# Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism

## Definition

Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids.

Let $\phi: S \to T$ be a (monoid) homomorphism.

Then $\phi$ is a monoid isomorphism if and only if $\phi$ is a bijection.

That is, $\phi$ is a monoid isomorphism if and only if $\phi$ 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).

## Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.