{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *strong Mal'cev condition* is given by finitely many operation symbols $g_1, \\dots, g_k$ with arities and a finite set $\\Sigma$ of [[identities|PKD440G2XW]] in these symbols.\nA [[variety|AJNZJD8442]] $\\mathcal{V}$ *satisfies* it if $\\mathcal{V}$ has terms $t_1, \\dots, t_k$ of the same arities with the following property.\nEach identity of $\\Sigma$ holds in $\\mathcal{V}$ after each $g_j$ is replaced by $t_j$.\nA *Mal'cev condition* is a sequence $U_1, U_2, \\dots$ of strong Mal'cev conditions such that every variety that satisfies $U_n$ also satisfies $U_{n+1}$, for each $n$.\nA variety satisfies the Mal'cev condition if it satisfies $U_n$ for some $n$.\nA property of varieties is *Mal'cev definable* if some Mal'cev condition is satisfied by exactly the varieties with that property.\n\n## Remarks\n\nExample: the existence of a [[Mal'cev term|Y203PPTS61]] is a strong Mal'cev condition, with one ternary symbol $p$ and $\\Sigma = \\set{p(x, y, y) = x, \\ p(x, x, y) = y}$.\nThe variety of groups satisfies it with $t(x, y, z) = x y^{-1} z$, since $x y^{-1} y = x$ and $x x^{-1} z = z$.\nThe existence of [[Jónsson terms|3M8Y76E4AG]] for some $n$ is a Mal'cev condition, with one strong Mal'cev condition $U_n$ for each $n$.\nNon-example: local finiteness of a variety is not Mal'cev definable.\nThe variety of sets, with no operations, is locally finite, and its terms are the projections.\nIf the projections satisfy $\\Sigma$, then the projections of every variety satisfy $\\Sigma$, so every variety would satisfy the condition.\nBut the variety of groups is not locally finite.\n\n## Explanations\n\n### L1 human\n\nA Mal'cev condition says that a variety has some terms that satisfy a fixed list of equations, possibly with the length of the list allowed to grow.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6R0CA4JWWY",
    "kind": "definition",
    "msc": [
      "08B05"
    ],
    "name": "Mal'cev condition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Congruence-permutable_algebra"
      }
    ],
    "tag": "7SP03R55HW",
    "uses": [
      "AJNZJD8442",
      "PKD440G2XW",
      "N9M2N27END"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/7SP03R55HW/",
  "links": [
    {
      "field": "defines",
      "target": "6R0CA4JWWY"
    },
    {
      "field": "uses",
      "target": "AJNZJD8442"
    },
    {
      "field": "uses",
      "target": "PKD440G2XW"
    },
    {
      "field": "uses",
      "target": "N9M2N27END"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7SP03R55HW.md",
  "msc": [
    "08B05"
  ],
  "name": "Mal'cev condition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7SP/7SP03R55HW.md",
  "status": null,
  "tag": "7SP03R55HW",
  "uri": "m2://7SP03R55HW",
  "url": "https://tamnd.github.io/math2/wiki/malcev-condition-definition/",
  "writing": {
    "tag": "7SP03R55HW",
    "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
      }
    ]
  }
}