Jónsson terms

Remarks

Jónsson's theorem says that $\mathcal{V}$ is a congruence-distributive variety if and only if $\mathcal{V}$ has Jónsson terms for some $n$. Example: lattices have Jónsson terms with $n = 2$: $d_0 = x$, $d_1 = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, $d_2 = z$. The checks are $d_1(x, y, x) = x$, $d_0(x, x, z) = x = d_1(x, x, z)$, and $d_1(x, z, z) = z = d_2(x, z, z)$. More generally, a majority term $m$ always gives Jónsson terms $x, m, z$. Non-example: for groups, the terms $x$, $x y^{-1} z$, $z$ are not Jónsson terms. Clause 2 fails for the middle term: in the group $\Z$, the value $x - y + x$ differs from $x$ when $y \ne x$.