Homomorphism on Induced Structure to Commutative Semigroup
Jump to navigation
Jump to search
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 $13.7: \ 1^\circ$