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UniversidaddeCádiz
International Meeting on Commutative Monoids 2023, 18-20 July

Program

Evora, 18-20 July 2023

18/07/2023
9:00-9:15 Opening
Titles
9:30-10:15 Evelia R. García Barroso Conductors of Abhyankar-Moh semigroups of even degrees
10:15-11:00 David Llena Betti elements on affine semigroups
Coffe break
11:30-12:15 Carlos-Jesús Moreno-Ávila Some combinatorial thoughts around semigroups at infinity
12:15-13:00 Isaac Guzmán-Maya Walnut: Introduction and applications
19/07/2023
9:30-10:15 Manuel Delgado Still counting numerical semigroups
10:15-11:00 Daniel Marín-Aragón A family of non-Weierstrass semigroups.
11:00-11:30 Coffe break
11:30-12:15 Ignacio Ojeda On the depth of simplicial affine semigroup algebras
12:15-13:00 María Ángeles Moreno-Frías Frobenius pseudo-variety of numerical semigroups with a given multiplicity and ratio
13:00-13:45 Adrián Sánchez-Loureiro On affine C-semigroups
20:00 Meeting dinner
20/07/2023
9:30-10:15 Luis José Santana Sánchez The p-Frobenius vector of affine semigroups
10:15-11:00 Márcio André Traesel (on-line) Frobenius R-variety in numerical semigroups containing a given semigroup
11:00-11:30 Coffe break
11:30-13:00 Open problem session

Manuel Delgado, Universidade do Porto , Still counting numerical semigroups

Abstract

There are two kinds of results related to counting numerical semigroups:

experimental results, usually relying on some deep computational and algorithmic work;

asymptotic results, generally depending upon combinatorial and statistical methods.

There are many results involving the following two distinct ways of counting numerical semigroups:

counting by Frobenius number (representing by N(f) the number of numerical semigroups of Frobenius number f, what is N(f) for some values of f, and what is the order of magnitude of N(f) when f tends to infinity?);

counting by genus (representing by n(g) the number of numerical semigroups with genus g, what is n(g) for some values of g, and what can be said about the behaviour of the sequence (n(g))?).

I will give an overview of the existing results, some of which are recent.

Partially supported by Proyecto “Monoides y semigrupos afines (ProyExcel_00868)“, Proyecto financiado en la convocatoria 2021 de Ayudas a Proyectos de Excelencia, en régimen de concurrencia competitiva, destinadas a entidades calificadas como Agentes del Sistema Andaluz del Conocimiento, en el ámbito del Plan Andaluz de Investigación, Desarrollo e Innovación (PAIDI 2020). Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía.

 

Evelia R. García Barroso, Universidad de La Laguna, Conductors of Abhyankar-Moh semigroups of even degrees

Abstract

In their paper [1] on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. In [2] were studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups and described such semigroups with the maximum conductor. In this talk I will show that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. The proof is constructive, explicitly describing families that achieve a given value as its conductor.

This talk is based on the results of [3].

Partially supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz, and by the grant PID2019-105896GB-I00 funded by MCIN/AEI/10.13039/501100011033.

References

[1] S. S. Abhyankar, T. T. Moh. Embeddings of the line in the plane, J. reine angew. Math., 276 (1975), 148Ð166. https://doi.org/10.1515/crll.1975.276.148.

[2] R. Barrolleta, E.R. García Barroso, A. P\l oski. On the Abhyankar-Moh inequality, Universitatis Iagellonicae Acta Mathematica, LII (2015), 7-14. doi:10.4467/20843828AM.15.001.3727.

[3] E.R. García Barroso, J.I. García-García, L.J. Santana Sánchez, A. Vigneron-Tenorio. Conductors of Abhyankar-Moh semigroups of even degrees, Electronic Research Archive, 31(4): 2213Ð2229. DOI: 10.3934/era.2023113

 

Isaac Guzmán-Maya, Universidad de Cádiz, Walnut: Introduction and applications

Abstract

Walnut software basics and operation. Working with automatic sequences: the evil numbers, the odious numbers, and the lower and upper Wythoff sequences.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

 

David Llena, Universidad de Almería, Betti elements on affine semigroups

Abstract

Let equation an ideal of equation, that is, equation, whereequation is a finite set. Consider  equation  which is a submonoid of equation (could be non finitely generated). However, if it is imposed that  equation has a finite number of elements, or equivalently equation has one (or more) element in every axis, we have a generalized numerical semigroup. The set of minimal generators of these semgiroups can be obtained from the set of gaps equation, but, en general, it is a very large set. In this work in progress, we try to give a characterization of the set of Betti elements, whitout using the set of minimal generators.

Partially supported by Proyecto “Monoides y semigrupos afines (ProyExcel_00868)“, Proyecto financiado en la convocatoria 2021 de Ayudas a Proyectos de Excelencia, en régimen de concurrencia competitiva, destinadas a entidades calificadas como Agentes del Sistema Andaluz del Conocimiento, en el ámbito del Plan Andaluz de Investigación, Desarrollo e Innovación (PAIDI 2020). Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía.

 

Daniel Marín-Aragón, Universidad de Cádiz, A family of non-Weierstrass semigroups.

