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.