Definition:External Direct Product

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

The external direct product $\left({S \times T, \circ}\right)$ of two algebraic structures $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ is the set of ordered pairs:
 * $\left({S \times T, \circ}\right) = \left\{{\left({s, t}\right): s \in S, t \in T}\right\}$

where the operation $\circ$ is defined as:
 * $\left({s_1, t_1}\right) \circ \left({s_2, t_2}\right) = \left({s_1 \circ_1 s_2, t_1 \circ_2 t_2}\right)$

$\circ$ is the operation induced on $S \times T$ by $\circ_1$ and $\circ_2$.

Another notation sometimes seen for $\left({S \times T, \circ}\right)$ is $\left({S \oplus T, \circ}\right)$.

Also known as
Some authors refer to this as the cartesian product of $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$.

Others (whose expositions are not concerned with the Internal Direct Product) call it just the direct product.

Also see

 * Internal Direct Product
 * Ring Direct Sum