Definition:Internal Direct Product

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

Let $\struct {A, \circ {\restriction_A} }, \struct {B, \circ {\restriction_B} }$ 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 $B$ :
 * 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 Cartesian product $\struct {A, \circ {\restriction_A} } \times \struct {B, \circ {\restriction_B} }$ onto $\struct {S, \circ}$.

The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_A}$ and $\circ {\restriction_B}$.

It can be seen that the mapping $\phi$ is the restriction of the mapping $\circ: S \times S \to S$ to the subset $A \times B$.

Also see

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