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 $\struct {S, \circ}$.
Pages in category "Definitions/Internal Direct Products"
The following 4 pages are in this category, out of 4 total.