Correct Terminology for Semi Inverse Mapping Suppose we have two finite sets $X$ and $Y$ and a many to one mapping $f:X\rightarrow Y$. 
Now let me define another mapping $g:Y\rightarrow\mathcal{P}(X)$ where $\mathcal{P}$ denotes the power set. We define $g(y)=\{x|x\in X, f(x)=y\} \forall y$. 
Can we call $g$ as an inverse of $x$ in any meaningful way? I know it is stretching the definition, but is there any terminology for such a mapping? 

More generally, there is a function $$f^{-1}:\mathcal{P}(Y) \to \mathcal{P}(X):S \mapsto \{x \in X \ \vert \ f(x) \in S\}$$ called the inverse image (or preimage) of $f$. 
The map you described is the special case where you only consider singletons. This could be called the fiber map (this is not usual terminology though). 
