Congruence-distributive variety

Remarks

Example: the variety of lattices is congruence distributive. Every lattice $L$ has the majority term $m(x, y, z) = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, which satisfies $m(x, x, y) = m(x, y, x) = m(y, x, x) = x$, and a variety with a majority term is congruence distributive. Non-example: the variety of groups is not congruence distributive. In the Klein four-group $V = \Z/2\Z \times \Z/2\Z$, every subgroup is normal, so $\operatorname{Con} V$ is isomorphic to the lattice of subgroups. For three distinct subgroups $A, B, C$ of order $2$, $A \wedge (B \vee C) = A \wedge V = A$, while $(A \wedge B) \vee (A \wedge C)$ is the trivial subgroup. Every congruence-distributive variety is congruence modular.