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:
- $\tuple {s_1, t_1} \circ \tuple {s_2, t_2} = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}$
General 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 $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
Structures with Two Operations
Let $\struct {S_1, +_1, \circ_1}, \struct {S_2, +_2, \circ_2}, \ldots, \struct {S_n, +_n, \circ_n}$ be algebraic structures with two operations.
Let $\ds \SS = \prod_{k \mathop = 1}^n S_k$ be as defined in cartesian product.
The operation $+$ induced on $\SS$ by $+_1, \ldots, +_n$ is defined as:
- $\tuple {s_1, s_2, \ldots, s_n} + \tuple {t_1, t_2, \ldots, t_n} = \tuple {s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}$
The operation $\circ $ induced on $\SS$ by $\circ_1, \ldots, \circ_n$ is defined as:
- $\tuple {s_1, s_2, \ldots, s_n} \circ \tuple {t_1, t_2, \ldots, t_n} = \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}$
for all ordered $n$-tuples in $\SS$.
The algebraic structure $\struct {\SS, +, \circ}$ is called the (external) direct product of $\struct {S_1, +_1, \circ_1}, \struct {S_2, +_2, \circ_2}, \ldots, \struct {S_n, +_n, \circ_n}$.
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 $\circ_2$.
- Results about external direct products can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces