Category:Quotient Epimorphisms

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