{
  "id": "1508.03252",
  "version": "v1",
  "published": "2015-08-13T15:42:39.000Z",
  "updated": "2015-08-13T15:42:39.000Z",
  "title": "Transferring symmetry downward and applications",
  "authors": [
    "Monica M. VanDieren",
    "Sebastien Vasey"
  ],
  "comment": "25 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "For $K$ an abstract elementary class satisfying the amalgamation property, we prove a downward transfer of the symmetry property for splitting (previously isolated by the first author). This allows us to deduce uniqueness of limit models from categoricity in a cardinal of high-enough cofinality, improving on a 16-year-old result of Shelah: $\\mathbf{Theorem}$ Suppose $\\lambda$ and $\\mu$ are cardinals so that $\\text{cf}(\\lambda)>\\mu\\geq\\text{LS}(K)$ and assume that $K$ has no maximal models and is categorical in $\\lambda$. If $M_0,M_1,M_2 \\in K_\\mu$ are such that both $M_1$ and $M_2$ are limit models over $M_0$, we have that $M_1\\cong_{M_0}M_2$. Another application of the symmetry transfer utilizes tameness (a locality property for types) and improves on the work of Will Boney and the second author: $\\mathbf{Theorem}$ Let $\\mu \\ge \\text{LS} (K)$. If $K$ is $\\mu$-superstable and $\\mu$-tame, then: * If $M_0, M_1, M_2 \\in K_\\mu$ are such that both $M_1$ and $M_2$ are limit models over $M_0$, then $M_1 \\cong_{M_0} M_2$. * For any $\\lambda > \\mu$, the union of an increasing chain of $\\lambda$-saturated models is $\\lambda$-saturated. * There exists a unique type-full good $\\mu^+$-frame with underlying class the saturated models in $K_{\\mu^+}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-13T15:42:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55"
    ],
    "keywords": [
      "transferring symmetry downward",
      "limit models",
      "application",
      "symmetry transfer utilizes tameness",
      "saturated models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150803252V"
    }
  }
}