{
  "id": "0812.2300",
  "version": "v1",
  "published": "2008-12-12T06:48:44.000Z",
  "updated": "2008-12-12T06:48:44.000Z",
  "title": "A characterization of well-founded algebraic lattices",
  "authors": [
    "Ilham Chakir",
    "Maurice Pouzet"
  ],
  "comment": "19 pages, 2 pictures, submitted",
  "categories": [
    "math.CO",
    "math.LO"
  ],
  "abstract": "We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice $L$ is well-founded if and only if $K(L)$, the join-semilattice of compact elements of $L$, is well-founded and contains neither $[\\omega]^{<\\omega}$, nor $\\underline\\Omega(\\omega^*)$ as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice $L$ is well-founded if and only if $K(L)$ is well-founded and contains no infinite independent set. If $K(L)$ is a join-subsemilattice of $I_{<\\omega}(Q)$, the set of finitely generated initial segments of a well-founded poset $Q$, then $L$ is well-founded if and only if $K(L)$ is well-quasi-ordered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-12T06:48:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A12",
      "06B35"
    ],
    "keywords": [
      "characterization",
      "compact elements",
      "infinite independent set",
      "algebraic modular lattice",
      "join-semilattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.2300C"
    }
  }
}