# Category:Congruence Relations

This category contains results about Congruence Relations.

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:

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Congruence Relations"

The following 23 pages are in this category, out of 23 total.

### A

### C

- Congruence Modulo Normal Subgroup is Congruence Relation
- Congruence Relation and Ideal are Equivalent
- Congruence Relation iff Compatible with Operation
- Congruence Relation iff Compatible with Operation/Proof 1
- Congruence Relation iff Compatible with Operation/Proof 2
- 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