Suppose $(S, \leq)$ is a poset. Let $C$ be a maximal chain in $S$. Suppose that the largest antichain of $S$ is of size $M$. Now consider the poset $(S \setminus C, \leq)$. Is it possible that the largest antichain in $S \setminus C$ is still of size $M$?
I am trying to find it in $\mathcal P(X)$ for some non-empty set $X$ when the partially ordered relation is "$\subseteq$". But I couldn't find one such. Please give me some example (if any).
Thank you very much.