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:

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

Also known as

Such a 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.

• Results about congruence relations can be found here.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 11$