Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism

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

Also see

 * Definition:Isomorphism (Abstract Algebra)