{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "7SP03R55HW"
    }
  ],
  "body": "\n## Summary\n\nA Mal'cev condition is a property of a [[variety|AJNZJD8442]] that asserts the existence of terms satisfying certain identities.\nMany properties of the congruence lattices of all algebras in a variety are equivalent to Mal'cev conditions.\nExamples are congruence permutability, given by a [[Mal'cev term|Y203PPTS61]], congruence distributivity, given by [[Jónsson terms|3M8Y76E4AG]], and congruence modularity, given by [[Day terms|B0YKA8401M]].\nMal'cev conditions turn questions about all algebras of a variety into questions about finitely many equations.\n\n## Explanations\n\n### L1 human\n\nSome properties of a whole class of algebras come from a single formula that works in all of them.\nIn groups, the formula $x y^{-1} z$ returns $x$ when $y = z$ and returns $z$ when $x = y$.\nThe existence of such a formula already forces every group to have well-behaved congruences.\nA Mal'cev condition is a statement of this kind: there are formulas that satisfy given equations.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08B05"
    ],
    "name": "Mal'cev condition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Congruence-permutable_algebra"
      }
    ],
    "tag": "6R0CA4JWWY"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/6R0CA4JWWY/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/6R0CA4JWWY.md",
  "msc": [
    "08B05"
  ],
  "name": "Mal'cev condition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6R0/6R0CA4JWWY.md",
  "status": null,
  "tag": "6R0CA4JWWY",
  "uri": "m2://6R0CA4JWWY",
  "url": "https://tamnd.github.io/math2/wiki/malcev-condition/",
  "writing": {
    "tag": "6R0CA4JWWY",
    "kind": "concept",
    "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
      }
    ]
  }
}