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Torsion theories and coverings of preordered groups

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Marino Gran at Catholic University of Louvain
Marino Gran
Aline Michel
Aline Michel
Aline Michel
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Abstract

We explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of a Galois groupoid. Finally, we prove that in the category of preordered groups there is also a pretorsion theory, whose torsion subcategory can be identified with a category of internal groups. This latter is precisely the subcategory of protomodular objects in the category of preordered groups, as recently discovered by Clementino, Martins-Ferreira, and Montoli.
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Algebra Univers. (2021) 82:22
c
2021 The Author(s), under exclusive licence to Springer
Nature Switzerland AG part of Springer Nature
1420-8911/21/020001-30
published online February 19, 2021
https://doi.org/10.1007/s00012-021-00709-6 Algebra Universalis
Torsion theories and coverings of preordered
groups
Marino Gran and Aline Michel
Abstract. We explore a non-abelian torsion theory in the category of pre-
ordered groups: the objects of its torsion-free subcategory are the par-
tially ordered groups, whereas the objects of the torsion subcategory are
groups (with the total order). The reflector from the category of pre-
ordered groups to this torsion-free subcategory has stable units, and we
prove that it induces a monotone-light factorization system. We describe
the coverings relative to the Galois structure naturally associated with
this reflector, and explain how these coverings can be classified as inter-
nal actions of a Galois groupoid. Finally, we prove that in the category of
preordered groups there is also a pretorsion theory, whose torsion subcat-
egory can be identified with a category of internal groups. This latter is
precisely the subcategory of protomodular objects in the category of pre-
ordered groups, as recently discovered by Clementino, Martins-Ferreira,
and Montoli.
Mathematics Subject Classification. 18E50, 06F15, 18E40, 18G50, 18A40.
Keywords. Preordered group, Torsion theory, Categorical Galois theory,
Pretorsion theory, Factorization system, Covering.
1. Introduction
The category PreOrdGrp of preordered groups is the category whose objects
(G, ≤) are groups Gendowed with a preorder relation ≤on Gwhich is com-
patible with the group structure +: a≤cand b≤dimplies a+b≤c+d,
for all a, b, c, d ∈G. The morphisms in this category are preorder preserving
group morphisms.
Presented by V. Marra.
The second author’s research is funded by a FRIA doctoral grant of the Communaut´e
fran¸caise de Belgique.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... The aim of this paper consists in generalizing to any category of V -groups (up to an additional assumption on the quantale V ) the results on torsion theories and coverings developed in [18] in the particular setting of preordered groups. We start the article with some prerequisites needed to understand its content. ...
... In the paper [18], torsion theories and coverings have been studied in the category PreOrdGrp of preordered groups [11]. A preordered group is a group (G, +, 0) endowed with a preorder ≤ which is compatible with the addition law + of the group G: for any a, b, c, d ∈ G, a ≤ c and b ≤ d implies that a + b ≤ c + d. ...
... As for the morphisms in the class E , they are the normal epimorphisms in PreOrdGrp whose kernel is in the subcategory Grp. We therefore recover, from Theorem 3.10, the result developed in [18,Theorem 3.12] for preordered groups. (2). ...
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