\begin{definition}[Definition:Homomorphism (Abstract Algebra)/Cartesian Product]
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$.
We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as:
:$\forall \tuple {x, y} \in S \times S: \map {\paren {\phi \times \phi} } {x, y} = \tuple {\map \phi x, \map \phi y}$
Hence we can state that $\phi$ is a homomorphism {{iff}}:
:$\map \ast {\map {\paren {\phi \times \phi} } {x, y} } = \map \phi {\map \circ {x, y} }$
using the notation $\map \circ {x, y}$ to denote the operation $x \circ y$.
The point of doing this is so we can illustrate what is going on in a commutative diagram:
::$\begin{xy} \xymatrix@L+2mu@+1em{
 S \times S \ar[r]^*{\circ}
     \ar[d]_*{\phi \times \phi}
&
 S \ar[d]^*{\phi}
\\
 T \times T  \ar[r]_*{\ast} &
 T
}\end{xy}$
Thus we see that $\phi$ is a homomorphism {{iff}} both of the composite mappings from $S \times S$ to $T$ have the same effect on all elements of $S \times S$.
\end{definition}