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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nA *Mal'cev term* for $\\mathcal{V}$ is a ternary term $p(x, y, z)$ such that $\\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]\n$$p(x, y, y) = x \\quad \\text{and} \\quad p(x, x, y) = y.$$\nAn algebra $A$ has a *Mal'cev operation* if some ternary [[term operation|N9M2N27END]] of $A$ satisfies these two equations for all elements of $A$.\n\n## Remarks\n\nExample: for groups, $p(x, y, z) = x y^{-1} z$ is a Mal'cev term, since $x y^{-1} y = x$ and $x x^{-1} z = z$.\nFor modules and rings, $p(x, y, z) = x - y + z$ is a Mal'cev term.\nNon-example: the variety of semilattices has no Mal'cev term.\nEvery semilattice term in the variables $x, y, z$ equals the join of the set $S$ of variables that occur in it.\nIf $p(x, y, y) = x$, then neither $y$ nor $z$ is in $S$, so $p = x$, and then $p(x, x, y) = x \\ne y$ in a semilattice with two elements.\nBy Mal'cev's theorem, a variety has a Mal'cev term if and only if it is a [[congruence-permutable variety|1B3N9T68YV]].\n\n## Explanations\n\n### L1 human\n\nA Mal'cev term is a three-input formula that returns the outer input when the other two inputs agree.\n",
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