Structure Induced by Idempotent Operation is Idempotent

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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 $\circ$ be an idempotent operation.

Then the pointwise operation $\oplus$ induced on $T^S$ by $\circ$ is also idempotent.

Proof

Let $f \in T^S$.

Then:

\(\ds \forall x \in S: \, \)

\(\ds \map {\paren {f \oplus f} } x\)

\(=\)

\(\ds \map f x \circ \map f x\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \map f x\)

$\circ$ is idempotent operation

From Equality of Mappings:

$f \oplus f = f$

Since $f$ was arbitrary:

$\forall f \in T^S : f \oplus f = f$

Hence $\oplus$ is idempotent by definition.

$\blacksquare$