Category:Definitions/Internal Direct Products

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This category contains definitions related to Internal Direct Products.
Related results can be found in Category:Internal Direct Products.

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 $B$ if and only if:

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 $\struct {S, \circ}$.