{
  "id": "1502.04144",
  "version": "v1",
  "published": "2015-02-13T22:36:21.000Z",
  "updated": "2015-02-13T22:36:21.000Z",
  "title": "Superstability in Tame Abstract Elementary Classes",
  "authors": [
    "Monica VanDieren"
  ],
  "comment": "22 pages, 6 figures, author partially supported for this work by grant DMS 0801313 of the National Science Foundation",
  "categories": [
    "math.LO"
  ],
  "abstract": "In this paper we address a problem posed by Shelah in 1999 to find a suitable notion for superstability for abstract elementary classes in which limit models of cardinality $\\mu$ are saturated. Theorem 1. Suppose that $\\mathcal{K}$ is a $\\chi$-tame abstract elementary class with no maximal models satisfying the joint embedding property and the amalgamation property. Suppose $\\mu$ is a cardinal with $\\mu\\geq\\beth_{(2^{LS(\\mathcal{K})+\\chi})^+}$. Let $M$ be a model of cardinality $\\mu$. If $\\mathcal{K}$ is both $\\chi$-stable and $\\mu$-stable and satisfies the $\\mu$-superstability assumptions, then any two $\\mu$-limit models over $M$ are isomorphic over $M$. Moreover, we identify sufficient conditions for superlimit models of cardinality $\\mu$ to exist, for model homogeneous models to be superlimit, and for a union of saturated models to be saturated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-13T22:36:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C95"
    ],
    "keywords": [
      "tame abstract elementary classes",
      "limit models",
      "cardinality",
      "amalgamation property",
      "maximal models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}