1. Examples

(1)

In Z, a congruence relation is the same as congruence mod $n$ for some $n$. The case $n=0$ is allowed, giving the equality relation.

(2)

In a group, a congruence relation is the same thing as the coset decomposition for a normal subgroup.

(3)

In a commutative ring, a congruence relation is the same thing as the coset decomposition for an ideal.

(4)

In a finite chain $C$, a congruence relation is any decomposition into intervals, as in Figure (a).

(5)

Lattices in general can have congruence relations, as in Figure (b).

(6)

For a homomorphism $\varphi: {\cal A} \rightarrow {\cal B}$, the kernel of $\varphi$ is a congruence relation.

Here the kernel of a homomorphism means the equivalence relation that $\varphi$ induces on its domain: $a \equiv a' \Leftrightarrow \varphi(a)=\varphi(a')$. This is a contrast with the specific cases of groups and rings, where the kernel is a normal subgroup. However, Example () shows that the two definitions are equivalent.

Figure: Congruence relations on lattices