Structure Induced by Associative Operation is Associative
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Theorem
Let $\struct {T, \circ}$ be an algebraic structure.
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 |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Theorem $13.6: \ 1^\circ$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.4$