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The multiples of a numerical semigroup

Ignacio Ojeda, José Carlos Rosales

Abstract

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.

The multiples of a numerical semigroup

Abstract

Given two numerical semigroups and we say that is a multiple of if there exists an integer such that . 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 without any quotient of embedding dimension less than , and provide new families with this property.
Paper Structure (6 sections, 28 theorems, 41 equations)

This paper contains 6 sections, 28 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The set $\max \operatorname{M}_d(S)$
  3. The fibers of $\Theta_S^d$
  4. $\operatorname{M}_d(S)-$monoides
  5. The $d-$multiples of $S$ with $\operatorname{M}_d(S)-$embedding dimension one
  6. On numerical semigroups with full quotient rank

Key Result

Lemma 2

Let $S$ and $T$ be two numerical semigroups. Given $d \in \mathbb{N} \setminus \{0\}$, one has that $T$ is $d-$multiple of $S$ if and only if Specifically, in this case, $\operatorname{F}(T) \geq d \operatorname{F}(S)$.

Theorems & Definitions (65)

  • Definition 1
  • Lemma 2
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • ...and 55 more