{
  "id": "1908.04566",
  "version": "v1",
  "published": "2019-08-13T10:15:46.000Z",
  "updated": "2019-08-13T10:15:46.000Z",
  "title": "On the lattice of weak topologies on the bicyclic monoid with adjoined zero",
  "authors": [
    "Serhii Bardyla",
    "Oleg Gutik"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "A Hausdorff topology $\\tau$ on the bicyclic monoid $\\mathcal{C}^0$ is called {\\em weak} if it is contained in the coarsest inverse semigroup topology on $\\mathcal{C}^0$. We show that the lattice $\\mathcal{W}$ of all weak shift-continuous topologies on $\\mathcal{C}^0$ is isomorphic to the lattice of all shift-invariant filters on $\\omega$ with an attached element $1$ endowed with the following partial order: $\\mathcal{F}\\leq \\mathcal{G}$ iff $\\mathcal{G}=1$ or $\\mathcal{F}\\subset \\mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\\mathcal{W}$. In particular, we proved that $\\mathcal{W}$ contains an antichain of cardinality $2^{\\mathfrak{c}}$ and a well-ordered chain of cardinality $\\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\\mathfrak{t}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-13T10:15:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bicyclic monoid",
      "adjoined zero",
      "coarsest inverse semigroup topology",
      "well-ordered chain",
      "weak shift-continuous topologies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}