\begin{definition}[Definition:Internal Direct Product/General Definition]
Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {S_1, \circ {\restriction_{S_1} } }, \ldots, \struct {S_n, \circ {\restriction_{S_n} } }$ be closed algebraic substructures of $\struct {S, \circ}$, 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.
$\struct {S, \circ}$ is the '''internal direct product of $\sequence {S_n}$''' {{iff}}:
:the mapping $\ds \phi: \prod_{k \mathop = 1}^n S_k \to S$ defined as:
::$\ds \forall i \in \set {1, 2, \ldots, n}: \forall s_i \in S_i: \map \phi {s_1, \ldots, s_n} = \prod_{k \mathop = 1}^n s_k$
:is an isomorphism from the '''(external) direct product''' $\struct {S_1, \circ {\restriction_{S_1} } } \times \cdots \times \struct {S_n, \circ {\restriction_{S_n} } }$ onto $\struct {S, \circ}$.
{{ExtractTheorem|The below: it's simple and obvious but it expresses a fact that is not part of the definition}}
The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_{S_1} }, \circ {\restriction_{S_2} }, \ldots, \circ {\restriction_{S_n} }$.
\end{definition}