{
  "id": "1306.2839",
  "version": "v3",
  "published": "2013-06-12T14:36:44.000Z",
  "updated": "2014-06-10T09:09:55.000Z",
  "title": "Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality",
  "authors": [
    "Mai Gehrke",
    "Samuel J. van Gool",
    "Vincenzo Marra"
  ],
  "comment": "36 pages, 1 table",
  "categories": [
    "math.LO",
    "math.GN",
    "math.GR",
    "math.RA"
  ],
  "abstract": "We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-06-10T09:09:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06D35",
      "06F20",
      "06D50",
      "18F20",
      "54B40"
    ],
    "keywords": [
      "lattice-ordered abelian groups",
      "mv-algebras",
      "partial commutative ordered semigroups",
      "decompositions yield sheaf representations",
      "maximal mv-spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.2839G"
    }
  }
}