\begin{definition}[Definition:Internal Direct Product/Decomposition]
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 $S$.
\end{definition}