Induced Structure Identity

From ProofWiki
Jump to navigation Jump to search

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