A multiring is a ring-like structure where the sum is multivalued and a hyperring is a multiring with a strong distributive property. With every multiring we associate a structural presheaf, and when that presheaf is a sheaf, we say that the multiring is geometric. A characterization of geometric von Neumann hyperrings is presented. And we build a von Neumann regular hull for multirings which is used in applications to algebraic theory of quadratic forms. Namely, we describe the functor Q, introduced by M. Marshall in [J. Pure Appl. Alg., 205, No. 2, 452-468 (2006)], as a left adjoint functor for the natural inclusion of the category of real reduced multirings (similar to real semigroups) into the category of preordered multirings and explore some of its properties. Next, we employ sheaf-theoretic methods to characterize real reduced hyperrings as certain geometric von Neumann regular real hyperrings and construct the functor V , a geometric von Neumann regular hull for a multiring. Finally, we look at some interesting logical and algebraic interactions between the functors Q and V that are useful for describing hyperrings in the image of the functor Q and that will allow us to explore the theory of quadratic forms for (formally) real semigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Marshall, “Real reduced multirings and multifields,” J. Pure Appl. Alg., 205, No. 2, 452-468 (2006).
H. R. O. Ribeiro, K. M. A. Roberto, and H. L. Mariano, “Functorial relationships between multirings and the various abstract theories of quadratic forms,” São Paulo J. Math. Sci., 16, No. 1, 5-42 (2022).
M. Dickmann and A. Petrovich, “Real semigroups and abstract real spectra. I,” in Contemp. Math., 344, R. Baeza et al. (Eds.), Am. Math. Soc., Providence, RI (2004), pp. 99-119.
M. A. Dickmann and F. Miraglia, Special Groups. Boolean-Theoretic Methods in the Theory of Quadratic Forms, Mem. Am. Math. Soc., 689, Am. Math. Soc., Providence, RI (2000).
M. A. Marshall, Spaces of Orderings and Abstract Real Spectra, Lect. Notes Math., 1636, Springer, Berlin (1996).
M. Dickmann and F. Miraglia, “Representation of reduced special groups in algebras of continuous functions,” in Contemp. Math., 493, R. Baeza et al. (Eds.), Am. Math. Soc., Providence, RI (2009), pp. 83-97.
J. Jun, “Algebraic geometry over hyperrings,” Adv. Math., 323, 142-192 (2018).
M. Dickmann and A. Petrovich, “Real semigroups, real spectra and quadratic forms over rings”; https://www.ime.usp.br/~miraglia/textos/RS-fev-18.pdf
P. Arndt and H. L. Mariano, “The von Neumann-regular hull of (preordered) rings and quadratic forms,” South Am. J. Log., 2, No. 2, 201-244 (2016).
C. C. Chang and H. J. Keisler, Model Theory, 3rd rev. ed., Stud. Logic Found. Math., 73, North-Holland, Amsterdam (1990).
F. Miraglia, An Introduction to Partially Ordered Structures and Sheaves, Contemp. Log., Polimetrica, Monza (2006).
M. Dickmann and F. Miraglia, “Quadratic form theory over preordered von Neumann-regular rings,” J. Alg., 319, No. 4, 1696-1732 (2008).
M. Dickmann, N. Schwartz, and M. Tressl, Spectral Spaces, New Math. Monogr., 35, Cambridge Univ. Press, Cambridge (2019).
H. R. O. Ribeiro and H. L. Mariano, “Witt rings for real semigroups,” in preparation.
H. R. O. Ribeiro and H. L. Mariano, “Hulls for real semigroups and applications,” in preparation.
K. M. A. Roberto, H. R. O. Ribeiro, and H. L. Mariano, “Quadratic extensions of special hyperfields and the general Arason–Pfister Hauptsatz,” submitted; https://arxiv.org/abs/2210.03784
K. M. A. Roberto, H. R. O. Ribeiro, and H. L. Mariano, “Quadratic structures associated to (multi)rings,” Categ. Gen. Algebr. Struct. Appl., 16, No. 1, 105-141 (2022).
Acknowledgements
We express our sincere gratitude to the referee for his/her careful reading and valuable suggestions that significantly improved the presentation of the text.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 62, No. 3, pp. 323-386, May-June, 2023. Russian DOI: https://doi.org/10.33048/alglog.2023.62.302.
Supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). (H. R. O. Ribeiro)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ribeiro, H.R.O., Mariano, H.L. Von Neumann Regular Hyperrings and Applications to Real Reduced Multirings. Algebra Logic 62, 215–256 (2023). https://doi.org/10.1007/s10469-024-09739-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-024-09739-0