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.