Homomorphism Preserves Subsemigroups

Theorem

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

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

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

Then $\phi \left({S\,'}\right)$ is a subsemigroup of $T$.

Proof

• A homomorphism is an epimorphism onto its image by Epimorphism by Restriction of Codomain.

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

The result follows.

$\blacksquare$