Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical semigroups with monotone Apéry set and fixed multiplicity and ratio

Aureliano M. Robles-Pérez, José Carlos Rosales

Abstract

We characterise the numerical semigroups with a monotone Apéry set (MANS-semigroups for abbreviate). Moreover, we describe the families of MANS-semigroups when we set the multiplicity and the ratio.

Numerical semigroups with monotone Apéry set and fixed multiplicity and ratio

Abstract

We characterise the numerical semigroups with a monotone Apéry set (MANS-semigroups for abbreviate). Moreover, we describe the families of MANS-semigroups when we set the multiplicity and the ratio.
Paper Structure (10 sections, 30 theorems, 13 equations, 2 figures)

This paper contains 10 sections, 30 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two-embedding dimensional case
  3. Three-embedding dimensional case
  4. Minimal generators
  5. Frobenius problem
  6. Pseudo-Frobenius numbers
  7. Irreducibility
  8. General case
  9. Apéry sets
  10. The tree of MANS-semigroups with multiplicity and ratio fixed

Key Result

Lemma 2.1

Let $S=\langle n_1<n_2<\cdots<n_e \rangle$ be a numerical semigroup with $n_1\geq 2$. If $S$ is a MANS-semigroup, then $n_2 \bmod n_1 < n_3 \bmod n_1 < \cdots < n_e \bmod n_1$.

Figures (2)

  • Figure 1: $G(\mathcal{MA}(5,6))$
  • Figure 2: $G(\mathcal{MA}(5,11))$

Theorems & Definitions (66)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 56 more