What is this property called for a function? $f(f(x))=f(x)$

Question

I am looking for a name for the following types of functions. Suppose that for function f, we have: $$f(x)=y_x$$ and $$f(f(x))=y_x$$ Is there any name for this property?

Answer 1

Such a function is called idempotent.

Some examples:

• The absolute value: $\operatorname{abs}: \mathbb{R}\to [0,\infty)$.
• Any lattice operation like $\max$, $\min$, $\cap$, $\cup$, $\gcd$, etc., can be transformed into an idempotent function. For example $f_1(x)=\max(0,x)$, $f_2(n) = \gcd(n, 60)$ and so on.
• Projections, e.g. $\pi_1\big((x,y,z)\big) = (0,y,z)$.

I hope this helps $\ddot\smile$

Answer 2

A function that has the same result when applied once as it does when applied more than once is known as an idempotent function.

Answer 3

Idempotent functions

http://en.wikipedia.org/wiki/Idempotence

It is common in computer science, in which you may want the result of an operation to be the same no matter how many times someone performs it. This is especially important in web-apps.

In maths an example would be a projection of a vector into a lower dimensional space.