{
  "id": "1709.03700",
  "version": "v1",
  "published": "2017-09-12T05:59:58.000Z",
  "updated": "2017-09-12T05:59:58.000Z",
  "title": "Uniqueness of directed complete posets based on Scott closed set lattices",
  "authors": [
    "Dongsheng Zhao",
    "Luoshan Xu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "In analogy to a result due to Drake and Thron about topological spaces, this paper studies the dcpos (directed complete posets) which are fully determined, among all dcpos, by their lattices of all Scott-closed subsets (such a dcpo will be called a $C_{\\sigma}$-unique dcpo, or $C_{\\sigma}$-unique in short). We shall introduce the notions of down-linear element and quasicontinuous element in dcpos, and use them to prove that dcpos of certain class, including all quasicontinuous dcpos as well as Johnstone's and Kou's examples, are $C_{\\sigma}$-unique. As a consequence, $C_{\\sigma}$-unique dcpos with their Scott topologies need not be bounded sober. These results will help to obtain a complete characterization of $C_{\\sigma}$-unique dcpos in the future.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-12T05:59:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scott closed set lattices",
      "directed complete posets",
      "unique dcpo",
      "uniqueness",
      "paper studies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}