{
  "id": "1909.13100",
  "version": "v1",
  "published": "2019-09-28T13:54:44.000Z",
  "updated": "2019-09-28T13:54:44.000Z",
  "title": "On proximal relations in transformation semigroups arising from generalized shifts",
  "authors": [
    "Fatemah Ayatollah Zadeh Shirazi",
    "Amir Fallahpour",
    "Mohammad Reza Mardanbeigi",
    "Zahra Nili Ahmadabadi"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "For a finite discrete topological space $X$ with at least two elements, a nonempty set $\\Gamma$, and a map $\\varphi:\\Gamma\\to\\Gamma$, $\\sigma_\\varphi:X^\\Gamma\\to X^\\Gamma$ with $\\sigma_\\varphi((x_\\alpha)_{\\alpha\\in\\Gamma})= (x_{\\varphi(\\alpha)})_{\\alpha\\in\\Gamma}$ (for $(x_\\alpha)_{\\alpha\\in\\Gamma}\\in X^\\Gamma$) is a generalized shift. In this text for $\\mathcal{S}=\\{\\sigma_\\psi:\\psi\\in\\Gamma^\\Gamma\\}$ and $\\mathcal{H}=\\{\\sigma_\\psi: \\Gamma\\mathop{\\rightarrow}\\limits^{\\psi}\\Gamma$ is bijective$\\}$ we study proximal relations of transformation semigroups $(\\mathcal{S},X^\\Gamma)$ and $(\\mathcal{H},X^\\Gamma)$. Regarding proximal relation we prove: \\[P({\\mathcal S},X^\\Gamma)=\\{((x_\\alpha)_{\\alpha\\in\\Gamma},(y_\\alpha)_{\\alpha\\in\\Gamma}) \\in X^\\Gamma\\times X^\\Gamma: \\exists\\beta\\in\\Gamma\\:(x_\\beta=y_\\beta)\\}\\] and $P({\\mathcal H},X^\\Gamma)\\subseteq \\{((x_\\alpha)_{\\alpha\\in\\Gamma},(y_\\alpha)_{\\alpha\\in\\Gamma}) \\in X^\\Gamma\\times X^\\Gamma: \\{\\beta\\in\\Gamma:x_\\beta=y_\\beta\\}$ is infinite~$\\}\\cup\\{ (x,x):x\\in \\mathcal{X}\\}$. \\\\ Moreover, for infinite $\\Gamma$, both transformation semigroups $({\\mathcal S},X^\\Gamma)$ and $({\\mathcal H},X^\\Gamma)$ are regionally proximal, i.e., $Q({\\mathcal S},X^\\Gamma)=Q({\\mathcal H},X^\\Gamma)=X^\\Gamma \\times X^\\Gamma$, also for sydetically proximal relation we have $L({\\mathcal H},X^\\Gamma)=\\{((x_\\alpha)_{\\alpha\\in\\Gamma},(y_\\alpha)_{\\alpha\\in\\Gamma}) \\in X^\\Gamma\\times X^\\Gamma: \\{\\gamma\\in\\Gamma:x_\\gamma\\neq y_\\gamma\\}$ is finite$\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-28T13:54:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H20"
    ],
    "keywords": [
      "transformation semigroups arising",
      "generalized shift",
      "study proximal relations",
      "finite discrete topological space",
      "regarding proximal relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}