# Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism

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## Definition

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

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

Then $\phi$ is a semigroup isomorphism iff $\phi$ is a bijection.

That is, $\phi$ is a semigroup 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).

## 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.