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.

$W$ 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 $\left\{\left\{0\right\}\right\}$. We show

• $HSP𝒢=HSP\left(HS𝒢\cap Sℝ\right)$, and
• $W\ne 𝒢=HSP𝒢$ if and only if $\exists V\left(𝒢=HSP\left\{\frac{1}{v}ℤ|v\in V\right\}\right).$

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

DOI: 10.5802/afst.1272

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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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/

[BH] R. Ball, A. Hager, A new characterization of the continuous functions on a locale, Positivity 10, 165-199 (2006) | MR | Zbl

[B1] B. Banaschewski, On the function ring functor in point-free topology, Appl. Categ. Str. 13(2005), 305-328. | MR | Zbl

[B2] B. Banaschewski, On the function rings of point-free topology, Kyungpook Math. J. 48 (2008), 195-206. | MR | Zbl

[COM] R. Cignoli, I. D’Ottavigno, D. Mundici,Algebraic foundations of many-solved reasoning, Kluwer (2000). | Zbl

[D] M. Darnel, Theory of lattice -ordered groups, Dekker (1995). | MR | Zbl

[GJ] L. Gillman, M. Jerison, Rings of continuous functions, Van Nostrand (1960). | MR | Zbl

[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 | Zbl

[HK] A. Hager, C. Kimber, Uniformly hyperarchimedean lattice-ordered groups, Order 24 (2007), 121-131. | MR | Zbl

[HM] A. Hager, J. Martinez, Singular archimedean lattice-ordered groups, Alg. Univ. 40(1998), 119-147. | MR | Zbl

[HR] A. Hager, L. Robertson, Representing and ringifying a Riesz space, Symp. Math. 21 (1977), 411-431. | MR | Zbl

[HI] M. Henriksen, J. lsbell, Lattice-ordered rings and function rings, Pac. J. Math. 12 (1962), 533-565.r | MR | Zbl

[HIJ] M. Henriksen, J. Isbell, D. Johnson, Residue class fields of lattice-ordered algebras, Fund. Math. 50 (1965), 107-117 | MR | Zbl

[I] J. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 5 - 32. | MR | Zbl

[LZ] W. Luxemburg, A. Zaanen, Riesz spaces I, North-Holland (1971). | Zbl

[P] R. Pierce, Introduction to the theory of abstract algebras, Holt, Rinehart and Winston (1968). | MR

[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). | MR | Zbl

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