{
  "id": "0911.2778",
  "version": "v1",
  "published": "2009-11-14T15:11:01.000Z",
  "updated": "2009-11-14T15:11:01.000Z",
  "title": "On $\\ell^{p}$-like equivalence relations",
  "authors": [
    "Tamás Mátrai"
  ],
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "For $f \\colon [0,1] \\rar \\real^{+}$, consider the relation $\\mathbf{E}_{f}$ on $[0,1]^{\\omega}$ defined by $(x_{n}) \\mathbf{E}_{f} (y_{n}) \\Leftrightarrow \\sum_{n < \\omega} f(|y_{n} - x_{n}|) < \\infty.$ We study the Borel reducibility of Borel equivalence relations of the form $\\mathbf{E}_{f}$. Our results indicate that for every $1 \\leq p < q < \\infty$, the order $\\leq_{B}$ of Borel reducibility on the set of equivalence relations $\\{\\bE \\colon \\bE_{\\Id^{p}} \\leq_{B} \\bE \\leq_{B} \\bE_{\\Id^{q}}\\}$ is more complicated than expected, e.g. consistently every linear order of cardinality continuum embeds into it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-14T15:11:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "46A45"
    ],
    "keywords": [
      "borel reducibility",
      "cardinality continuum embeds",
      "borel equivalence relations",
      "linear order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.2778M"
    }
  }
}