Definition:Congruence Relation

Definition
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\RR$ be an equivalence relation on $S$.

Then $\RR$ is a congruence relation for $\circ$ :


 * $\forall x_1, x_2, y_1, y_2 \in S: \paren {x_1 \mathrel \RR x_2} \land \paren {y_1 \mathrel \RR y_2} \implies \paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$

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

Also see

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