Monoid Homomorphisms
A homomorphism between two monoids (M,*) and (M′,•) is a function f : M → M′ such that
- f(x*y) = f(x)•f(y) for all x, y in M
- f(e) = e′
where e and e′ are the identities on M and M′ respectively. Monoid homomorphisms are sometimes simply called monoid morphisms.
Not every semigroup homomorphism is a monoid homomorphism since it may not preserve the identity. Contrast this with the case of group homomorphisms: the axioms of group theory ensure that every semigroup homomorphism between groups preserves the identity. For monoids this isn't always true and it is necessary to state it as a separate requirement.
A bijective monoid homomorphism is called a monoid isomorphism. Two monoids are said to be isomorphic if there is an isomorphism between them.
Read more about this topic: Monoid