\begin{definition}[Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism]
Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids.
Let $\phi: S \to T$ be a (monoid) homomorphism.
Then $\phi$ is a monoid isomorphism {{iff}} $\phi$ is a bijection.
That is, $\phi$ is a monoid isomorphism {{iff}} $\phi$ is both a monomorphism and an epimorphism.
If $S$ is isomorphic to $T$, then the notation $S \cong T$ can be used (although notation varies).
\end{definition}