\begin{definition}[Definition:Internal Direct Product]
Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictions of $\circ$ to $A$ and $B$ respectively.
$\struct {S, \circ}$ is the '''internal direct product of $A$ and $B$''' {{iff}}:
:the mapping $\phi: A \times B \to S$ defined as:
::$\forall a \in A, b \in B: \map \phi {a, b} = a \circ b$
:is an isomorphism from the '''(external) direct product''' $\struct {A, \circ {\restriction_A} } \times \struct {B, \circ {\restriction_B} }$ onto $\struct {S, \circ}$.
\end{definition}