# Definition:Congruence Relation

## Definition

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

Let $\RR$ be an equivalence relation on $S$.

Then $\RR$ is a **congruence relation for $\circ$** if and only if:

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

## Also known as

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

## Examples

### Equal Fourth Powers over $\C$ for Multiplication

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

- $\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is a **congruence relation for multiplication on $\C$**.

### Equal Fourth Powers over $\C$ for Addition

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

- $\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is *not* a **congruence relation for addition on $\C$**.

### $\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ over $\Z$ for Multiplication

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

- $\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then $\RR$ is *not* a **congruence relation for multiplication on $\Z$**.

### $\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ over $\Z$ for Addition

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

- $\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then $\RR$ is *not* a **congruence relation for addition on $\Z$**.

## Also see

- Definition:Relation Compatible with Operation
- Equivalence Relation is Congruence 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): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups