{
  "id": "1704.03392",
  "version": "v1",
  "published": "2017-04-11T16:13:58.000Z",
  "updated": "2017-04-11T16:13:58.000Z",
  "title": "The complexity of the embeddability between torsion-free abelian groups of uncountable size",
  "authors": [
    "Filippo Calderoni"
  ],
  "comment": "14 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that for every uncountable cardinal $\\kappa$ such that $\\kappa^{<\\kappa}=\\kappa$, the quasi-order of embeddability on the $\\kappa$-space of $\\kappa$-sized graphs Borel reduces to the embeddability on the $\\kappa$-space of $\\kappa$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $\\kappa$-space of $\\kappa$-sized $R$-modules, for every $\\mathbb{S}$-cotorsion-free ring $R$ of cardinality less than the continuum. As a consequence we get that all the previous are complete $\\Sigma^1_1$ quasi-orders.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-11T16:13:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15"
    ],
    "keywords": [
      "embeddability",
      "uncountable size",
      "complexity",
      "sized torsion-free abelian groups",
      "sized graphs borel reduces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}