# Definition:Congruence Relation

## Contents

## 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:

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

## Also known as

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

## Also see

- 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