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