Definition:Internal Direct Product

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Definition

Let $\left({S_1, \circ {\restriction_{S_1}} }\right), \left({S_2, \circ {\restriction_{S_2}} }\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$

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 $\left({S, \circ}\right)$ is the internal direct product of $S_1$ and $S_2$ if the mapping:

$C: S_1 \times S_2 \to S: C \left({\left({s_1, s_2}\right)}\right) = s_1 \circ s_2$

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


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 $C$ is the restriction of the mapping $\circ$ of $S \times S$ to the subset $S_1 \times S_2$.


General Definition

Let $\left({S_1, \circ {\restriction_{S_1}}}\right), \ldots, \left({S_n, \circ {\restriction_{S_n}}}\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$

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.


The structure $\left({S, \circ}\right)$ is the internal direct product of $\left \langle {S_n} \right \rangle$ if the mapping:

$\displaystyle C: \prod_{k \mathop = 1}^n S_k \to S: C \left({s_1, \ldots, s_n}\right) = \prod_{k \mathop = 1}^n s_k$

is an isomorphism from the cartesian product $\left({S_1, \circ {\restriction_{S_1}}}\right) \times \cdots \times \left({S_n, \circ {\restriction_{S_n}}}\right)$ onto $\left({S, \circ}\right)$.


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 $\left({S_1, \circ {\restriction_{S_1}}}\right), \left({S_2, \circ {\restriction_{S_2}}}\right), \ldots, \left({S_n, \circ {\restriction_{S_n}}}\right)$ whose direct product is isomorphic with $\left({S, \circ}\right)$ is called a decomposition of $S$.


Also known as

Some authors call this just the direct product.

Some authors call it the direct composite.


Also see

  • Results about internal direct products can be found here.


Sources