cnrs   Université de Toulouse

Table of contents for this issue | Previous article | Next article
Bernhard Banaschewski; Anthony Hager
The $HSP$-Classes of Archimedean $l$-groups with Weak Unit
Annales de la faculté des sciences de Toulouse Sér. 6, 19 no. S1: Special issue: Proceedings of the Conference on Ordered Rings in Honor of Melvin Henriksen, Louisiana State University at Baton Rouge, 2007 (2010), p. 13-24, doi: 10.5802/afst.1272
Article PDF | Reviews MR 2675718 | Zbl pre05799078

Résumé - 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 $W$. $\mathbb{Z}$ 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.


[BH] R. Ball, A. Hager, A new characterization of the continuous functions on a locale, Positivity 10, 165-199 (2006)  MR 2223592 |  Zbl 1101.06007
[B1] B. Banaschewski, On the function ring functor in point-free topology, Appl. Categ. Str. 13(2005), 305-328.  MR 2175953 |  Zbl 1158.54308
[B2] B. Banaschewski, On the function rings of point-free topology, Kyungpook Math. J. 48 (2008), 195-206.  MR 2429308 |  Zbl 1155.06008
[COM] R. Cignoli, I. D’Ottavigno, D. Mundici,Algebraic foundations of many-solved reasoning, Kluwer (2000).  Zbl 0937.06009
[D] M. Darnel, Theory of lattice -ordered groups, Dekker (1995).  MR 1304052 |  Zbl 0810.06016
[GJ] L. Gillman, M. Jerison, Rings of continuous functions, Van Nostrand (1960).  MR 116199 |  Zbl 0093.30001
[H] A. Hager, Algebraic closures of $l$-groups of continuous functions, pp. 165 - 193 in Rings of Continuous Functions (C. Aull, Editor), Dekker Notes 95 (1985).  MR 789270 |  Zbl 0616.06017
[HK] A. Hager, C. Kimber, Uniformly hyperarchimedean lattice-ordered groups, Order 24 (2007), 121-131.  MR 2367346 |  Zbl 1128.06007
[HM] A. Hager, J. Martinez, Singular archimedean lattice-ordered groups, Alg. Univ. 40(1998), 119-147.  MR 1651866 |  Zbl 0936.06015
[HR] A. Hager, L. Robertson, Representing and ringifying a Riesz space, Symp. Math. 21 (1977), 411-431.  MR 482728 |  Zbl 0382.06018
[HI] M. Henriksen, J. lsbell, Lattice-ordered rings and function rings, Pac. J. Math. 12 (1962), 533-565.r Article |  MR 153709 |  Zbl 0111.04302
[HIJ] M. Henriksen, J. Isbell, D. Johnson, Residue class fields of lattice-ordered algebras, Fund. Math. 50 (1965), 107-117 Article |  MR 133350 |  Zbl 0101.33401
[I] J. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 5 - 32.  MR 358725 |  Zbl 0246.54028
[LZ] W. Luxemburg, A. Zaanen, Riesz spaces I, North-Holland (1971).  Zbl 0231.46014
[P] R. Pierce, Introduction to the theory of abstract algebras, Holt, Rinehart and Winston (1968).  MR 227070
[W] E. Weinberg, Lectures on ordered groups and rings, Univ. of Illinois (1968).
[Y] K. Yosida, On the representation of the vector lattice, Proc. Imp. Acad. (Tokyo) 18, 339-343, (1942). Article |  MR 15378 |  Zbl 0063.09070
Search for an article
Search within the site