In my recent work I stumbled across a problem of this type:


G with two partial oders $\leq$ and $\preceq$ on every set, i.e. for every $n \in \mathbb{N}$ $A\_n \subset A\_{n+1}$ and $(A\_n, \leq) $ and $(A\_n, \preceq)$ are partially ordered sets (partial orders do not depend on $n$).


The question now is whether there exist for any $n \in \mathbb{N}$ non-trivial sets $B \in \mathcal{P}(A\_n)$ which are maximal antichains with respect to $(A\_n,\leq)$ and $(A\_n,\preceq)$? If so, how many?


Thank you very much in advance!



(maximal) antichains with respect to two different partial orders on the same set
Almost a trivial counterexample:


Let $A\_n$ be $\{0,\dots,n+2\}$ with one order being the usual $\leq$, that is a linear order, and another being $=$, that is the discrete order.


Note that all the $A\_n$ have at least two elements. But the only sets which are antichains in *any* $A\_n$ are singletons, which are never maximal with respect to $=$. 


