{
  "id": "2207.08962",
  "version": "v1",
  "published": "2022-07-18T22:18:47.000Z",
  "updated": "2022-07-18T22:18:47.000Z",
  "title": "$p$-numerical semigroups with $p$-symmetric properties",
  "authors": [
    "Takao Komatsu",
    "Haotian Ying"
  ],
  "categories": [
    "math.CO",
    "math.AC",
    "math.NT"
  ],
  "abstract": "e largest integer such that the linear equation $a_1 x_1+\\cdots+a_k x_k=n$ ($a_1,\\dots,a_k$ are given positive integers with $\\gcd(a_1,\\dots,a_k)=1$) does not have a non-negative integer solution $(x_1,\\dots,x_k)$. The generalized Frobenius number (called the $p$-Frobenius number) is the largest integer such that this linear equation has at most $p$ solutions. That is, when $p=0$, the $0$-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss $p$-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer $p$, $p$-gaps, $p$-symmetric semigroups, $p$-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When $p=0$, they correspond to the original gaps, symmetric semigroups, and pseudo-symmetric semigroups, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-18T22:18:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical semigroups",
      "symmetric properties",
      "pseudo-symmetric semigroups",
      "linear equation",
      "largest integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}