Definition:External Direct Product

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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:

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


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 $\displaystyle \mathcal S_n = \prod_{k \mathop = 1}^n S_k$ 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 $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.


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

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

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

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

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

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


The algebraic structure $\left({S, +, \circ}\right)$ is called the (external) direct product of $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$.


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.


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


Also see

  • Results about external direct products can be found here.


Sources