Definition:Congruence Relation

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

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

Then $\mathcal R$ is a congruence relation for $\circ$ if and only if:

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

Also known as

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

Also see

  • Results about congruence relations can be found here.