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M. A. Moreno-Frías

Publications and source records attributed to M. A. Moreno-Frías.

16 recordsLinked to original sources
arXiv2024

Semi-covariety of numerical semigroups

The main aim of this work is to introduce and justify the study of semi-covarities. A {\it semi-covariety} is a non-empty family $\mathcal{F}$ of numerical semigroups such that it is closed under finite intersections, has a minimum, $\min(\mathcal{F}),$ and if $S\in \mathcal{F}$ being $S\neq \min(\mathcal{F})$, then there is $x\in S$ such that $S\backslash \{x\}\in \mathcal{F}$. As examples, we will study the semi-covariety formed by all the numerical semigroups containing a fixed numerical semigroup, and the semi-covariety composed by all the numerical semigroups of coated odd elements and fixed Frobenius number.

math.AC↗
arXiv2023

The covariety of perfect numerical semigroups with fixed Frobenius number

Let $S$ be a numerical semigroup. We will say that $h\in {\mathbb{N}} \backslash S$ is an {\it isolated gap }of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called perfect numerical semigroup. Denote by ${\mathrm m}(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family ${\mathscr{C}}$ of numerical semigroups that fulfills the following conditions: there is the minimum of ${\mathscr{C}},$ the intersection of two elements of ${\mathscr{C}}$ is again an element of ${\mathscr{C}}$ and $S\backslash \{{\mathrm m}(S)\}\in {\mathscr{C}}$ for all $S\in {\mathscr{C}}$ such that $S\neq \min({\mathscr{C}}).$ In this work we prove that the set ${\mathscr{P}}(F)=\{S\mid S \mbox{ is a perfect numerical}\ \mbox{semigroup with Frobenius number }F\}$ is a covariety. Also, we describe three algorithms which compute: the set ${\mathscr{P}}(F),$ the maximal elements of ${\mathscr{P}}(F)$ and the elements of ${\mathscr{P}}(F)$ with a given genus. A ${\mathrm{Parf}}$-semigroup (respectively, ${\mathrm{Psat}}$-semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (respectively, saturated numerical semigroup). We will prove that the sets: ${\mathrm{Parf}}(F)=\{S\mid S \mbox{ is a ${\mathrm{Parf}}$-numerical semigroup with Frobenius number} F\}$ and ${\mathrm{Psat}}(F)=\{S\mid S \mbox{ is a ${\mathrm{Psat}}$-numerical semigroup with Frobenius number } F\}$ are covarieties. As a consequence we present some algorithms to compute ${\mathrm{Parf}}(F)$ and ${\mathrm{Psat}}(F)$.

math.AC↗
arXiv2023

The covariety of saturated numerical semigroups with fixed Frobenius number

In this work we will show that if $F$ is a positive integer, then ${\mathrm{Sat}}(F)=\{S\mid S \mbox{ is a saturated numerical semigroup with Frobenius number } F\}$ is a covariety. As a consequence, we present two algorithms: one that computes ${\mathrm{Sat}}(F),$ and the other which computes all the elements of ${\mathrm{Sat}}(F)$ with a fixed genus. If $X\subseteq S\backslash Δ(F)$ for some $S\in {\mathrm{Sat}}(F),$ then we will see that there is the least element of ${\mathrm{Sat}}(F)$ containing a $X$. This element will denote by ${\mathrm{Sat}}(F)[X].$ If $S\in{\mathrm{Sat}}(F),$ then we define the ${\mathrm{Sat}}(F)$-rank of $S$ as the minimum of $\{\mbox{cardinality}(X)\mid S={\mathrm{Sat}}(F)[X]\}.$ In this paper, also we present an algorithm to compute all the element of ${\mathrm{Sat}}(F)$ with a given ${\mathrm{Sat}}(F)$-rank.

math.AC↗
arXiv2023

The ratio-covariety of numerical semigroups with fixed multiplicity and Frobenius number

In this work we will introduce the concept of ratio-covariety, as a nonempty family $\mathscr{R}$ of numerical semigroups verifying certain properties. This concept will allow us to: \begin{enumerate} \item Describe an algorithmic process to compute $\mathscr{R}.$ \item Prove the existence of the smallest element of $\sR$ that contains a set of positive integers. \item Talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. \end{enumerate} In addition, in this paper we will apply the previous results to the study of the ratio-covariety $\mathscr{R}(F,m)=\{S\mid S \mbox{ is a numerical semigroup with Fro-} \mbox {benius number }F \mbox{ and multiplicity }m\}.$

math.AC↗
arXiv2023

The set of Arf numerical semigroups with given Frobenius number

In this work we will show that if $F$ is a positive integer, then the set ${\mathrm{Arf}}(F)=\{S\mid S \mbox{ is an Arf numerical semigroup with Frobenius number } F\}$ verifies the following conditions: 1) $Δ(F)=\{0,F+1,\rightarrow\}$ is the minimum of ${\mathrm{Arf}}(F),$ 2) if $\{S, T\} \subseteq {\mathrm{Arf}}(F)$, then $S \cap T \in {\mathrm{Arf}}(F),$ 3) if $S \in {\mathrm{Arf}}(F),$ $S\neq Δ(F)$ and ${\mathrm m}(S)=\min (S \backslash \{0\})$, then $S\backslash \{{\mathrm m}(S)\} \in {\mathrm{Arf}}(F)$. The previous results will be used to give an algorithm which calculates the set ${\mathrm{Arf}}(F).$ Also we will see that if $X\subseteq S\backslash Δ(F)$ for some $S\in {\mathrm{Arf}}(F),$ then there is the smallest element of ${\mathrm{Arf}}(F)$ containing $X.$

