Definition:Idempotence/Operation

Definition

Let $S$ be a set.

Let $\circ: S \times S \to S$ be a binary operation on $S$.

If all the elements of $S$ are idempotent under $\circ$, then the term can be applied to the operation itself:

The binary operation $\circ$ is idempotent if and only if:

$\forall x \in S: x \circ x = x$

Also see

• Set Union is Idempotent
• Set Intersection is Idempotent

• Results about idempotence can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.17$