# Category:Quotient Epimorphisms

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This category contains results about **Quotient Epimorphisms**.

Definitions specific to this category can be found in Definitions/Quotient Epimorphisms.

Let $\RR$ be a congruence relation on an algebraic structure $\struct {S, \circ}$.

Let $q_\RR: \struct {S, \circ} \to \struct {S / \RR, \circ_\RR}$ denote the quotient mapping from $\struct {S, \circ}$ to the quotient structure $\struct {S / \RR, \circ_\RR}$:

- $\forall x \in S: \map {q_\RR} x = \eqclass x \RR$

where $\eqclass x \RR$ denotes the equivalence class of $x$ under $\RR$.

Then $q_\RR$ is referred to as the **quotient epimorphism** from $\struct {S, \circ}$ to $\struct {S / \RR, \circ_\RR}$.

## Pages in category "Quotient Epimorphisms"

The following 9 pages are in this category, out of 9 total.

### Q

- Quotient Epimorphism from Integers by Principal Ideal
- Quotient Epimorphism is Epimorphism
- Quotient Epimorphism is Epimorphism/Group
- Quotient Epimorphism is Epimorphism/Ring
- Quotient Group Epimorphism is Epimorphism
- Quotient Mapping on Structure is Epimorphism
- Quotient Ring Epimorphism is Epimorphism
- Quotient Theorem for Epimorphisms
- Quotient Theorem for Group Epimorphisms