{
  "id": "1507.01990",
  "version": "v1",
  "published": "2015-07-07T23:46:30.000Z",
  "updated": "2015-07-07T23:46:30.000Z",
  "title": "Superstability and Symmetry",
  "authors": [
    "Monica M. VanDieren"
  ],
  "doi": "10.13140/RG.2.1.2619.5040",
  "categories": [
    "math.LO"
  ],
  "abstract": "In this paper we extend the work of \\cite{ShVi}, \\cite{Va1}, \\cite{Va2}, and \\cite{GVV} by weakening the structural assumptions on $\\mathcal{K}_{\\mu^+}$ to something more closely resembling superstability from first order logic in order to derive the uniqueness of limit models of cardinality $\\mu$. Moreover, we identify a necessary and sufficient condition for symmetry of non-splitting that involves reduced towers. This new condition is particularly interesting since it does not have a pre-established first-order analog. Additionally, this condition provides a mechanism for deriving symmetry in abstract elementary classes without having to assume set-theoretic assumptions or tameness: Corollary: Suppose that $\\mathcal{K}$ satisfies the amalgamation and joint embedding properties and $\\mu$ is a cardinal $\\geq\\beth_{(2^{Hanf(\\mathcal{K})})^+}$. If $\\mathcal{K}$ is categorical in $\\lambda=\\mu^+$, then $\\mathcal{K}$ has symmetry for non-$\\mu$-splitting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-07T23:46:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first order logic",
      "abstract elementary classes",
      "assume set-theoretic assumptions",
      "pre-established first-order analog",
      "limit models"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150701990V"
    }
  }
}