Is there a name for this type of inverse?

Question

I have a function $f : A \to B$ and an inverse $f^{-1} : B \to A$, and the only property of the inverse is that $(f \circ f^{-1} \circ f)(x) = f(x)$. In particular, it is not necessarily true that $(f^{-1} \circ f)(x) = x$. I normally associate this property with inverses, so what should I call $f^{-1}$?

Answer 1

Such things come up in the theory of von Neumann regular rings. The object is sometimes called a pseudoinverse, but several different items are called by that name. It is called a pseudoinverse in the theory of regular semigroups.