Structure Induced by Commutative Operation is Commutative

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 a commutative operation.

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

Let $\struct {T, \circ}$ be a commutative algebraic structure.

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

Then:

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

$=$

$\ds \map f x \circ \map g x$

Definition of Pointwise Operation

$\ds $

$=$

$\ds \map g x \circ \map f x$

$\circ$ is commutative

$\ds $

$=$

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

Definition of Pointwise Operation

$\blacksquare$

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 $13.4$