{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \\ge 1$.\n*Jónsson terms* for $\\mathcal{V}$ are ternary terms $d_0, d_1, \\dots, d_n$ such that $\\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:\n\n1. $d_0(x, y, z) = x$ and $d_n(x, y, z) = z$.\n2. $d_i(x, y, x) = x$ for every $i \\le n$.\n3. $d_i(x, x, z) = d_{i+1}(x, x, z)$ for every even $i \u003c n$.\n4. $d_i(x, z, z) = d_{i+1}(x, z, z)$ for every odd $i \u003c n$.\n\n## Remarks\n\nJónsson's theorem says that $\\mathcal{V}$ is a [[congruence-distributive variety|9GZ7F6ETC6]] if and only if $\\mathcal{V}$ has Jónsson terms for some $n$.\nExample: lattices have Jónsson terms with $n = 2$: $d_0 = x$, $d_1 = (x \\wedge y) \\vee (y \\wedge z) \\vee (x \\wedge z)$, $d_2 = z$.\nThe checks are $d_1(x, y, x) = x$, $d_0(x, x, z) = x = d_1(x, x, z)$, and $d_1(x, z, z) = z = d_2(x, z, z)$.\nMore generally, a majority term $m$ always gives Jónsson terms $x, m, z$.\nNon-example: for groups, the terms $x$, $x y^{-1} z$, $z$ are not Jónsson terms.\nClause 2 fails for the middle term: in the group $\\Z$, the value $x - y + x$ differs from $x$ when $y \\ne x$.\n\n## Explanations\n\n### L1 human\n\nJónsson terms are a chain of three-input formulas that witness, by equations alone, that a variety has distributive congruence lattices.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "3M8Y76E4AG",
    "kind": "definition",
    "msc": [
      "08B05",
      "08B10"
    ],
    "name": "Jónsson terms",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Distributive_lattice"
      }
    ],
    "tag": "HS7GJ1C86M",
    "uses": [
      "AJNZJD8442",
      "PKD440G2XW",
      "9GZ7F6ETC6"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HS7GJ1C86M/",
  "links": [
    {
      "field": "defines",
      "target": "3M8Y76E4AG"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "PKD440G2XW"
    },
    {
      "field": "uses",
      "target": "9GZ7F6ETC6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HS7GJ1C86M.md",
  "msc": [
    "08B05",
    "08B10"
  ],
  "name": "Jónsson terms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HS7/HS7GJ1C86M.md",
  "status": null,
  "tag": "HS7GJ1C86M",
  "uri": "m2://HS7GJ1C86M",
  "url": "https://tamnd.github.io/math2/wiki/jonsson-terms-definition/",
  "writing": {
    "tag": "HS7GJ1C86M",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}