# 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

• 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.