Some doubt about minimal antichain cover of poset.

by John Watson   Last Updated January 14, 2018 17:20 PM

Suppose $P$ is finite partially ordered set (poset) with $\preceq$. Suppose it's width is $n$ i.e the minimal number od antichains which covers $P$. Say $$\mathcal{A} = \{A_1,A_2,...A_n\}\;\;\;\;\;\; {\rm and}\;\;\;\;\;\;\mathcal{A}' = \{A'_1,A'_2,...A'_n\}$$ are two family of antichains which covers $P$. Suppose that $$|A_1|\leq |A_2|\leq ...\leq |A_n|$$ and $$|A'_1|\leq |A'_2|\leq ...\leq |A'_n|$$ Can we say that $|A_i|=|A'_i|$ for each $i\leq n$?

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