Definition:External Direct Product/General Definition
Definition
Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ be algebraic structures.
Let $\ds \SS_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$.
Let $\circledcirc_n$ be the operation induced on $\SS_n$ by $\circ_1, \ldots, \circ_n$ defined as:
- $\tuple {s_1, s_2, \ldots, s_n} \circledcirc_n \tuple {t_1, t_2, \ldots, t_n} := \begin{cases}
s_1 \circ_1 t_1 & : n = 1 \\ \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2} & : n = 2 \\ \tuple {\tuple {s_1, s_2, \ldots, s_{n - 1} } \circledcirc_{n - 1} \tuple {t_1, t_2, \ldots, t_{n - 1} }, s_n \circ_n t_n} & : n > 2 \end{cases}$ for all ordered $n$-tuples in $\SS_n$.
That is:
- $\tuple {s_1, s_2, \ldots, s_n} \circledcirc_n \tuple {t_1, t_2, \ldots, t_n} := \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}$
The algebraic structure $\struct {\SS_n, \circledcirc_n}$ is called the (external) direct product of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
Also known as
Some authors refer to this as the cartesian product of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
Others (whose expositions are not concerned with the internal direct product) call it just the direct product.
Also see
- Results about external direct products can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations