Pointwise Operation on Distributive Structure is Distributive

Theorem

Let $S$ be a set.

Let $\struct {T, +, \circ}$ be an algebraic structure with two operations $+$ and $\circ$.

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

Let $\struct {T^S, +', \circ'}$ 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 $T^S$.

Suppose $S$ is the empty set. Suppose $T^S$ is the set of all mappings from the empty set, $S$, to $T$.

Suppose $\struct {T^S, +', \circ'}$ is the structure induced on $T^S$ by $+$ and $\circ$.

Suppose $\circ$ is distributive over $+$ in $T$.

This needs considerable tedious hard slog to complete it. In particular: Prove that the statement is true when S is the empty set To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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

Suppose $S$ is non-empty.

Let $x \in S$.

Then:

\(\ds \map {\paren {f \circ' \paren {g +' h} } } x\)

\(=\)

\(\ds \map f x \circ \paren {\map g x + \map h x}\)

Definition of Induced Structure

\(\ds \)

\(=\)

\(\ds \paren {\map f x \circ \map g x} + \paren {\map f x \circ \map h x}\)

$\circ$ is distributive over $+$ on $T$

\(\ds \)

\(=\)

\(\ds \map {\paren {\paren {f \circ' g} +' \paren {f \circ' h} } } x\)

Definition of Induced Structure

Similarly:

$\map {\paren {\paren {g +' h} \circ' f} } x = \map {\paren {\paren {g \circ' f} +' \paren {h \circ' f} } } x$

Hence the result.

$\blacksquare$