Structure Induced by Associative Operation is Associative

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Theorem

Let $\struct {T, \circ}$ be an algebraic structure, and let $S$ be a set.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.

Let $\circ$ be associative.


Then the pointwise operation $\oplus$ induced on $T^S$ by $\circ$ is also associative.


Proof

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

Let $\struct {T, \circ}$ be an associative algebraic structure.

Then:

\(\ds \map {\paren {\paren {f \oplus g} \oplus h} } x\) \(=\) \(\ds \map {\paren {f \oplus g} } x \circ \map h x\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \paren {\map f x \circ \map g x} \circ \map h x\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \map f x \circ \paren {\map g x \circ \map h x}\) $\circ$ is associative
\(\ds \) \(=\) \(\ds \map f x \circ \map {\paren {g \oplus h} } x\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \map {\paren {f \oplus \paren {g \oplus h} } } x\) Definition of Pointwise Operation

$\blacksquare$


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