{
  "id": "1311.4053",
  "version": "v2",
  "published": "2013-11-16T12:56:41.000Z",
  "updated": "2019-09-17T15:06:53.000Z",
  "title": "A dichotomy for $D$-rank 1 types in simple theories",
  "authors": [
    "Ziv Shami"
  ],
  "comment": "One corollary added at the end of the article, more details added in proofs and minor correction made in the proof of Corollary 3.21",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove a dichotomy for $D$-rank 1 types in simple theories that generalizes Buechler's dichotomy for $D$-rank 1 minimal types: every such type is either 1-based or its algebraic closure, by a single formula, almost contains a non-algebraic formula that belongs to a non-forking extension of the type. In addition we prove that a densely-1 based type of $D$-rank 1 is 1-based. We also observe that for a hypersimple unidimensional theory the existence of a non-algebraic stable type implies stability (and thus superstability).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-16T12:56:41.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2019-09-17T15:06:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple theories",
      "non-algebraic stable type implies stability",
      "hypersimple unidimensional theory",
      "generalizes buechlers dichotomy",
      "algebraic closure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4053S"
    }
  }
}