Congruence-permutable variety

Remarks

Example: the variety of groups is congruence permutable. Congruences of a group $G$ correspond to normal subgroups, and the composition of the congruences of $M$ and $N$ is the congruence relation $a^{-1} b \in MN$. Since $MN = NM$ for normal subgroups, the two compositions are equal. Non-example: the variety of lattices is not congruence permutable. In the chain $0 < 1 < 2$, let $\alpha$ identify $0$ and $1$, and let $\beta$ identify $1$ and $2$. The pair $(0, 2)$ is in one composition, through $0 \mathrel{\alpha} 1 \mathrel{\beta} 2$. It is not in the other, because the only element $\beta$-related to $0$ is $0$, and $0$ is not $\alpha$-related to $2$. For a single algebra, the congruences $\alpha$ and $\beta$ permute if and only if $\alpha \circ \beta$ is the join $\alpha \vee \beta$ in the congruence lattice.