\begin{definition}[Definition:Homomorphism (Abstract Algebra)]
Let $\struct {S, \circ}$ and $\struct {T, *}$ be magmas.
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from $\struct {S, \circ}$ to $\struct {T, *}$.
Let $\circ$ have the morphism property under $\phi$, that is:
:$\forall x, y \in S: \map \phi {x \circ y} = \map \phi x * \map \phi y$
Then $\phi$ is a '''homomorphism'''.
This can be generalised to magmas with more than one operation:
Let:
:$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
:$\struct {T, *_1, *_2, \ldots, *_n}$
be magmas.
Let $\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {T, *_1, *_2, \ldots, *_n}$ be a mapping from $\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$ to $\struct {T, *_1, *_2, \ldots, *_n}$.
Let, $\forall k \in \closedint 1 n$, $\circ_k$ have the morphism property under $\phi$, that is:
:$\forall x, y \in S: \map \phi {x \circ_k y} = \map \phi x *_k \map \phi y$
Then $\phi$ is a '''homomorphism'''.
\end{definition}