{
  "id": "1510.00596",
  "version": "v1",
  "published": "2015-09-30T20:45:09.000Z",
  "updated": "2015-09-30T20:45:09.000Z",
  "title": "Length of an intersection",
  "authors": [
    "Christian Delhommé",
    "Maurice Pouzet"
  ],
  "comment": "13 pages",
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "A poset $\\bfp$ is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\\em length}, $\\ell(\\bfp)$ of $\\bfp$. We prove that if the vertex set $X$ of $\\bfp$ is infinite, of cardinality $\\kappa$, and the ordering $\\leq$ is the intersection of finitely many partial orderings $\\leq_i$ on $X$, $1\\leq i\\leq n$, then, letting $\\ell(X,\\leq_i)=\\kappa\\multordby q_i+r_i$, with $r_i<\\kappa$, denote the euclidian division by $\\kappa$ (seen as an initial ordinal) of the length of the corresponding poset~:\\[ \\ell(\\bfp)< \\kappa\\multordby\\bigotimes_{1\\leq i\\leq n}q_i+ \\Big|\\sum_{1\\leq i\\leq n} r_i\\Big|^+ \\] where $|\\sum r_i|^+$ denotes the least initial ordinal greater than the ordinal $\\sum r_i$. This inequality is optimal (for $n\\geq 2$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-30T20:45:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A06",
      "06A07",
      "03F15"
    ],
    "keywords": [
      "intersection",
      "linear extensions",
      "initial ordinal greater",
      "partial orderings",
      "euclidian division"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151000596D"
    }
  }
}