{
  "id": "1707.07130",
  "version": "v1",
  "published": "2017-07-22T09:43:00.000Z",
  "updated": "2017-07-22T09:43:00.000Z",
  "title": "On a locally compact semitopological $α$-bicyclic monoid",
  "authors": [
    "Serhii Bardyla"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper for every ordinal $\\alpha<\\omega+1$ we describe all locally compact Hausdorff topologies which make $\\alpha$-bicyclic monoid $\\mathcal{B}_{\\alpha}$ a semitopological semigroup. In particular, we prove that there exist exactly $k$ distinct locally compact Hausdorff topologies which make $\\mathcal{B}_{k}$ a semitopological semigroup and the set of all distinct locally compact Hausdorff topologies which make $\\mathcal{B}_{\\omega}$ a semitopological semigroup is countable. Moreover, for every ordinal $\\alpha<\\omega+1$ the set of all locally compact Hausdorff topologies on $\\mathcal{B}_{\\alpha}$ which make $\\mathcal{B}_{\\alpha}$ a semitopological semigroup is linearly ordered by the inclusion. Also we prove that for each ordinal $\\alpha$ the $\\alpha+1$-bicyclic semigroup $\\mathcal{B}_{\\alpha+1}$ is isomorphic to the Bruck extension of the $\\alpha$-bicyclic semigroup $\\mathcal{B}_{\\alpha}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-22T09:43:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bicyclic monoid",
      "distinct locally compact hausdorff topologies",
      "locally compact semitopological",
      "semitopological semigroup",
      "bicyclic semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}