Definition:Semigroup Homomorphism

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

Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.

That is, $\forall a, b \in S$:
 * $\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$

Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a semigroup homomorphism.

Also see

 * Definition:Homomorphism (Abstract Algebra)
 * Definition:Group Homomorphism
 * Definition:Ring Homomorphism


 * Definition:Semigroup Epimorphism: a surjective semigroup homomorphism


 * Definition:Semigroup Monomorphism: an injective semigroup homomorphism


 * Definition:Semigroup Isomorphism: a bijective semigroup homomorphism


 * Definition:Semigroup Endomorphism: a semigroup homomorphism from a semigroup to itself


 * Definition:Semigroup Automorphism: a semigroup isomorphism from a semigroup to itself