Mal'cev term

Remarks

Example: for groups, $p(x, y, z) = x y^{-1} z$ is a Mal'cev term, since $x y^{-1} y = x$ and $x x^{-1} z = z$. For modules and rings, $p(x, y, z) = x - y + z$ is a Mal'cev term. Non-example: the variety of semilattices has no Mal'cev term. Every semilattice term in the variables $x, y, z$ equals the join of the set $S$ of variables that occur in it. If $p(x, y, y) = x$, 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. By Mal'cev's theorem, a variety has a Mal'cev term if and only if it is a congruence-permutable variety.