{
  "id": "2009.07978",
  "version": "v1",
  "published": "2020-09-16T23:57:47.000Z",
  "updated": "2020-09-16T23:57:47.000Z",
  "title": "Semidistributivity and Whitman Property in Implication Zroupoids",
  "authors": [
    "Juan M. Cornejo",
    "Hanamantagouda P. Sankappanavar"
  ],
  "comment": "11 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "In 2012, the second author introduced and studied the variety $\\mathcal{I}$ of implication zroupoids that generalize De Morgan algebras and $\\lor$-semilattices with $0$. An algebra $\\mathbf A = \\langle A, \\to, 0 \\rangle$, where $\\to$ is binary and $0$ is a constant, is called an \\emph{implication zroupoid} ($\\mathcal{I}$-zroupoid, for short) if $\\mathbf A$ satisfies: $(x \\to y) \\to z \\approx [(z' \\to x) \\to (y \\to z)']'$, where $x' : = x \\to 0$, and $ 0'' \\approx 0$. Let $\\mathcal{I}$ denote the variety of implication zroupoids and $\\mathbf A \\in \\mathcal{I}$. For $x,y \\in \\mathbf A$, let $x \\land y := (x \\to y')'$ and $x \\lor y := (x' \\land y')'$. In an earlier paper we had proved that if $\\mathbf A \\in \\mathcal{I}$, then the algebra $\\mathbf A_{mj} = \\langle A, \\lor, \\land \\rangle$ is a bisemigroup. In this paper we generalize the notion of semi-distributivity from lattices to bisemigroups and prove that, for every $\\mathbf A \\in \\mathcal{I}$, the bisemigroup $\\mathbf A_{mj}$ is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety $\\mathcal{MEJ}$ of $\\mathcal I$, defined by the identity: $x \\land y \\approx x \\lor y$, satisfies the Whitman Property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-16T23:57:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06D30",
      "06E75",
      "08B15",
      "20N02",
      "03G10"
    ],
    "keywords": [
      "implication zroupoids",
      "whitman property",
      "semidistributivity",
      "bisemigroup",
      "morgan algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}