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

• Results about semigroup homomorphisms can be found here.

Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$: Definition $2.1$