Quotient Mapping on Structure is Epimorphism

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Theorem

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


Then the quotient mapping from $\struct {S, \circ}$ to the quotient structure $\struct {S / \RR, \circ_\RR}$ is an epimorphism:

$q_\RR: \struct {S, \circ} \to \struct {S / \RR, \circ_\RR}: \forall x, y \in S: \map {q_\RR} {x \circ y} = \map {q_\RR} x \circ_\RR \map {q_\RR} y$


Proof

The quotient mapping $q_\RR: S \to S / \RR$ is the canonical surjection from $S$ to $S / \RR$.


Next we show that this is a homomorphism:

\(\ds \map {q_\RR} x\) \(=\) \(\ds \eqclass x \RR\) Definition of Quotient Mapping
\(\ds \map {q_\RR} y\) \(=\) \(\ds \eqclass y \RR\) Definition of Quotient Mapping
\(\ds \map {q_\RR} {x \circ y}\) \(=\) \(\ds \eqclass {x \circ y} \RR\) Definition of Quotient Mapping
\(\ds \eqclass {x \circ y} \RR\) \(=\) \(\ds \eqclass x \RR \circ_\RR \eqclass y \RR\) Definition of Operation Induced on Quotient Set
\(\ds \leadsto \ \ \) \(\ds \map {q_\RR} {x \circ y}\) \(=\) \(\ds \map {q_\RR} x \circ_\RR \map {q_\RR} y\) Definition of Quotient Mapping


Thus the morphism property is shown to hold.


So the quotient mapping $q_\RR: \struct {S, \circ} \to \struct {S / \RR, \circ_\RR}$ has been shown to be a homomorphism which is a surjection, and is thus an epimorphism.

$\blacksquare$


Also see


Sources

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