# Category:Internal Direct Products

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This category contains results about **Internal Direct Products**.

Definitions specific to this category can be found in Definitions/Internal Direct Products.

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 $\struct {S, \circ}$.

## Subcategories

This category has only the following subcategory.

### I

## Pages in category "Internal Direct Products"

The following 7 pages are in this category, out of 7 total.