{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety of algebras|AJNZJD8442]] and let $n \\ge 3$ be an integer.\nA *near-unanimity term* of arity $n$ for $\\mathcal{V}$ is a [[term|N9M2N27END]] $t(x_1, \\dots, x_n)$ such that $\\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]\n$$t(y, x, x, \\dots, x) \\approx t(x, y, x, \\dots, x) \\approx \\cdots \\approx t(x, x, \\dots, x, y) \\approx x.$$\nHere the $i$-th term in the chain has $y$ in position $i$ and $x$ in all other positions.\nAn algebra $A$ has a near-unanimity term when the variety it generates has one, that is, when the identities hold in $A$.\n\n## Remarks\n\nExample: in the variety of lattices, $t(x, y, z) = (x \\wedge y) \\vee (y \\wedge z) \\vee (x \\wedge z)$ is a near-unanimity term of arity $3$.\nCheck: $t(y, x, x) = (y \\wedge x) \\vee x \\vee (y \\wedge x) = x$ by absorption, and the other two identities follow in the same way by symmetry of $t$.\nA near-unanimity term of arity $3$ is called a [[majority term|9879ZMW5AA]].\nNon-example: the variety of groups has no near-unanimity term of any arity.\nThe reason is that a variety with a near-unanimity term is congruence-distributive, while the group $\\Z_2 \\times \\Z_2$ has a non-distributive lattice of normal subgroups.\nIf $t$ is a near-unanimity term of arity $n$, then $s(x_1, \\dots, x_{n + 1}) = t(x_1, \\dots, x_n)$ is one of arity $n + 1$.\nThe abbreviation NU term is common.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "4CRH8NNFEF",
    "kind": "definition",
    "msc": [
      "08B05",
      "08B10"
    ],
    "name": "Near-unanimity term",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Majority_function"
      }
    ],
    "tag": "7N5NDD77MS",
    "uses": [
      "AJNZJD8442",
      "N9M2N27END",
      "PKD440G2XW"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/7N5NDD77MS/",
  "links": [
    {
      "field": "defines",
      "target": "4CRH8NNFEF"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "N9M2N27END"
    },
    {
      "field": "uses",
      "target": "PKD440G2XW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7N5NDD77MS.md",
  "msc": [
    "08B05",
    "08B10"
  ],
  "name": "Near-unanimity term",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7N5/7N5NDD77MS.md",
  "status": null,
  "tag": "7N5NDD77MS",
  "uri": "m2://7N5NDD77MS",
  "url": "https://tamnd.github.io/math2/wiki/near-unanimity-term-definition/",
  "writing": {
    "tag": "7N5NDD77MS",
    "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
      }
    ]
  }
}