotes the class of abstract algebras of the title (with homomorphisms preserving unit). The familiar $H, S,$ and $P$ from universal algebra are here meant in $W$. $\mathbb{Z}$ and $\mathbb{R}$ denote the integers and the reals, with unit 1, qua $W$-objects. $V$ denotes a non-void finite set of positive integers. Let $\mathcal{G}\subseteq W$ be non-void and not $\lbrace \lbrace 0\rbrace \rbrace $. We show (1) $HSP\mathcal{G}= HSP(HS\mathcal{G}\cap S\mathbb{R})$, and (2) $W \ne \mathcal{G}= HSP\mathcal{G}$ if and only if $\exists V (\mathcal{G}= HSP\lbrace \frac{1}{v}\mathbb{Z}|v\in V \rbrace ).$ 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 that $HSP\mathbb{R}= W$ (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 $HSP\mathbb{Z}= \mathcal{C}$ in each case.


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