# Definition:Congruence Relation

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## Contents

## Definition

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

- 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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 11$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups