# 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 $\displaystyle \mathcal S_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$.

Let $\circledcirc_n$ be the operation induced on $\mathcal S_n$ by $\circ_1, \ldots, \circ_n$ defined as:

$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \begin{cases} s_1 \circ_1 t_1 & : n = 1 \\ \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right) & : n = 2 \\ \left({\left({s_1, s_2, \ldots, s_{n-1} }\right) \circledcirc_{n-1} \left({t_1, t_2, \ldots, t_{n-1}}\right), s_n \circ_n t_n}\right) & : n > 2 \end{cases}$

for all ordered $n$-tuples in $\mathcal S_n$.

That is:

$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$

The algebraic structure $\left({\mathcal S_n, \circledcirc_n}\right)$ 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.