# Definition:Internal Direct Product

## Definition

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

Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictions of $\circ$ to $A$ and $B$ respectively.

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

- the mapping $\phi: A \times B \to S$ defined as:

- $\forall a \in A, b \in B: \map \phi {a, b} = a \circ b$

- is an isomorphism from the
**(external) direct product**$\struct {A, \circ {\restriction_A} } \times \struct {B, \circ {\restriction_B} }$ onto $\struct {S, \circ}$.

### General 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 known as

Some sources refer to the **internal direct product** simply as the **direct product**.

Others call it the **direct composite**.

It is also commonplace for sources which evolve their expositions on abstract algebra purely from an arithmetical perspective to refer to the **direct sum** when the operation is, or derives from, addition.

## Examples

### External Direct Product which is not Internal Direct Product

Let $m$ and $n$ be integers such that $m, n > 1$.

Let $S$ be a set with $n$ elements.

Let $A$ and $B$ be subsets of $S$ which have $m$ and $n$ elements respectively.

Let $\struct {S, \gets}$ be the algebraic structure formed from $S$ with the left operation.

Then:

- $\struct {S, \gets}$ is isomorphic with the
**external direct product**of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$

but:

- $\struct {S, \gets}$ is not the
**internal direct product**of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$.

## Also see

- Definition:External Direct Product
- Definition:Internal Group Direct Product
- Definition:Ring Direct Sum

- Results about
**internal 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