# Category:Congruence Relations

This category contains results about Congruence Relations.

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:

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

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### C

### Q

## Pages in category "Congruence Relations"

The following 20 pages are in this category, out of 20 total.

### A

### C

- Congruence Modulo Normal Subgroup is Congruence Relation
- Congruence Relation and Ideal are Equivalent
- Congruence Relation iff Compatible with Operation
- Congruence Relation induces Normal Subgroup
- Congruence Relation on Group induces Normal Subgroup
- Congruence Relation on Ring induces Ideal
- Congruence Relation on Ring induces Ring
- Congruences on Rational Numbers
- Construction of Inverse Completion/Congruence Relation