Say I have a function $f: A \to 2^B$. Given an element $b \in B$, I want to refer to the set $f_b := \{a \in A: b \in f(a)\}$.
Is there a standard name for such sets?
Notice it's not technically correct to call it the inverse image of $b$, because of a type mismatch -- the range of $f$ is the subsets of $B$, whereas $b$ is an element of $B$. Also note I'm not asking about $f^{-1}(\{b\})$.
As far as I know there is no established terminology for this. If we denote by $\uparrow b = \{U \subseteq B : b \in U\}$, then $f_b = f^{-1}(\uparrow b)$. This set $\uparrow b$ is called the principal filter at $b$ (more on filters on Wikipedia).
So maybe something like principal pre-image at $b$ would make sense? You would have to define what that means, but if you use it a lot it can shorten your notation. Or since you have to introduce some new notation anyway, you can use the notation $\uparrow b$ and simply talk about $f^{-1}(\uparrow b)$ or "the pre-image of $\uparrow b$".