Definition:Semigroup Endomorphism

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Let $\left({S, \circ}\right)$ be a semigroups.

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

Then $\phi$ is a semigroup endomorphism.

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Linguistic Note

The word endomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix endo- (from ἔνδον') meaning inner or internal.

Thus endomorphism means internal structure.