{
  "id": "1507.02118",
  "version": "v1",
  "published": "2015-07-08T12:13:42.000Z",
  "updated": "2015-07-08T12:13:42.000Z",
  "title": "Uniqueness of Limit Models in Classes with Amalgamation",
  "authors": [
    "Rami Grossberg",
    "Monica VanDieren",
    "Andres Villaveces"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove: Main Theorem: Let $\\mathcal{K}$ be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality $\\mu$. Let $\\mu$ be a cardinal above the the L\\\"owenheim-Skolem number of the class. If $\\mathcal{K}$ is $\\mu$-Galois-stable, has no $\\mu$-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two $(\\mu,\\sigma_\\ell)$-limits over $M$, for $\\ell\\in\\{1,2\\}$, are isomorphic over $M$. This theorem extends results of Shelah from \\cite{Sh394}, \\cite{Sh576}, \\cite{Sh600}, Kolman and Shelah in \\cite{KoSh} and Shelah and Villaveces from \\cite{ShVi}. A preliminary version of our uniqueness theorem, which was circulated in 2006, was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes in \\cite{GrVa2}. Preprints of this paper have also influenced the Ph.D. theses of Drueck \\cite{Dr} and Zambrano \\cite{Za}. This paper also serves the expository role of presenting together the arguments in \\cite{Va1} and \\cite{Va2} in a more natural context in which the amalgamation property holds and this work provides an approach to the uniqueness of limit models that does not rely on Ehrenfeucht-Mostowski constructions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-08T12:13:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "limit models",
      "tame abstract elementary classes",
      "shelahs categoricity conjecture",
      "amalgamation property holds",
      "theorem extends results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150702118G"
    }
  }
}