{
  "id": "2407.12533",
  "version": "v1",
  "published": "2024-07-17T13:14:50.000Z",
  "updated": "2024-07-17T13:14:50.000Z",
  "title": "Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces",
  "authors": [
    "Qianxue Liu",
    "Shoufeng Wang"
  ],
  "comment": "40pages",
  "categories": [
    "math.GR",
    "math.QA"
  ],
  "abstract": "As generalizations of inverse semibraces introduced by Catino, Mazzotta and Stefanelli, Miccoli has introduced regular $\\star$-semibraces under the name of involution semibraces and given a sufficient condition under which the associated map to a regular $\\star$-semibrace is a set-theoretic solution of the Yang-Baxter equation. From the viewpoint of universal algebra, regular $\\star$-semibraces are (2,2,1)-type algebras. In this paper we continue to study set-theoretic solutions of the Yang-Baxter equation and regular $\\star$-semibraces. We first consider several kinds of (2,2,1)-type algebras that induced by regular $\\star$-semigroups and give some equivalent characterizations of the statement that they form regular $\\star$-semibraces. Then we give sufficient and necessary conditions under which the associated maps to these (2,2,1)-type algebras are set-theoretic solutions of the Yang-Baxter equation. Finally, as analogues of weak braces defined by Catino, Mazzotta, Miccoli and Stefanelli, we introduce weak $\\star$-braces in the class of regular $\\star$-semibraces, describe their algebraic structures and prove that the associated maps to weak $\\star$-braces are always set-theoretic solutions of the Yang-Baxter equation. The result of the present paper shows that the class of completely regular, orthodox and locally inverse regular $\\star$-semigroups is a source of possibly new set-theoretic solutions of the Yang-Baxter equation. Our results establish the close connection between the Yang-Baxter equation and the classical structural theory of regular $\\star$-semigroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-17T13:14:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16T25",
      "20M17",
      "81R50",
      "16Y99"
    ],
    "keywords": [
      "yang-baxter equation",
      "associated map",
      "semigroups",
      "study set-theoretic solutions",
      "involution semibraces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}