Is there any standard terminology for this property?

Question

Let $f$ be a map whose domain is $X$. If $f$ satisfies the property that for all $x\in X$, $$f(f(x))=f(x)\text{,}$$ is there any standard name for such a function? Not sure if "projection" is the answer.

Answer 1

This is called an idempotent map. More generally, one can talk about idempotent elements: given a set $S$ with a binary operator $*: S \times S \to S$, an element $x \in S$ is called idempotent if $x * x = x$. (Idempotent maps are idempotent elements in the endomorphism monoid of some object.)

In linear algebra, idempotent linear operators on a vector space are sometimes called "projections" or "projectors".

Answer 2

I think the term for that would be "idempotent."