Congruence-modular variety

Remarks

Example: the variety of groups is congruence modular, since the normal subgroups of a group form a modular lattice. More generally, every congruence-permutable variety and every congruence-distributive variety is congruence modular. Non-example: the variety of sets, with no operations, is not congruence modular. The congruences of a set are all its equivalence relations. On $\set{1, 2, 3, 4}$, let $\alpha$ have the classes $\set{1, 2}, \set{3}, \set{4}$, let $\gamma$ have the classes $\set{1, 2}, \set{3, 4}$, and let $\beta$ have the classes $\set{1, 3}, \set{2, 4}$. Then $\alpha \le \gamma$, $\beta \wedge \gamma$ is the identity relation and $\alpha \vee \beta$ is the full relation. So $\alpha \vee (\beta \wedge \gamma) = \alpha$, while $(\alpha \vee \beta) \wedge \gamma = \gamma \ne \alpha$.