# Definition:Internal Direct Product/General Definition

## Definition

Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.

Let $\struct {S_1, \circ {\restriction_{S_1} } }, \ldots, \struct {S_n, \circ {\restriction_{S_n} } }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_{S_1} }, \ldots, \circ {\restriction_{S_n} }$ are the operations induced by the restrictions of $\circ$ to $S_1, \ldots, S_n$ respectively.

$\struct {S, \circ}$ is the **internal direct product of $\sequence {S_n}$** if and only if:

- the mapping $\ds \phi: \prod_{k \mathop = 1}^n S_k \to S$ defined as:

- $\ds \forall i \in \set {1, 2, \ldots, n}: \forall s_i \in S_i: \map \phi {s_1, \ldots, s_n} = \prod_{k \mathop = 1}^n s_k$

- is an isomorphism from the
**(external) direct product**$\struct {S_1, \circ {\restriction_{S_1} } } \times \cdots \times \struct {S_n, \circ {\restriction_{S_n} } }$ onto $\struct {S, \circ}$.

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The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_{S_1} }, \circ {\restriction_{S_2} }, \ldots, \circ {\restriction_{S_n} }$.

### Decomposition

The set of algebraic substructures $\struct {S_1, \circ {\restriction_{S_1} } }, \struct {S_2, \circ {\restriction_{S_2} } }, \ldots, \struct {S_n, \circ {\restriction_{S_n} } }$ whose **(external) direct product** is isomorphic with $\struct {S, \circ}$ is called a **decomposition** of $S$.

## Also see

- Results about
**internal 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