Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism

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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 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).


Also see


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.


Sources