{
  "id": "2309.13783",
  "version": "v1",
  "published": "2023-09-25T00:23:38.000Z",
  "updated": "2023-09-25T00:23:38.000Z",
  "title": "Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem",
  "authors": [
    "Gábor Czédli"
  ],
  "comment": "11 pages, 1 figure",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "Let FD(3) denote the free distributive lattice on three generators. For each positive integer $k$, we determine the minimum size $m$ of a generating set of the $k$-th direct power FD(3)$^k$ of FD(3) up to ``accuracy $1/2$'' in the sense that we can give only the set $\\{m,m+1\\}$ rather than $m$. For infinitely many values of $k$, we give $m$ exactly. Furthermore, to provide a tool for determining $m$ above, we prove a Sperner (type) theorem for the 3-crown, which is the 6-element poset (partially ordered set) formed by the proper but nonempty subsets of a 3-element set. Namely, we give lower and upper bounds for the maximum number of pairwise unrelated copies of the 3-crown in the powerset lattice of an $n$-element finite set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-25T00:23:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "06D99"
    ],
    "keywords": [
      "free distributive lattice",
      "minimum-sized generating sets",
      "sperner theorem",
      "generators",
      "th direct power fd"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}