Gumm terms

Remarks

A variety has Gumm terms if and only if it is a congruence-modular variety. Example: if $p$ is a Mal'cev term, then $n = 0$, $d_0 = x$ and $p$ are Gumm terms, since $p(x, z, z) = x = d_0(x, z, z)$ and $p(x, x, z) = z$. Example: lattices have Gumm terms with $n = 1$, $d_0 = x$, $d_1 = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ and $p(x, y, z) = z$. Here $d_0(x, x, z) = x = d_1(x, x, z)$, $d_1(x, z, z) = z = p(x, z, z)$, and $p(x, x, z) = z$. These examples show the two extreme cases: a pure Mal'cev step, and a chain of Jónsson terms with $p$ the third projection. Non-example: with $n = 0$, the terms $d_0 = x$ and $p = x$ are not Gumm terms in a nontrivial variety, since clause 6 reads $x = z$.