# Definition:Internal Direct Product/General Definition

## 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

Such a set of algebraic substructures $\left({S_1, \circ {\restriction_{S_1}}}\right), \ldots, \left({S_n, \circ {\restriction_{S_n}}}\right)$ whose direct product forms $\left({S, \circ}\right)$ is called a decomposition of $S$.