Monoid Homomorphism

Definition 1. A monoid homomorphism is a magma homomorphism between monoids.

In addition, a monoid homomorphism $f$ between two monoids $M$ and $N$ is neutral-preserving if

\[f(e) = i,\]

where $e$ and $i$ are respectively the neutral elements of $M$ and $N$.

Definition 2. The kernel of a monoid homomorphism $f : M \to N$ is

\[\ker f = f^{-1}[\{i\}],\]

where $i$ is the neutral element of $N$.