Rocky Mountain Journal of Mathematics

*-Maximum lattice-ordered groups

Anthony W. Hager

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Abstract

The category $W$ is archimedean $l$-groups $G$ with distinguished weak order unit $e_G$, with unit-preserving $l$-group homomorphisms. For $G$ in $W$, $G^*$ denotes the convex sub-$l$-group generated by $e_G$. If $G$ satisfies [any isomorphism of $H^*$ with $G^*$ extends to an embedding of $H$ in $G$], then $G$ is called $\ast$-maximum. ($H^*$ and $G^*$ are isomorphic if and only if the unit intervals $[0, e_H]$ and $[0, e_G]$ are isomorphic MV-algebras.) This paper analyzes the property ``$\ast$-maximum'': Several characterizations are given. It is shown that any $\ast$-maximum $G$ has quasi-F Yosida space; this applies to prove a conjecture of the author and J\. Martinez about ``rings of $\omega_1$-quotients.'' It is shown that each $G$ in $W$ has a $\ast$-maximum hull, and this hull is described.

Article information

Source
Rocky Mountain J. Math., Volume 43, Number 6 (2013), 1901-1930.

Dates
First available in Project Euclid: 25 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1393336662

Digital Object Identifier
doi:10.1216/RMJ-2013-43-6-1901

Mathematical Reviews number (MathSciNet)
MR3178449

Zentralblatt MATH identifier
1309.06010

Subjects
Primary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] 46E05: Lattices of continuous, differentiable or analytic functions 54G05: Extremally disconnected spaces, $F$-spaces, etc.
Secondary: 03B50: Many-valued logic 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 54C30: Real-valued functions [See also 26-XX]

Keywords
Lattice-ordered group Archimedean order unit hull pseudo-adjoint quasi-F space uniform approximation ring of quotients MValgebra

Citation

Hager, Anthony W. *-Maximum lattice-ordered groups. Rocky Mountain J. Math. 43 (2013), no. 6, 1901--1930. doi:10.1216/RMJ-2013-43-6-1901. https://projecteuclid.org/euclid.rmjm/1393336662


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