{
  "id": "2312.04162",
  "version": "v1",
  "published": "2023-12-07T09:24:33.000Z",
  "updated": "2023-12-07T09:24:33.000Z",
  "title": "Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order",
  "authors": [
    "Gianluca Paolini",
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete, in fact, more strongly, we can $\\mathfrak{L}_{\\omega_1, \\omega}$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\\mathrm{Th}(\\mathbb{Z}^{(\\omega)})$ is a complete analytic quasi-order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-07T09:24:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "20K20"
    ],
    "keywords": [
      "complete analytic quasi-order",
      "faithfully borel complete",
      "pure embeddability",
      "countable torsion-free abelian groups",
      "additional properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}