{
  "id": "math/0703045",
  "version": "v5",
  "published": "2007-03-01T20:33:32.000Z",
  "updated": "2021-09-14T14:58:58.000Z",
  "title": "Definable groups for dependent and 2-dependent theories",
  "authors": [
    "Saharon Shelah"
  ],
  "comment": "Version of 2017",
  "journal": "Sarajevo J. Math., 13(25)(1), (2017) 3-25",
  "doi": "10.5644/SJM.13.1.01",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $T$ be a (first order complete) dependent theory, ${\\mathfrak{C}}$ a $\\bar\\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\\bar\\kappa$ type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is $\\leq 2^{|T|}$. We then deal with 2-dependent theories, a wider class of first order theories.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-09-08T02:56:21.000Z",
      "abstract": "Let T be a (first order complete) dependent theory, C a kappa-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are the union of < kappa type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is <= 2^{|T|} . We then deal with 2-dependent theories, a wider class of first order theories.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2021-09-14T14:58:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "definable groups",
      "first order complete",
      "kappa type definable subsets",
      "bounded index",
      "first order theories"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3045S"
    }
  }
}