Definition:External Direct Product

Definition
Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.

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

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

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

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

Another notation sometimes seen for $\struct {S \times T, \circ}$ is $\struct {S \oplus T, \circ}$.

Also see

 * $\circ$ is the 'operation induced on $S \times T$ by $\circ_1$ and $\circ_2..


 * Definition:Internal Direct Product


 * Definition:Group Direct Product
 * Definition:Internal Group Direct Product


 * Definition:Ring Direct Sum