correct name of mathematical property

Question

I am developing a program that transforms artifacts in one (computer) language to artifacts in another language. In my program there are certain border line situations where the result of applyin the default transformation of an artifact A in language X produces B in language Y. But after transforming B back to language X, the result is not the same as A.

So I have (in certain border cases):

$x \neq f^{-1} (f(x))$

I am looking for the best way to describe this in the documentation of my program. My question is: What is the name of the mathematical property of certain function that implies ?:

$x = f^{-1} (f(x))$

I need this to clearly explain that this property does not always hold in my program.

Answer 1

If $f^{-1}(f(x))=x$ for all $x$ in the domain of $f$ you could say: $f$ has a left inverse, or $f$ is injective, or that $f$ is a monomorphism ($f$ is monic).

In the program you could say that that $f$ does not have a left inverse or that $f^{-1}$ is not $f$'s left inverse.

Answer 2

Ideally, the notation $f^{-1}$ is reserved for "the function such that $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$ for all (x,y) in the (domain,codomain) of f". Sometimes the notation is still used when only one of these two properties holds, but if neither of them holds you really shouldn't call the function $f^{-1}$ in the first place.

As for terminology, if f and g are functions and $g(f(x)) = x$ for all x in the domain of f, then:

• g is a a left inverse or retraction of f;

• f is a right inverse or section of g.