Let $A, B$ be two sets such that there is a function $2^A\to2^B$ preserving unions and $\emptyset\mapsto\emptyset$ and $A\mapsto B$ and no singleton maps to $\emptyset$.
Does there exist a "canonical" function $B\to A$?
If not, what are the conditions that guarantee the existence of a such function?
I have no idea to attempt to this problem. First I tried to work with some explicit examples, but that does not help me to understand what is actually happening here. Can anybody help me to solve this problem? I do not need a complete solution, but enough explanation to understand the problem statement and some hint (if necessary).
Thank you in advance.
I think the idea would be to define $$s_b = \{b\} \quad \forall b \in B$$ and notice that if $f:2^A\to2^B$ then you want to define $g:B \to A$ relating $g(b) $ to $f^{-1}(s_b)$.
You have to make sure this is well-defined as a function, though...