math.AC↗
arXiv2023

The covariety of numerical semigroups with fixed Frobenius number

Denote by $\mathrm m(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathscr{C}$ of numerical semigroups that fulfills the following conditions: there is the minimum of $\mathscr{C},$ the intersection of two elements of $\mathscr{C}$ is again an element of $\mathscr{C}$ and $S\backslash \{\mathrm m(S)\}\in \mathscr{C}$ for all $S\in \mathscr{C}$ such that $S\neq \min(\mathscr{C}).$ In this work we describe an algorithmic procedure to compute all the elements of $\mathscr{C}.$ We prove that there exists the smallest element of $\mathscr{C}$ containing a set of positive integers. We show that $\mathscr{A}(F)=\{S\mid S \mbox{ is a numerical semigroup with Frobenius number }F\}$ is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.

math.AC↗
arXiv2022

The complexity of a numerical semigroup

Let $S$ and $Δ$ be numerical semigroups. A numerical semigroup $S$ is an $\mathbf{I}(Δ)$-{\it semigroup} if $S\backslash \{0\}$ is an ideal of $Δ$. We will denote by $\mathcal{J}(Δ)=\{S \mid S \text{ is an $\mathbf{I}(Δ)$-semigroup} \}.$ We will say that $Δ$ is {\it an ideal extension of } $S$ if $S\in \mathcal{J}(Δ).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by $\mathcal{J}^0(\mathbb{N})=\mathbb{N},$ $\mathcal{J}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$ and $\mathcal{J}^{k+1}(\mathbb{N})=\mathcal{J}(\mathcal{J}^{k}(\mathbb{N}))$ for all $k\in \mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\{k\in \mathbb{N}\mid S \in \mathcal{J}^k(\mathbb{N})\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.

math.NT↗
arXiv2017

Semigroups with fixed multiplicity and embedding dimension

Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf's conjecture is checked in the tree associated to some packed numerical semigroups.

math.AC↗
arXiv2016

On divisor-closed submonoids and minimal distances in finitely generated monoids

We study the lattice of divisor-closed submonoids of finitely generated cancellative commutative monoids. In case the monoid is an affine semigroup, we give a geometrical characterization of such submonoids in terms of its cone. Finally, we use our results to give an algorithm for computing $Δ^*(H)$ the set of minimal distance of $H$.

math.AC↗
arXiv2016

Proportionally modular affine semigroups

This work introduces a new kind of semigroup of $\N^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We prove they are finitely generated and we give an algorithm to compute their minimal generating sets. We also specialise on the case $p=2$. For this case, we provide a faster algorithm to compute their minimal system of generators and we prove they are Cohen-Macaulay and Buchsbaum. Besides, the Gorenstein property is charactized, and their (minimal) Fröbenius vectors are determinated.

math.AC↗
arXiv2014

Computation of Delta sets of numerical monoids

Let $\{a_1,\dots,a_p\}$ be the minimal generating set of a numerical monoid $S$. For any $s\in S$, its Delta set is defined by $Δ(s)=\{l_{i}-l_{i-1}|i=2,\dots,k\}$ where $\{l_1<\dots<l_k\}$ is the set $\{\sum_{i=1}^px_i\,|\, s=\sum_{i=1}^px_ia_i \textrm{ and } x_i\in \N \textrm{ for all }i\}.$ The Delta set of $S$, denoted by $Δ(S)$, is the union of all the sets $Δ(s)$ with $s\in S.$ As proved in [S.T. Chapman, R. Hoyer, and N. Kaplan. Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77 (2009), no. 3, 273--279], there exists a bound $N$ such that $Δ(S)$ is the union of the sets $Δ(s)$ with $s\in S$ and $s<N$. In this work, by using geometrical tools, we obtain a sharpened bound and we give an algorithm to compute $Δ(S)$ from the factorizations of only $a_1$ elements.

math.AC↗
arXiv2012

Affine convex body semigroups

In this paper we present a new kind of semigroups called convex body semigroups which are generated by convex bodies of R^k. They generalize to arbitrary dimension the concept of proportionally modular numerical semigroup of [7]. Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of R^2 are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them.

math.AC↗
arXiv2001

Gröbner $δ$-bases and Gröbner bases for differential operators

This paper deals with the notion of Gröbner $δ$-base for some rings of linear differential operators by adapting the works of W. Trinks, A. Assi, M. Insa and F. Pauer. We compare this notion with the one of Gröbner base for such rings. As an application, and following a previous work of A. Assi, we give some results on finiteness and on flatness of finitely generated left modules over these rings.

math.AG↗