Quotient Mapping on Structure is Epimorphism

Theorem
Let $$\mathcal R$$ be a congruence relation on an algebraic structure $$\left({S, \circ}\right)$$.

Then the quotient mapping from $$\left({S, \circ}\right)$$ to the quotient structure $$\left({S / \mathcal R, \circ_{\mathcal R}}\right)$$ is an epimorphism:


 * $$q_{\mathcal R}: \left({S, \circ}\right) \to \left({S / \mathcal R, \circ_{\mathcal R}}\right): \forall x, y \in S: q_{\mathcal R} \left({x \circ y}\right) = q_{\mathcal R} \left({x}\right) \circ_{\mathcal R} q_{\mathcal R} \left({y}\right)$$

This is sometimes called the canonical epimorphism from $$\left({S, \circ}\right)$$ to $$\left({S / \mathcal R, \circ_{\mathcal R}}\right)$$.

Proof

 * The quotient mapping $$q_{\mathcal R}: S \to S / \mathcal R$$ is the canonical surjection from $$S$$ to $$S / \mathcal R$$.


 * Next we show that this is a homomorphism:

$$ $$ $$ $$ $$

Thus the morphism property is shown to hold.

So the quotient mapping $$q_{\mathcal R}: \left({S, \circ}\right) \to \left({S / \mathcal R, \circ_{\mathcal R}}\right)$$ has been shown to be a homomorphism which is a surjection, and is thus an epimorphism.