{
  "id": "1804.00932",
  "version": "v2",
  "published": "2018-04-03T12:36:49.000Z",
  "updated": "2018-07-16T16:33:06.000Z",
  "title": "Countable models of the theories of Baldwin-Shi hypergraphs and their regular types",
  "authors": [
    "Danul K. Gunatilleka"
  ],
  "comment": "12 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$. Restricting our attention to when the rank $\\delta$ is rational valued, we show that each countable model of the theory of a given Baldwin-Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class of finite structures with the inherited notion of strong substructure. We introduce a notion of dimension for a model and show that there is a an elementary chain $\\{\\mathfrak{M}_{\\beta}:\\beta<\\omega+1\\}$ of countable models of the theory of a fixed Baldwin-Shi hypergraph with $\\mathfrak{M}_{\\beta}\\preccurlyeq\\mathfrak{M}_\\gamma$ if and only if the dimension of $\\mathfrak{M}_\\beta$ is at most the dimension of $\\mathfrak{M}_\\gamma$ and that each countable model is isomorphic to some $\\mathfrak{M}_\\beta$. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on the work of Brody and Laskowski, we use these structures to give an example of a pseudofinite, $\\omega$-stable theory with a non-locally modular regular type, answering a question of Pillay in $[9]$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-07-16T16:33:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C65",
      "03C45"
    ],
    "keywords": [
      "countable model",
      "non-locally modular regular type",
      "generic structure built",
      "finite structures",
      "strong substructure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}