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

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