Definition:Monoid Homomorphism

Definition

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

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

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

$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$

Suppose further that $\phi$ preserves identities, i.e.:

$\phi \left({e_S}\right) = e_T$

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

Also see

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

• Definition:Monoid Epimorphism: a surjective monoid homomorphism

• Definition:Monoid Monomorphism: an injective monoid homomorphism

• Definition:Monoid Isomorphism: a bijective monoid homomorphism

• Definition:Monoid Endomorphism: a monoid homomorphism from a monoid to itself

• Definition:Monoid Automorphism: a monoid isomorphism from a monoid to itself

• Results about monoid 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

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.13$