The HSP-Classes of Archimedean l-groups with Weak Unit
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. S1, pp. 13-24.

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

  • HSP𝒢=HSP(HS𝒢S), and
  • W𝒢=HSP𝒢 if and only if V(𝒢=HSP{1 v|vV}).

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

DOI : 10.5802/afst.1272
Banaschewski, Bernhard 1 ; Hager, Anthony 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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     title = {The $HSP${-Classes} of {Archimedean} $l$-groups with {Weak} {Unit}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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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, Série 6, Tome 19 (2010) no. S1, pp. 13-24. doi : 10.5802/afst.1272. http://www.numdam.org/articles/10.5802/afst.1272/

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