Sesión Lógica y ComputabilidadOn the class of algebraic closed hyperfields and its model-theory
Hugo Luiz Mariano
Universidade de São Paulo, Brasil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
The concept of hyperrings was introduced by Marc Krasner in 1957, as an expansion of the concept of ring were the sum is multivalued. The concept of polynomials (and its roots) over hyper compositional structures that generalizes commutative rings are considered in [5], [1], [2], [6].
The goal of the present work is twofold:
(i) To expand the basic theory of algebraic extensions to the realm of superfields –a field with multivalued operations of sum and product–, showing that every superfield has a (unique up to isomorphism) full algebraic extension to a superfield that is algebraically closed.
(ii) To show that every infinite algebraically closed superfield admits a quantifier elimination procedure that, in some sense corresponds to a model-theoretic statement which cannot be the traditional quantifier elimination, as established in [4].
Trabajo en conjunto con: Kaique Matias de Andrade Roberto (Unicamp, Brasil).
Referencias
[1] R. Ameri, M. Eyvazi, S. Hoskova-Mayerova. Superring of polynomials over a hyperring. \emph{Mathematics}, 7(10):902, 2019.
[2] M. Baker, O. Lorscheid. Descartes' rule of signs, Newton polygons, and polynomials over hyperfields. \emph{J. Algebra}, 569:416–441, 2021.
[3] B. Davvaz, T. Musavi. Codes over hyperrings. \emph{Matematicki Vesnik},721 68(1):26–38, 2016.
[4] A. Linzi. On algebraically closed Krasner hyperfields. \emph{Italian J. Pure Appl. Math.} 54:105–112, 2025.
[5] J. Mittas. Sur certaines classes de structures hypercompositionnelles. \emph{ Proceedings of the Academy of Athens}, 48: 298–318, 1973.
[6] K. M. A. Roberto, H. L. Mariano. On superrings of polynomials and algebraically closed multifields. \emph{J. Applied Logics}, 9(1):419–444, 2022.
[7] K. M. A. Roberto, H. R. O. Ribeiro, H. L. Mariano. On algebraic extensions and algebraic closures of superfields. Preprint.