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Torsion theories and coverings of preordered groups

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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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References

  1. Borceux, F.: Handbook of Categorical Algebra, vol. 2. Encyclopedia Math. Appl., vol. 51. Cambridge University Press, Cambridge (1994)

  2. Borceux, F., Bourn, D.: Mal’cev, protomodular, homological and semi-abelian categories. Mathematics and its applications, vol. 566. Kluwer, Dordrecht (2004)

  3. Borceux, F., Clementino, M.M.: Topological semi-abelian algebras. Adv. Math. 190, 425–453 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourn, D., Gran, M.: Torsion theories in homological categories. J. Algebra 305, 18–47 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourn, D., Janelidze, Z.: Pointed protomodularity via natural imaginary subtractions. J. Pure Appl. Algebra 213, 1835–1851 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourn, D., Martins-Ferreira, N., Montoli, A., Sobral, M.: Schreier split epimorphisms in monoids and in semirings. Textos de Matemática (Série B), vol. 45. Departamento de Matemática da Universidade de Coimbra (2013)

  7. Burris, S., Sankappanavar, H.P.: A course in universal algebra. Springer, New York (1981)

    Book  MATH  Google Scholar 

  8. Carboni, A., Janelidze, G., Kelly, G.M., Paré, R.: On localization and stabilization for factorization systems. Appl. Categorical Struct. 5, 1–58 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cassidy, C., Hébert, M., Kelly, G.M.: Reflective subcategories, localizations and factorization systems. J. Aust. Math. Soc. 38, 237–329 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Clementino, M.M., Dikranjan, D., Tholen, W.: Torsion theories and radicals in normal categories. J. Algebra 305, 98–129 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Clementino, M.M., Martins-Ferreira, N., Montoli, A.: On the categorical behaviour of preordered groups. J. Pure Appl. Algebra 223, 4226–4245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Everaert, T., Gran, M.: Monotone-light factorisation systems and torsion theories. Bull. Sci. Math. 137, 996–1006 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Everaert, T., Gran, M.: Protoadditive functors, derived torsion theories and homology. J. Pure Appl. Algebra 219, 3629–3676 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Facchini, A.: Commutative monoids, noncommutative rings and modules. In: Clementino, M.M., Facchini, A., Gran, M. (eds.) New Perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts (Accepted for publication)

  15. Facchini, A., Finocchiaro, C.: Pretorsion theories, stable category and preordered sets. Ann. Mat. Pura Appl. (2019). https://doi.org/10.1007/s10231-019-00912-2

  16. Facchini, A., Finocchiaro, C., Gran, M.: A new Galois structure in the category of preordered sets. Theory Appl. Categories 35(11), 326–349 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Facchini, A., Finocchiaro, C., Gran, M.: Pretorsion theories in general categories. J. Pure Appl. Algebra 225, 106503 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fichtner, K.: Varieties of universal algebras with ideals. Mat. Sb. N. S. 75, 445–453 (1968)

    MathSciNet  Google Scholar 

  19. Gran, M., Rossi, V.: Torsion theories and Galois coverings of topological groups. J. Pure Appl. Algebra 208, 135–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gran, M., Sterck, F., Vercruysse, J.: A semi-abelian extension of a theorem of Takeuchi. J. Pure Appl. Algebra 223, 4171–4190 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grandis, M., Janelidze, G.: From torsion theories to closure operators and factorization systems. Categories Gen. Algebraic Struct. Appl. 12(1), 89–121 (2020)

    MathSciNet  MATH  Google Scholar 

  22. Janelidze, G., Márki, L., Tholen, W.: Locally semisimple coverings. J. Pure Appl. Algebra 128, 281–289 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Janelidze, G.: Pure Galois theory in categories. J. Algebra 132, 270–286 (1990)

    Article  MathSciNet  Google Scholar 

  24. Janelidze, G., Kelly, G.M.: Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97, 135–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Janelidze, G., Sobral, M., Tholen, W.: Beyond Barr exactness: effective descent morphisms. In: Pedicchio, M.C., Tholen , W. (eds.) Categorical foundations. encyclopedia of mathematics and its applications, pp. 359–405. Cambridge University Press, Cambridge (2004)

  26. Janelidze, G., Tholen, W.: Characterization of torsion theories in general categories. Categories in algebra, geometry and mathematical physics, 249256, Contemp. Math., vol. 431. American Mathematical Society, Providence, RI (2007)

  27. Janelidze, Z.: The pointed subobject functor, \(3\times 3\) lemmas, and subtractivity of spans. Theory Appl. Categories 23(11), 221–242 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Johnstone, P.T.: A note on the Semi-abelian Variety of Heyting Semilattices. Fields Inst. Commun. 43, 317–318 (2004)

    MATH  Google Scholar 

  29. Mantovani, S.: Torsion theories for crossed modules. Invited talk at the “Workshop on category theory and topology”. Université catholique de Louvain (2015)

  30. Montoli, A., Rodelo, D., Van der Linden, T.: Two characterisations of groups amongst monoids. J. Pure Appl. Algebra 222(4), 747–777 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xarez, J.: The monotone-light factorization for categories via preordered and ordered sets. Ph.D. thesis, University of Aveiro (2003)

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Acknowledgements

The authors are grateful to the anonymous referee for some very useful suggestions on a preliminary version of the article.

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  1. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, 2, Chemin du Cyclotron, 1348, Louvain-la-Neuve, Belgium

    Marino Gran & Aline Michel

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  1. Marino Gran
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  2. Aline Michel
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Correspondence to Marino Gran.

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The second author’s research is funded by a FRIA doctoral grant of the Communauté française de Belgique.

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Gran, M., Michel, A. Torsion theories and coverings of preordered groups. Algebra Univers. 82, 22 (2021). https://doi.org/10.1007/s00012-021-00709-6

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