Definition:Congruence Relation

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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