Composition of Commuting Idempotent Mappings is Idempotent

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Let $S$ be a set.

Let $f, g: S \to S$ be idempotent mappings from $S$ to $S$.


$f \circ g = g \circ f$

where $\circ$ denotes composition.

Then $f \circ g$ is idempotent.

Proof 1

\(\ds \paren {f \circ g} \circ \paren {f \circ g}\) \(=\) \(\ds f \circ \paren {g \circ f} \circ g\) Composition of Mappings is Associative
\(\ds \) \(=\) \(\ds f \circ \paren {f \circ g} \circ g\) by hypothesis
\(\ds \) \(=\) \(\ds \paren {f \circ f} \circ \paren {g \circ g}\) Composition of Mappings is Associative
\(\ds \) \(=\) \(\ds f \circ g\) $f$ and $g$ are idempotent by hypothesis


Proof 2

By Set of all Self-Maps under Composition forms Semigroup, the set of all self-maps on $S$ forms a semigroup under composition.

The result follows from Product of Commuting Idempotent Elements is Idempotent.