Pointwise Operation on Distributive Structure is Distributive

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Theorem

Let $S$ be a set.

Let $\left({T, +, \circ}\right)$ be an algebraic structure with two operations $+$ and $\circ$.

Let $T^S$ be the set of all mappings from $S$ to $T$.

Let $\left({T^S, +', \circ'}\right)$ be the structure induced on $T^S$ by $+$ and $\circ$.

Let $\circ$ be distributive over $+$ in $T$.


Then $\circ'$ is distributive over $+'$ in $T$.


Proof

Let $f, g, h: S \to T$ be elements of $T^S$.

Let $x \in S$.

Then:

\(\displaystyle \left ({f \circ' \left({g +' h}\right)}\right) \left({x}\right)\) \(=\) \(\displaystyle f \left({x}\right) \circ \left({g \left({x}\right) + h \left({x}\right)}\right)\) Definition of Induced Structure
\(\displaystyle \) \(=\) \(\displaystyle \left({f \left({x}\right) \circ g \left({x}\right)}\right) + \left({f \left({x}\right) \circ h \left({x}\right)}\right)\) $\circ$ is distributive over $+$ on $T$
\(\displaystyle \) \(=\) \(\displaystyle \left ({\left ({f \circ' g}\right) +' \left ({f \circ' h}\right)}\right) \left({x}\right)\) Definition of Induced Structure

Similarly:

$\left ({\left({g +' h}\right) \circ' f}\right) \left({x}\right) = \left ({\left ({g \circ' f}\right) +' \left ({h \circ' f}\right)}\right) \left({x}\right)$

Hence the result.

$\blacksquare$