Abstract

The equation-semigroups are a family of numerical semigroups defined by the position of their pseudo-Frobenius numbers in their sets of gaps. We study this family and show that they can be written as the interesection of Weierstrass semigroups but they are non-Weierstrass.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

 

Carlos-Jesús Moreno-Ávila, Universitat Jaume I, Some combinatorial thoughts around semigroups at infinity

Abstract

Let equation be a curve on equation with only one place at infinity at a point equation, and let equation be its semigroup at infinity, i.e. the additive submonoid of equation consisting of the orders —with negative sign— of the poles of the regular functions around (but not in) equation After a theorem by Abhyankar and Moh, we can associate to equation a so-called equation-sequence in equation which is a system of generators of equation, by no means unique.

Curves with only one place at infinity are relevant, for instance they play an important role in the study of the Jacobian conjecture. However, not so much is known about equation from a combinatorial point of view. In this talk we review the previous concepts and results and we see some properties of the equation-sequences; in particular, we introduce the notion of minimal equation-sequence as that generated only by the minimal elements of the semigroup at infinity. In addition, we show how we can compute the remaining equation-sequences associating with the same semigroup at infinity. Finally, we see algorithm procedures to compute minimal equation-sequences.

This talk is based on a joint work in progress with C. Galindo, F. Monserrat and J.-J. Moyano-Fernández.

 

María Ángeles Moreno-Frías, Universidad de Cádiz, Frobenius pseudo-variety of numerical semigroups with a given multiplicity and ratio

Abstract

Let equation be a numerical semigroup. Denote by equation, equation and equation the minimal system of generators, the muliplicity and the ratio of equation respectively.

In this work, we study the set of all numerical semigroups with multiplicity equation and ratio equation, denoted by equation. In particular, we prove that equation is a Frobenius pseudo-variety (see [2]) and we present some algorithms which compute all the elements of equation with a given genus or with a given Frobenius number.

Let equation. The results of [3] allow us to state that an element of equation is determined by a subset of equation. As a consequence, we give formulas to obtain the Frobenius number and the genus of an element of equation from its subset of equation associated.

This talk is based on a work which has been recently published in Appl. Algebr. Eng. Commun. Comput. (see [1]).

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

References

[1] M.A. Moreno-Frías and J. C. Rosales. Frobenius pseudo-varieties of numerical semigroups with a given multiplicity and ratio, Appl. Alg. Eng. Comm. Comp. https://doi.org/10.1007/s00200-023-00610-w.

[2] A. M. Robles-Pérez and J. C. Rosales. Frobenius pseudo-varieties in numerical semigroups, Ann. Mat. Pura Appl. 194 (2015), 275–287.

[3] J. C. Rosales and M. B. Branco. The Frobenius problem for numerical semigorups, J. Number Theory 131 (2011), 2310–2319.

 

Ignacio Ojeda, Universidad de Extremadura, On the depth of simplicial affine semigroup algebras

Abstract

In this talk, we recall the different characterizations of the depth of a monomial algebra. We conjecture that the property of a monomial algebra of having extreme depth is determined by the existence of maximal elements in certain Avery sets when the monomial structure comes from simplicial affine semigroups, exhibiting relevant evidences in low-dimensional cases.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

 

Adrián Sánchez-Loureiro, Universidad de Cádiz, On affine C-semigroups

Abstract

Let equation be an integer cone. An affine semigroup equation is called equation-semigroup whenever equation is finite. We show the generalization of several properties of numerical semigroups to equation-semigroups.

Joint work with J. I. García-García, D. Marín-Aragón and A. Vigneron-Tenorio.

Supported by “Monoides afines y sus aplicaciones (PR2022-011)”, Proyecto de investigación del Plan Propio – UCA 2022-2023, Universidad de Cádiz.

Reference

García-García, J. I.; Marín-Aragón, D.; Sánchez-Loureiro, A.; Vigneron-Tenorio, A. Some properties of affine equation-semigroups. arXiv:2305.05044

 

José Luis Santana Sánchez, Universidad de Valladolid, The p-Frobenius vector of affine semigroups

Abstract

Given a numerical semigroup equation, minimally generated by equation, we denote by equation. In this notation, the Frobenius number of equation is the maximum integer equation such that equation. A natural generalization of the Frobenius number presented in the literature is the so called equation-Frobenius number, which is the maximum integer equation such that equation.

In this talk we consider a further generalization, and we define the equation-Frobenius vector of an affine semigroup equation with a monomial order equation on equation in an analogous way. Our goal is to present some algorithms that compute the equation-Frobenius vectors as well as characterizing how these vectors behave when considering gluing of affine semigroups.

 

Márcio André Traesel, Instituto Federal de São Paulo, Frobenius R-variety in numerical semigroups containing a given semigroup

Abstract

Let equation be a numerical semigroup and equation. We prove that equation is Frobenius R-variety that can be arranged in a tree rooted in equation. We introduce the concepts of Frobenius and genus number of equation restricted to equation (respectively equation and equation). We give formulas for equation, equation and generalizations of the Amorós’s and Wilf’s conjecture. Moreover, we will show that most of the results of irreducibility can be generalized to equation-irreducibility.