Arithmetical variety

Remarks

Equivalently, $\mathcal{V}$ is arithmetical if and only if it has a Pixley term $p$, with $p(x, y, y) = x$, $p(x, x, y) = y$ and $p(x, y, x) = x$. Example: the variety of Boolean algebras is arithmetical. It is congruence distributive because Boolean algebras are lattices, and it is congruence permutable because the term $p(x, y, z) = (x \wedge \neg y) \vee (x \wedge z) \vee (\neg y \wedge z)$ is a Mal'cev term. Non-example: the variety of groups is congruence permutable but not congruence distributive, so it is not arithmetical. Non-example: the variety of lattices is congruence distributive but not congruence permutable.