{
  "id": "1304.7714",
  "version": "v1",
  "published": "2013-04-29T17:00:54.000Z",
  "updated": "2013-04-29T17:00:54.000Z",
  "title": "Forcing With Copies of Countable Ordinals",
  "authors": [
    "Milos Kurilic"
  ],
  "comment": "15 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let \\alpha be a countable ordinal and \\P(\\alpha) the collection of its subsets isomorphic to \\alpha. We show that the separative quotient of the set \\P (\\alpha) ordered by the inclusion is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(\\omega ^\\gamma)/I(\\omega ^\\gamma), where \\gamma is a limit ordinal or 1 and I(\\omega ^\\gamma) the corresponding ordinal ideal. Moreover, the poset \\P(\\alpha) is forcing equivalent to a two-step iteration P(\\omega)/Fin * \\pi, where \\pi is an \\omega_1-closed separative pre-order in each extension by P(\\omega)/Fin and, if the distributivity number is equal to\\omega_1, to P(\\omega)/Fin. Also we analyze the quotients over ordinal ideals P(\\omega ^\\delta)/I(\\omega ^\\delta) and their distributivity and tower numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-29T17:00:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E40",
      "03E10",
      "03E35",
      "03E17",
      "06A06"
    ],
    "keywords": [
      "countable ordinal",
      "distributivity number",
      "subsets isomorphic",
      "two-step iteration",
      "tower numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.7714K"
    }
  }
}