Epimorphism Preserves Semigroups

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

If $$\left({S, \circ}\right)$$ is a semigroup, then so is $$\left({T, *}\right)$$.

Proof
If $$\left({S, \circ}\right)$$ is a semigroup, then by definition it's closed.

From Morphism Property Preserves Closure, $$\left({T, *}\right)$$ is therefore also closed.

If $$\left({S, \circ}\right)$$ is a semigroup, then by definition $$\circ$$ is associative.

From Epimorphism Preserves Associativity, $$*$$ is therefore also associative.

So $$\left({T, *}\right)$$ is closed, and $$*$$ is associative, and therefore by definition, $$\left({T, *}\right)$$ is a semigroup.