The $HSP$-Classes of Archimedean $l$-groups with Weak Unit

Bernhard Banaschewski; Anthony Hager

doi:10.5802/afst.1272

The

H S P

-Classes of Archimedean

l

-groups with Weak Unit

Bernhard Banaschewski ¹; Anthony Hager ²

¹ Department of Mathematics, McMaster University, Hamilton, Ontario 68S4K1 Canada
² Department of Mathematics and Computer Science Wesleyan University, Middletown, CT 06459 USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 13-24.

Abstract

$W$ 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 $W$ . $ℤ$ and $ℝ$ denote the integers and the reals, with unit 1, qua $W$ -objects. $V$ denotes a non-void finite set of positive integers. Let $𝒢 \subseteq W$ be non-void and not ${{0}}$ . We show

$H S P 𝒢 = H S P (H S 𝒢 \cap S ℝ)$ , and
$W \neq 𝒢 = H S P 𝒢$ if and only if $\exists V (𝒢 = H S P {\frac{1}{v} ℤ | v \in V}) .$

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 that $H S P ℝ = W$ (which can be proved in several ways). Note that (2) contrasts $W$ with $𝒞 =$ archimedean $l$ -groups, and $𝒞 =$ abelian $l$ -groups, where $H S P ℤ = 𝒞$ in each case.

MR

DOI: 10.5802/afst.1272

Author's affiliations:

Bernhard Banaschewski ¹; Anthony Hager ²

¹ Department of Mathematics, McMaster University, Hamilton, Ontario 68S4K1 Canada
² Department of Mathematics and Computer Science Wesleyan University, Middletown, CT 06459 USA

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     author = {Bernhard Banaschewski and Anthony Hager},
     title = {The $HSP${-Classes} of {Archimedean} $l$-groups with {Weak} {Unit}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {13--24},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {S1},
     year = {2010},
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     mrnumber = {2675718},
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     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1272/}
}

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Bernhard Banaschewski; Anthony Hager. The $HSP$-Classes of Archimedean $l$-groups with Weak Unit. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 13-24. doi : 10.5802/afst.1272. https://afst.centre-mersenne.org/articles/10.5802/afst.1272/

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