Definition:Internal Direct Product

Definition
Let $\struct {S_1, \circ {\restriction_{S_1} } }, \struct {S_2, \circ {\restriction_{S_2} } }$ be closed algebraic substructures of an algebraic structure $\struct {S, \circ}$

where $\circ {\restriction_{S_1}}, \circ {\restriction_{S_2}}$ are the operations induced by the restrictions of $\circ$ to $S_1, S_2$ respectively.

The structure $\struct {S, \circ}$ is the internal direct product of $S_1$ and $S_2$ the mapping $\phi: S_1 \times S_2 \to S$ defined as:


 * $\forall s_1 \in S_1, s_2 \in S_2: \map \phi {s_1, s_2} = s_1 \circ s_2$

is an isomorphism from the cartesian product $\struct {S_1, \circ {\restriction_{S_1} } } \times \struct {S_2, \circ {\restriction_{S_2} } }$ onto $\struct {S, \circ}$.

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

It can be seen that the mapping $\phi$ is the restriction of the mapping $\circ: S \times S \to S$ to the subset $S_1 \times S_2$.

Also known as
Some authors call this just the direct product.

Some authors call it the direct composite.

Also see

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