Definition:Operation Induced by Direct Product

Definition
Let $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$ be algebraic structures.

Let $S = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.

Then the operation induced on $S$ by $\circ_1$ and $\circ_2$ is the operation $\circ$ defined as:
 * $\left({s_1, s_2}\right) \circ \left({t_1, t_2}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right)$

for all ordered pairs in $S$.

Also see

 * Definition:External Direct Product
 * Definition:Internal Direct Product