projecting antichains to middlemost levels

Question

Suppose I have an antichain $A\subset \mathcal{P}(n)$ such that $x\in A\Rightarrow |x|\leq k\leq n/2$. Is there a nice way to "project" my antichain to the middle level? I guess I'm looking for disjoint chains connecting $A$ to $A'$. I feel like this should be possible, but I'm not sure how to go about it.

Answer 1

By Dilworth's theorem, a finite poset has a decomposition into $k$ disjoint chains where $k$ is the size of the largest antichain. For examples such as $\mathcal{P}(n)$ there are explicit such decompositions. In your example, each chain will have a unique element of size $\lfloor n/2\rfloor$.