Definition:Homomorphism (Abstract Algebra)/Cartesian Product

Definition
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a mapping from one algebraic structure $\left({S, \circ}\right)$ to another $\left({T, *}\right)$.

We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as:
 * $\forall \left({x, y}\right) \in S \times S: \left({\phi \times \phi}\right) \left({x, y}\right) = \left({\phi \left({x}\right), \phi \left({y}\right)}\right)$

Hence we can state that $\phi$ is a homomorphism iff:
 * $\ast \left({\left({\phi \times \phi}\right) \left({x, y}\right)}\right) = \phi \left({\circ \left({x, y}\right)}\right)$

using the notation $\circ \left({x, y}\right)$ 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:


 * Homomorphism.png

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 $S \times S$.