Definition:External Direct Product/General Definition

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

Let $\displaystyle S = \prod_{k \mathop = 1}^n S_k$ be as defined in cartesian product.

Let $\circ$ be the operation induced on $S$ by $\circ_1, \ldots, \circ_n$ defined as:
 * $\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$

for all ordered $n$-tuples in $S$.

The algebraic structure $\left({S, \circ}\right)$ is called the (external) direct product of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.

Also known as
Some authors refer to this as the cartesian product of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.

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

Also see

 * Definition:Internal Direct Product
 * Definition:Ring Direct Sum