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.
$\blacksquare$