Definition:Semigroup Homomorphism

Definition

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups.

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

That is, $\forall a, b \in S$:

$\map \phi {a \circ b} = \map \phi a * \map \phi b$

Then $\phi: \struct {S, \circ} \to \struct {T, *}$ 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 $2.1$