\begin{definition}[Definition:Semigroup Homomorphism]
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups.
Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in S$:
:$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$
Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a semigroup homomorphism.
\end{definition}