Homomorphism on Induced Structure to Commutative Semigroup

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Theorem

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

Let $\struct {T, \oplus}$ be a commutative semigroup.

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

Let $f$ and $g$ be homomorphisms from $S$ into $T$.

Let $f \oplus' g$ be the pointwise operation on $T^S$ induced by $\oplus$.


Then $f \oplus' g$ is a homomorphism from $\struct {S, \circ}$ into $\struct {T, \oplus}$.


Proof

Let $\struct {T, \oplus}$ be a commutative semigroup.

Let $x, y \in S$.

Then:

\(\ds \map {\paren {f \oplus' g} } {x \circ y}\) \(=\) \(\ds \map f {x \circ y} \oplus \map g {x \circ y}\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \paren {\map f x \oplus \map f y} \oplus \paren {\map g x \oplus \map g y}\) $f$ and $g$ are homomorphisms
\(\ds \) \(=\) \(\ds \paren {\map f x \oplus \map g x} \oplus \paren {\map f y \oplus \map g y}\) $\oplus$ is Commutative and Associative
\(\ds \) \(=\) \(\ds \map {\paren {f \oplus' g} } x \oplus \map {\paren {f \oplus' g} } Y\) Definition of Pointwise Operation

$\blacksquare$


Notice that for this to work, $\oplus$ needs to be both associative and commutative.


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 $13.7: \ 1^\circ$