A Sperner family on a set $A$ is set $S$ of subsets of $A$ such that none of the subsets in $S$ contains another subset in $S$.
If ${f : B \rightarrow A}$ is a surjective function and $S$ is a Sperner family on $A$ then we can take the inverse images of the subsets in $S$ along $f$ to obtain a Sperner family ${f^{-1} S}$ on $B$.
Now fix a Sperner family $T$ on a set $B$. For set $A$ (necessarily of smaller cardinality than $B$): Is there a simple way to decide if there is a surjection ${B \rightarrow A}$ and a Sperner family $S$ on $A$ such that ${f^{-1} S = T}$ ?
For example, it is obvious that any binary partition on $B$ is the inverse image of a partition along a function ${B \rightarrow \{ 0, 1 \}}$. On the other hand, for an arbitrary Sperner $T$ on $B$ and a given set $A$, I don't know a simple way to answer the question. Maybe it is possible to give an answer by looking at the sizes of the subsets in $T$?
