Definition:External Direct Product/Structures with Two Operations

Definition
Let $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$ be algebraic structures with two operations.

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

The operation $+$ induced on $S$ by $+_1, \ldots, +_n$ is defined as:
 * $\left({s_1, s_2, \ldots, s_n}\right) + \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}\right)$

The operation $\circ $ induced on $S$ by $\circ_1, \ldots, \circ_n$ is 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 $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$.

Also known as
Some authors refer to this as the cartesian product of $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$.

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
 * External Direct Sum of Rings