{
  "id": "1408.3383",
  "version": "v3",
  "published": "2014-08-14T18:54:56.000Z",
  "updated": "2015-09-19T20:52:31.000Z",
  "title": "Tameness, Uniqueness and amalgamation",
  "authors": [
    "Adi Jarden"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking $\\lambda^+$-frame from a semi-good non-forking $\\lambda$-frame. But the classes $K_{\\lambda^+}$ and $\\preceq \\restriction K_{\\lambda^+}$ are replaced: $K_{\\lambda^+}$ is restricted to the saturated models and the partial order $\\preceq \\restriction K_{\\lambda^+}$ is restricted to the partial order $\\preceq^{NF}_{\\lambda^+}$. Here, we avoid the restriction of the partial order $\\preceq \\restriction K_{\\lambda^+}$, assuming that every saturated model (in $\\lambda^+$ over $\\lambda$) is an amalgamation base and $(\\lambda,\\lambda^+)$-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that $M \\preceq M^+$ if and only if $M \\preceq^{NF}_{\\lambda^+}M^+$, provided that $M$ and $M^+$ are saturated models. We present sufficient conditions for three good non-forking $\\lambda^+$-frames: one relates to all the models of cardinality $\\lambda^+$ and the two others relate to the saturated models only. By an `unproven claim' of Shelah, if we can repeat this procedure $\\omega$ times, namely, `derive' good non-forking $\\lambda^{+n}$ frame for each $n<\\omega$ then the categoricity conjecture holds. Vasey applies one of our main theorems in a proof of the categoricity conjecture under the above `unproven claim' of Shelah and more assumptions. In [Jrprime], we apply the main theorem in a proof of the existence of primeness triples.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-15T07:05:48.000Z",
      "abstract": "We combine two approaches to the study of classification theory of AECs: \\begin{enumerate} \\item that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and \\item that of Grossberg: (studying non-splitting) assuming the amalgamation property and tameness. \\end{enumerate} We study non-forking frames assuming the existence of uniqueness triples and tameness for non-forking types. However as in \\cite{shh}, we do not assume the amalgamation property in general (but in a specific cardinality $\\lambda$). In \\cite{jrsh875} we derive a good non-forking $\\lambda^+$-frame from a semi-good non-forking $\\lambda$-frame. But the classes $K_{\\lambda^+}$ and $\\preceq \\restriction K_{\\lambda^+}$ are replaced: $K_{\\lambda^+}$ is restricted to the saturated models and the partial order $\\preceq \\restriction K_{\\lambda^+}$ is restricted to the partial order $\\preceq^{NF}_{\\lambda^+}$. Here, we avoid the restriction of the partial order $\\preceq \\restriction K_{\\lambda^+}$, assuming $(\\lambda,\\lambda^+)$-tameness for non-forking types, (in addition to the hypotheses of \\cite{jrsh875}): Theorem \\ref{we can use NF} states that $M \\preceq M^+$ if and only if $M \\preceq^{NF}_{\\lambda^+}M^+$, provided that $M$ and $M^+$ are models of cardinality $\\lambda^+$. Our results have been applied by Jarden in: \\begin{enumerate} \\item \\cite{jrtame}: avoiding the restriction of the class $K_{\\lambda^+}$ and deriving a good non-forking $\\lambda^+$-frame, \\item Corollary \\ref{the relations are equivalent iff amalgamation holds}: proving the amalgamation property in $\\lambda^+$, \\item \\cite{jrprime}: proving the existence of primeness triples and \\item again \\cite{jrtame}: proving the serial $(\\lambda,\\lambda^+)$-continuity property. \\end{enumerate}",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-09-19T20:52:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "amalgamation property",
      "partial order",
      "uniqueness triples",
      "non-forking types",
      "primeness triples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3383J"
    }
  }
}