Definition:External Direct Product/Structures with Two Operations
Definition
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 $\struct {S_1, +_1, \circ_1}, \struct {S_2, +_2, \circ_2}, \ldots, \struct {S_n, +_n, \circ_n}$.
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 $\text {IV}$: Rings and Fields: $22$. New Rings from Old