Definition:Internal Direct Product/Decomposition

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

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 $\struct {S, \circ}$ be the internal direct product of $S_1$, $S_2, \ldots, S_n$.

The set of algebraic substructures $\struct {S_1, \circ {\restriction_{S_1} } }, \struct {S_2, \circ {\restriction_{S_2} } }, \ldots, \struct {S_n, \circ {\restriction_{S_n} } }$ whose (external) direct product is isomorphic with $\struct {S, \circ}$ is called a decomposition of $S$.

Also see

 * Definition:Group Decomposition