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:


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

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 $S$.

Also see

 * External Direct Product
 * Ring Direct Sum