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José Carlos Rosales

Publications and source records attributed to José Carlos Rosales.

At least 19 recordsLinked to original sources
arXiv2026

On Numerical Semigroups with Fixed Quotient

Let $Δ$ be a numerical semigroup and let $d\ge 2$ be an integer. We study the fiber of the quotient map \(S\mapsto S/d\) over $Δ$. We describe its elements as semigroups of the form $\langle X\rangle+dΔ$, for suitable finite sets $X\subseteqΔ$, and then analyze explicit and computable regions of this fiber. In particular, we introduce a family $Δ_d(a)$ of multiples with prescribed quotient and compute its generators, classical invariants, Apéry sets, and presentations. We also show that this construction preserves Wilf's inequality and controls the depth. Finally, we introduce the $\mathcal{M}_d(Δ)$-rank, determine its maximal value in the fiber, relate it to the ordinary embedding dimension, characterize the rank-one elements, and give closed formulas for their Frobenius-type invariants and pseudo-Frobenius numbers.

math.AC↗
arXiv2024

The multiples of a numerical semigroup

Given two numerical semigroups $S$ and $T$ we say that $T$ is a multiple of $S$ if there exists an integer $d \in \mathbb{N} \setminus \{0\}$ such that $S = \{x \in \mathbb{N} \mid d x \in T\}$. In this paper we study the family of multiples of a (fixed) numerical semigroup. We also address the open problem of finding numerical semigroups of embedding dimension $e$ without any quotient of embedding dimension less than $e$, and provide new families with this property.

math.GR↗
arXiv2023

Arithmetic varieties of numerical semigroups

In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the root three associated with an arithmetic variety. This tree is not locally finite; however, if the Frobenius number is fixed, the tree has finitely many nodes and algorithms can be developed. All algorithms provided in this article include their (non-debugged) implementation in GAP.

math.AC↗
arXiv2022

The extended Frobenius problem for Lucas series incremented by a Lucas number

We study the extended Frobenius problem for sequences of the form $\{l_a\}\cup\{l_a+l_n\}_{n\in\mathbb{N}}$ and $\{l_a+l_n\}_{n\in\mathbb{N}}$, where $\{l_n\}_{n\in\mathbb{N}}$ is the Lucas series and $l_a$ is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf's conjecture.

math.NT↗
arXiv2022

On sets related to integer partitions with quasi-required elements and disallowed elements

Given a set A of non-negative integers and a set B of positive integers,we are interested in computing all sets C (of positive integers) that are minimal in the family of sets K (of positive integers) such that (i) K contains no elements generated by non-negative integer linear combinations of elements in A and (ii) for any partition of an element in B there is at least one summand that belongs to K. To solve this question, we translate it into a numerical semigroups problem.

math.NT↗
arXiv2021

Modular Frobenius pseudo-varieties

If $m \in \mathbb{N}$ and $A$ is a finite subset of $\bigcup_{k \in \mathbb{N} \setminus \{0,1\}} \{1,\ldots,m-1\}^k$, then we denote by \begin{align*} \mathscr{C}(m,A) = \left\{S\in \mathscr{S}_m \mid s_1+\cdots+s_k-m \in S \mbox{ if } (s_1,\ldots,s_k)\in S^k \mbox{ and }\right. \\ \left.(s_1 \bmod m, \ldots, s_k \bmod m)\in A \right\}. \end{align*} In this work we prove that $\mathscr{C}(m,A)$ is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $\mathscr{C}(m,A)$ and to compute all the elements of $\mathscr{C}(m,A)$ with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $\mathscr{C}(m,A)$ when $A=\{1,\ldots,m-1\}^3$, $A=\{(1,1),\ldots,(m-1,m-1)\}$, and $A=\{1,\ldots,m-1\}^2 \setminus \{(1,1),\ldots,(m-1,m-1)\}$, respectively.

math.GR↗
arXiv2020

The arithmetic extensions of a numerical semigroup

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.

math.AC↗
arXiv2017

On the enumeration of the set of atomic numerical semigroups with fixed Frobenius number

A numerical semigroup is irreducible if it cannot be obtained as intersection of two numerical semigroups containing it properly. If we only consider numerical semigroups with the same Frobenius number, that concept is generalized to atomic numerical semigroup. Based on a previous one developed to obtain all irreducible numerical semigroups with a fixed Frobenius number, we present an algorithm to compute all atomic numerical semigroups with a fixed Frobenius number.

math.GR↗
arXiv2017

On a transport problem and monoids of non-negative integers

A problem about how to transport profitably a group of cars leads us to study the set $T$ formed by the integers $n$ such that the system of inequalities, with non-negative integer coefficients, $$a_1x_1 +\cdots+ a_px_p + α\leq n \leq b_1x_1 +\cdots+ b_px_p - β$$ has at least one solution in ${\mathbb N}^p$. We will see that $T\cup\{0\}$ is a submonoid of $({\mathbb N},+)$. Moreover, we show algorithmic processes to compute $T$.

math.GR↗
arXiv2016

Frobenius restricted varieties in numerical semigroups

The common behaviour of many families of numerical semigroups led up to defining, firstly, the Frobenius varieties and, secondly, the (Frobenius) pseudo-varieties. However, some interesting families are still out of these definitions. To overcome this situation, here we introduce the concept of restricted Frobenius variety (or R-variety). We will generalize most of the results for varieties and pseudo-varieties to R-varieties. In particular, we will study the tree structure that arise within them.

math.GR↗
arXiv2016

Numerical semigroups in a problem about economic incentives for consumers

Motivated by a promotion to increase the number of musical downloads, we introduce the concept of $C$-incentive and show an algorithm that compute the smallest $C$-incentive containing a subset $X \subseteq {\mathbb N}$. On the other hand, in order to study $C$-incentives, we see that we can focus on numerical $C$-incentives. Then, we establish that the set formed by all numerical $C$-incentives is a Frobenius pseudo-variety and we show an algorithmic process to recurrently build such a pseudo-variety.

math.GR↗
arXiv2016

A combinatorial problem and numerical semigroups

Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets $C$ of positive integers that fulfill the following conditions: 1) the cardinality of $C$ is equal to $g$; 2) if $x,y\in \mathbb{N} \setminus \{0\}$ and $x+y\in C$, then $C \cap \{x,y\} \neq \emptyset$; 3) if $x \in C$ and $\frac{x-b_i}{a_i} \in \mathbb{N} \setminus \{0\}$ for some $i\in \{1,\ldots,n\}$, then $\frac{x-b_i}{a_i} \in C$; 4) $X \cap C = \emptyset$.

math.CO↗