Distributivity is Preserved in Induced Structure
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Theorem
Let $\struct {T, \oplus, \otimes}$ be an algebraic structure, and let $S$ be a set.
Let $T^S$ denote the set of all mappings from $S$ to $T$.
Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\oplus$.
Let $\struct {T^S, \otimes}$ be the structure on $T^S$ induced by $\otimes$.
If $\otimes$ is distributive over $\oplus$, then the pointwise operation induced on $T^S$ by $\otimes$ is distributive over the operation induced by $\oplus$.
Proof
Let $\struct {T, \oplus, \otimes}$ be an algebraic structure in which $\otimes$ distributes over $\oplus$.
Let $f, g, h \in T^S$.
Then:
\(\ds \map {\paren {f \otimes \paren {g \oplus h} } } x\) | \(=\) | \(\ds \map f x \otimes \paren {\map g x \oplus \map h x}\) | Definition of Pointwise Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f x \otimes \map g x} \oplus \paren {\map f x \otimes \map h x}\) | because $\otimes$ distributes over $\oplus$ in $T$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\paren {f \otimes g} \oplus \paren {f \otimes h} } } x\) | Definition of Pointwise Operation |
This shows left distributivity.
The proof for right distributivity is identical.
$\blacksquare$