Quotient Theorem for Epimorphisms

Theorem
Let $$\left({S, \circ}\right)$$ and $$\left({T, *}\right)$$ be algebraic structures.

Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be an epimorphism.

Let $$\mathcal{R}_\phi$$ be the equivalence induced by $\phi$.

Let $$S / \mathcal{R}_\phi$$ be the quotient of $S$ by $\mathcal{R}_\phi$.

Let $$q_{\mathcal{R}_\phi}: S \to S / \mathcal{R}_\phi$$ be the quotient mapping induced by $\mathcal{R}_\phi$.

Let $$\left({S / \mathcal{R}_\phi}, {\circ_{\mathcal{R}_\phi}}\right)$$ be the quotient structure defined by $\mathcal{R}_\phi$.

Then:


 * The $$\mathcal{R}_\phi$$ is a congruence for $$\circ$$;
 * There is one and only one isomorphism $$\psi: \left({S / \mathcal{R}_\phi}, {\circ_{\mathcal{R}_\phi}}\right) \to \left({T, *}\right)$$ which satisfies $$\psi \bullet q_{\mathcal{R}_\phi} = \phi$$.

(where, in order not to cause notational confusion, $$\bullet$$ is used as the symbol for composition of mappings.

Proof

 * First we check that $$\mathcal{R}_\phi$$ is compatible with $$\circ$$:

Thus $$\mathcal{R}_\phi$$ is compatible with $$\circ$$.


 * From the Quotient Theorem for Surjections, there is a unique bijection from $$S / \mathcal{R}_\phi$$ onto $$T$$ satisfying $$\psi \bullet q_{\mathcal{R}_\phi} = \phi$$. Also:

Therefore $$\phi$$ is an isomorphism.