Induced Structure Identity
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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 $e$ be an identity for $\circ$.
Then the constant mapping $f_e: S \to T$ defined as:
- $\forall x \in S: \map {f_e} x = e$
is the identity for the pointwise operation $\oplus$ induced on $T^S$ by $\circ$.
Proof
Let $\struct {T, \circ}$ be an algebraic structure with an identity $e$.
Let $f \in T^S$.
Then:
\(\ds \map {\paren {f \oplus f_e} } x\) | \(=\) | \(\ds \map f x \circ \map {f_e} x\) | Definition of Pointwise Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x \circ e\) | Definition of Constant Mapping $f_e$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x\) | $e$ is the identity of $\struct {T, \circ}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds e \circ \map f x\) | $e$ is the identity of $\struct {T, \circ}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {f_e} x \circ \map f x\) | Definition of Constant Mapping $f_e$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {f \oplus f_e} } x\) | 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 $13.6: \ 2^\circ$
- 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 $13.4$