Homomorphism Preserves Subsemigroups

Theorem

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism.

Let $S'$ be a subsemigroup of $S$.

Then $\phi \paren {S'}$ is a subsemigroup of $T$.

Proof

By Restriction of Homomorphism to Image is Epimorphism, $\phi$ is an epimorphism onto its image .

Then by Epimorphism Preserves Semigroups, it follows that the image of $S'$ is a semigroup.

The result follows.

$\blacksquare$