{
  "id": "math/0509338",
  "version": "v3",
  "published": "2005-09-15T03:53:31.000Z",
  "updated": "2015-02-05T21:23:31.000Z",
  "title": "Uniqueness of Limit Models in Classes with Amalgamation",
  "authors": [
    "R. Grossberg",
    "M. VanDieren",
    "A. Villaveces"
  ],
  "comment": "29 pages, 1 figure",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the L\\\"owenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M. This theorem extends results of Shelah, Kolman and Shelah, and Shelah and Villaveces. A preliminary version of our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-04-13T09:20:14.000Z",
      "abstract": "We prove: [Main Theorem] Let K be an AEC and m > LS(K). Suppose K satisfies the disjoint amalgamation property for models of cardinality m. If K is m-Galois-stable, does not have long splitting chains, and satisfies locality of splitting, then any two (m,s_l)$-limits over a model M (for l in {1,2}) are isomorphic over M. This result extends results of Shelah from [Sh 394], [Sh 576], [Sh 600], Kolman and Shelah in [KoSh] and Shelah and Villaveces from [ShVi]. Our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame AEC in [GrVa 2].",
      "comment": "20 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-02-05T21:23:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45",
      "03C35",
      "03C48"
    ],
    "keywords": [
      "limit models",
      "shelahs categoricity conjecture",
      "disjoint amalgamation property",
      "result extends results",
      "main theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9338G"
    }
  }
}