{
  "id": "2202.00920",
  "version": "v1",
  "published": "2022-02-02T09:03:22.000Z",
  "updated": "2022-02-02T09:03:22.000Z",
  "title": "The complexity of a numerical semigroup",
  "authors": [
    "J. I. García-García",
    "M. A. Moreno-Frías",
    "J. C. Rosales",
    "A. Vigneron-Tenorio"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $S$ and $\\Delta$ be numerical semigroups. A numerical semigroup $S$ is an $\\mathbf{I}(\\Delta)$-{\\it semigroup} if $S\\backslash \\{0\\}$ is an ideal of $\\Delta$. We will denote by $\\mathcal{J}(\\Delta)=\\{S \\mid S \\text{ is an $\\mathbf{I}(\\Delta)$-semigroup} \\}.$ We will say that $\\Delta$ is {\\it an ideal extension of } $S$ if $S\\in \\mathcal{J}(\\Delta).$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-02T09:03:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical semigroup",
      "complexity",
      "ideal extension",
      "recursively denote"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}