Idempotence

Builds on Binary operation Function Function composition

Definition

Let $\ast$ be a binary operation on a set $S$ . An element $x \in S$ is idempotent for $\ast$ if $x \ast x = x$ . The operation $\ast$ is idempotent if every element of $S$ is idempotent for $\ast$ .
Let $X$ be a set. A function $f \colon X \to X$ is idempotent if $f \circ f = f$ , that is, $f(f(x)) = f(x)$ for all $x \in X$ , where $\circ$ is composition.

Remarks

Example: union is an idempotent operation on the power set of a set $X$ , because $Y \cup Y = Y$ for every $Y \subseteq X$ .

Non-example: multiplication on $\Z$ is not idempotent, because $2 \cdot 2 = 4 \ne 2$ ; its only idempotent elements are $0$ and $1$ .

The two notions agree. A function $f \colon X \to X$ is idempotent in the sense of clause 2 if and only if it is idempotent for composition on the set of maps $X \to X$ . A function is idempotent if and only if it fixes every point of its image.

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Binary operation concept
Function concept
Function composition concept
Idempotence definition

Tag	`QV3V8HJW2F`
URI	`m2://QV3V8HJW2F`
Kind	definition
Writing	W0 stub
MSC	08A02, 16U40
Defines	Idempotence
Uses	Binary operation, Function, Function composition
Changed	2 November 2026

Idempotence

Definition

Remarks

Reading path

Further reading

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