Mapping on Cartesian Product of Substructures is Restriction of Operation
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {A, \circ {\restriction_A} }, \struct {B, \circ {\restriction_B} }$ 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.
Let the mapping $\phi: A \times B \to S$ be defined as:
- $\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$
where $A \times B$ denotes the Cartesian product of $A$ and $B$.
Then $\phi$ is the restriction to $A \times B$ of the operation $\circ$ on $S \times S \to S$.
Proof
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Suppose the mapping $\phi: A \times B \to S$ is defined as:
- $\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$
where $A \times B$ denotes the Cartesian product of $A$ and $B$.
Then:
\(\ds \forall a \in A, b \in B: \, \) | \(\ds a \circ b\) | \(=\) | \(\ds \map \phi {a, b}\) | Definition of $\phi$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren{\phi \cap \paren {A \times B} } \tuple{a, b}\) | Definition of Restriction/Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi {\restriction_{A \times B} } \tuple{a, b }\) | Definition of Restriction/Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds a \mathbin {\circ {\restriction_{A \times B} } } b\) | Definition of Operation Induced by Restriction |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces