# 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

$\blacksquare$