Distributivity is Preserved in Induced Structure

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 $\oplus$.

Let $\struct {T, \oplus, \otimes}$ be an algebraic structure in which $\otimes$ distributes over $\oplus$.

Let $f, g, h \in T^S$.

Then:

$\ds \map {\paren {f \otimes \paren {g \oplus h} } } x$

$=$

$\ds \map f x \otimes \paren {\map g x \oplus \map h x}$

$\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 $

$=$

$\ds \map {\paren {\paren {f \otimes g} \oplus \paren {f \otimes h} } } x$

The proof for right distributivity is identical.

$\blacksquare$