Earlier quoted context omitted.
Mathematically, it's x^2 = x, which implies x^n = x for all positive integers n. Nilpotence (x^2 = 0) is also very helpful some times: it's a process which is self-reversing. Like the discrete Fourier transform (if you set up the constants properly).
X^2 is a weird way to describe it. A function f is idempotent if f(f(x)) = f(x).
f(x) = f(f(x)) = f f x = f^2 x = f x
And leaving out the x, because it is just a placeholder anyways:
f f = f^2 = f
And of course this means that
f^n = f because f^n-1 f = f^n-1 by induction.