Restriction of Idempotent Operation is Idempotent

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

$\blacksquare$