Restriction of Idempotent Operation is Idempotent
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $T \subseteq S$.
Let the operation $\circ$ be idempotent.
Then $\circ$ is also idempotent upon restriction to $\struct {T, \circ \restriction_T}$.
Proof
\(\ds T\) | \(\subseteq\) | \(\ds S\) | by hypothesis | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a \in T: \, \) | \(\ds a\) | \(\in\) | \(\ds S\) | Definition of Subset | |||||||||
\(\ds \leadsto \ \ \) | \(\ds a \mathop {\circ \restriction_T} a\) | \(=\) | \(\ds a \circ a\) | Definition of Restriction of Operation | ||||||||||
\(\ds \) | \(=\) | \(\ds a\) | Definition of Idempotence |
$\blacksquare$