Definition:Internal Direct Product/Decomposition

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

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

Let $\left({S, \circ}\right)$ be the internal direct product of $S_1$, $S_2, \ldots, S_n$.

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 see

 * Definition:Group Decomposition