Definition:Congruence Relation

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\mathcal R$ be an equivalence relation on $S$.

Then $\mathcal R$ is a congruence relation for $\circ$ iff:


 * $\forall x_1, x_2, y_1, y_2 \in S: \left({x_1 \mathrel{\mathcal R} x_2}\right) \land \left({y_1 \mathrel{\mathcal R} y_2}\right) \implies \left({x_1 \circ y_1}\right) \mathrel{\mathcal R} \left({x_2 \circ y_2}\right)$

Also known as
Such an equivalence relation $\mathcal R$ is also described as compatible with $\circ$.

Also see

 * Relation Compatible with Operation
 * Congruence Relation iff Compatible with Operation, justifying the terminology of calling such a relation compatible with an operation.