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Onthe computation of the MED closure of a numerical semigroup

Jorge Jiménez Urroz, José M. Tornero

TL;DR

This work studies maximal embedding dimension (MED) numerical semigroups and introduces the MED closure MED(S) as the smallest MED semigroup containing a given S with the same multiplicity $m(S)$. It provides two explicit computational methods to obtain MED(S): (i) an Apéry-saturation procedure based on the Apéry set with respect to $m(S)$, and (ii) an effective, residue-class based algorithm that yields a linear-time procedure in the size of the minimal generating set and offers a complexity bound. A key theoretical link established is that MED(S) occurs precisely when $m(S)$ is Arf, and that Arf semigroups are MED; the work also discusses controlled enlargements of semigroups and conductor bounds. The results supply practical algorithms and bounds for constructing MED closures, with illustrative examples, advancing the understanding of the interplay between Apéry sets, Arf property, and MED structure.

Abstract

Maximally embedding dimension (MED) numerical semigroups are a wide and interesting family, with some remarkable algebraic and combinatorial properties. Associated to any numerical semigroup one can construct a MED closure, as it is well known. This paper shows two different explicit methods to construct this closure which also sheds new light on the very nature of this object.

Onthe computation of the MED closure of a numerical semigroup

TL;DR

This work studies maximal embedding dimension (MED) numerical semigroups and introduces the MED closure MED(S) as the smallest MED semigroup containing a given S with the same multiplicity . It provides two explicit computational methods to obtain MED(S): (i) an Apéry-saturation procedure based on the Apéry set with respect to , and (ii) an effective, residue-class based algorithm that yields a linear-time procedure in the size of the minimal generating set and offers a complexity bound. A key theoretical link established is that MED(S) occurs precisely when is Arf, and that Arf semigroups are MED; the work also discusses controlled enlargements of semigroups and conductor bounds. The results supply practical algorithms and bounds for constructing MED closures, with illustrative examples, advancing the understanding of the interplay between Apéry sets, Arf property, and MED structure.

Abstract

Maximally embedding dimension (MED) numerical semigroups are a wide and interesting family, with some remarkable algebraic and combinatorial properties. Associated to any numerical semigroup one can construct a MED closure, as it is well known. This paper shows two different explicit methods to construct this closure which also sheds new light on the very nature of this object.
Paper Structure (6 sections, 16 theorems, 82 equations)

This paper contains 6 sections, 16 theorems, 82 equations.

Table of Contents

  1. Essential concepts
  2. MED closure of a numerical semigroup
  3. Interlude: How to enlarge a numerical semigroup (carefully)
  4. Computing $\operatorname{MED}(S)$: The Apéry saturation
  5. Computing $\operatorname{MED}(S)$: An effective solution
  6. Final remarks

Key Result

Proposition 1.3

Every numerical semigroup $S$ can be written in the form with $\gcd(a_1, \ldots ,a_k)=1$. We will call $\{a_1, \ldots ,a_k\}$ a set of generators of $S$. There exists a unique minimal set of generators of $S$.

Theorems & Definitions (48)

  • Definition 1.1
  • Example 1.2
  • Proposition 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 38 more