The Category of Hypergroups: seeking for a generalized version of Abelian Categories

Kaique Matias de Roberto; Ana Luiza da Conceição Tenório

doi:10.14244/lajm.v2i02.14

Autores

Kaique Matias de Roberto Instituto de Matemática e Estatística USP
Ana Luiza da Conceição Tenório IME USP

DOI:

https://doi.org/10.14244/lajm.v2i02.14

Palavras-chave:

Hypergroups, Homological Algebra, Abelian Category

Resumo

In this paper we study categorical properties of the category of abelian hypergroups that leads to the notion of hyper (almost) preadditive and hyper (almost) abelian categories. Our goal is to create a path towards a general theory of homological algebra for hyper-algebras. This is a first attempt to achieve this goal. We hope to improve the definitions and results, and provide more examples soon.

Referências

R Ameri. On categories of hypergroups and hypermodules. Journal of Discrete Mathematical Sciences and Cryptography, 6(2-3):121–132, 2003. DOI: https://doi.org/10.1080/09720529.2003.10697969

R. Ameri, M. Eyvazi, and S. Hoskova-Mayerova. Superring of polynomials over a hyperring. Mathematics, 7(10):902, 2019. DOI: https://doi.org/10.3390/math7100902

Francis Borceux. Handbook of categorical algebra: volume 1, Basic category the- ory, volume 1. Cambridge University Press, 1994.

Francis Borceux. Handbook of categorical algebra: volume 2, Categories and Structures, volume 1. Cambridge University Press, 1994. DOI: https://doi.org/10.1017/CBO9780511525858

P. Gladki and K. Worytkiewicz. Witt rings of quadratically presentable fields. Cat- egories and General Algebraic Structures, 12(1):1–23, 2020. DOI: https://doi.org/10.29252/CGASA.12.1.1

Ana Claudia Golzio. A brief historical survey on hyperstructures in algebra and logic. South American Journal of Logic, 2018.

George Gra ̈tzer. A representation theorem for multi-algebras. Arch. Math, 3:452– 456, 1962. DOI: https://doi.org/10.1007/BF01650093

Alexandre Grothendieck. Sur quelques points d’alge`bre homologique. Tohoku Mathematical Journal, Second Series, 9(2):119–183, 1957. DOI: https://doi.org/10.2748/tmj/1178244839

J. Jun. Algebraic geometry over hyperrings. Advances in Mathematics, 323:142– 192, 2018. DOI: https://doi.org/10.1016/j.aim.2017.10.043

M. Marshall. Real reduced multirings and multifields. Journal of Pure and Applied Algebra, 205(2):452–468, 2006. DOI: https://doi.org/10.1016/j.jpaa.2005.07.011

Christos Massouros and Gerasimos Massouros. An overview of the foundations of the hypergroup theory. Mathematics, 9(9):1014, 2021. DOI: https://doi.org/10.3390/math9091014

So Nakamura and Manuel L Reyes. Categories of hypermagmas, hypergroups, and related hyperstructures. arXiv preprint arXiv:2304.09273, 2023.

C. Pelea and I. Purdea. Multialgebras, universal algebras and identities. Journal of the Australian Mathematical Society, 81(1):121–140, 2006. DOI: https://doi.org/10.1017/S1446788700014671

H. R. de O. Ribeiro. Anel de Witt para semigrupos reais, envolto ́ria von Neumann e B-pares. PhD thesis, Universidade de São Paulo, 2021.

H. R. de O. Ribeiro, K. M. de A. Roberto, and H. L. Mariano. Functorial relation- ship between multirings and the various abstract theories of quadratic forms. Sa ̃o Paulo Journal of Mathematical Sciences, 2020. https://doi.org/10.1007/ s40863-020-00185-1.

Luis Ribes and Pavel Zalesskii. Profinite groups. Springer, 2000. DOI: https://doi.org/10.1007/978-3-662-04097-3

K. M. de A. Roberto and H. L. Mariano. K-theories and free inductive graded rings in abstract quadratic forms theories. Categories and General Algebraic Structures, 14(2), 2021.

K. M. de A Roberto, H. R. Ribeiro, and H. L. Mariano. Quadratic structures associated to (multi) rings. Categories and General Algebraic Structures, 14(2), 2021.

O Viro. Hyperfields for tropical geometry I. Hyperfields and dequantization. arXiv preprint arXiv:1006.3034, 2010.

Charles A Weibel. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1995.