Permuting congruences

Remarks

The congruences $\theta$ and $\psi$ permute if and only if $\theta \circ \psi$ is a congruence. In that case $\theta \circ \psi = \theta \vee \psi$, the join in the congruence lattice. A congruence-permutable algebra has a modular congruence lattice.

Example: let $G$ be a group and $N$, $M$ normal subgroups, with congruences $a \mathrel{\theta} b \iff a^{-1} b \in N$ and $a \mathrel{\psi} b \iff a^{-1} b \in M$. Then $(a, c) \in \theta \circ \psi$ if and only if $a^{-1} c \in NM$, and $(a, c) \in \psi \circ \theta$ if and only if $a^{-1} c \in MN$. Since $NM = MN$ for normal subgroups, $\theta$ and $\psi$ permute.

Non-example: on the set $\set{1, 2, 3}$ with no operations, let $\theta$ have classes $\set{1, 2}, \set{3}$ and $\psi$ have classes $\set{1}, \set{2, 3}$. Then $(1, 3) \in \theta \circ \psi$ through $b = 2$. But $(1, 3) \notin \psi \circ \theta$, since the only $b$ with $(1, b) \in \psi$ is $1$, and $(1, 3) \notin \theta$.