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New torsion theory in unital abelian $\ell$-groups

In: Mathematica Slovaca, vol. 61, no. 3
Jorge Martínez

Details:

Year, pages: 2011, 451 - 468
Keywords:
categoty of abelian lattice-ordered groups with designated strong order unit, functorial torsion class of archimedean lattice-ordered groups, monocoreflective subcategory
About article:
This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects. Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean lattice-ordered groups are characterized: for each subgroup $A$ of the rationals with the identity $1$, either $\mathfrak{T}=\mathfrak{S}(A)$, the class of all lattice-ordered groups of functions on a set $X$ which have finite range in $A$, or $\mathfrak{T}=\mathbb{T}(A)$, the class of all subgroups of $A$ with $1$. As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup $A$ of the reals with $1$, of rank two or greater, then $\mathfrak{T}$ contains all $\ell$-groups of the form $A\stackrel{\rightarrow}{×} G$, for all abelian lattice-ordered groups $G$. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\stackrel{\rightarrow}{×} G$, for all abelian lattice-ordered groups $G$.
How to cite:
ISO 690:
Martínez, J. 2011. New torsion theory in unital abelian $\ell$-groups. In Mathematica Slovaca, vol. 61, no.3, pp. 451-468. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0023-4

APA:
Martínez, J. (2011). New torsion theory in unital abelian $\ell$-groups. Mathematica Slovaca, 61(3), 451-468. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0023-4
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