# Is there any standard terminology for this property?

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.

• @MRK, it is not the identity function. Consider $f(x)=|x|$. – Zuriel Dec 22 '14 at 2:54
• You should probably accept one of the answers, if you're satisfied with them. – Nishant Dec 22 '14 at 4:23
• @Nishant, yours is correct but it seems to me that the other answer is equally correct and provides more information. – Zuriel Dec 22 '14 at 6:02

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.)
• "Projections" are usually special cases of idempotent maps. I think usually, a projection $\pi: X\to Y$ is defined so that $f(y)=y$ for all $y\in Y$ (assuming here that $Y\subseteq X$). – Nishant Dec 22 '14 at 2:58