Definition:External Direct Product

Definition

Let $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ be algebraic structures.

The external direct product $\left({S \times T, \circ}\right)$ of two algebraic structures $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ is the set of ordered pairs:

$\left({S \times T, \circ}\right) = \left\{{\left({s, t}\right): s \in S, t \in T}\right\}$

where the operation $\circ$ is defined as:

$\left({s_1, t_1}\right) \circ \left({s_2, t_2}\right) = \left({s_1 \circ_1 s_2, t_1 \circ_2 t_2}\right)$

$\circ$ is the operation induced on $S \times T$ by $\circ_1$ and $\circ_2$.

Another notation sometimes seen for $\left({S \times T, \circ}\right)$ is $\left({S \oplus T, \circ}\right)$.

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

The operation induced on $S$ by $\circ_1, \ldots, \circ_n$ is the operation $\circ$ defined by:

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

for all ordered $n$-tuples in $S$.

The algebraic structure $\left({S, \circ}\right)$ is called the (external) direct product of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.

Structures with Two Operations

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

• The operation induced on $S$ by $+_1, \ldots, +_n$ is the operation $+$ defined by:

$\left({s_1, s_2, \ldots, s_n}\right) + \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}\right)$

• The operation induced on $S$ by $\circ_1, \ldots, \circ_n$ is the operation $\circ$ defined by:

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

for all ordered $n$-tuples in S.

The algebraic structure $\left({S, +, \circ}\right)$ is called the (external) direct product of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.

Also known as

Some authors refer to this as the cartesian product of $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$.

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

• Results about external direct products can be found here.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 13$