\begin{definition}[Definition:Monoid Homomorphism]
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be monoids.
Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in S$:
:$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$
Suppose further that $\phi$ preserves identities, i.e.:
:$\phi \left({e_S}\right) = e_T$
Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a monoid homomorphism.
\end{definition}