Definition:Quotient Epimorphism

Definition
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 $\struct {S / \RR, \circ_\RR}$.

This is usually encountered in the context of specific algebraic structures thus:

Also known as
A quotient epimorphism is also known variously as:
 * a quotient morphism
 * a natural epimorphism
 * a natural morphism
 * a natural homomorphism
 * a canonical epimorphism
 * a canonical morphism.

Also see

 * Quotient Mapping on Structure is Epimorphism, where it is shown that $q_\RR$ is indeed an epimorphism.