Suppose that $P$ and $Q$ are graded posets (with rank function $r$) and suppose that all maximal chains of $P$ and $Q$ have length $n$.

Let $f:P \to Q$ be a surjective monotone function such that $r(x)=r(f(x))$.

Let $M(P)$ and $M(Q)$ be the sets of maximal chains of $P$ and $Q$ respectively and let $M_{f}:M(P) \to M(Q)$ be defined by $$M_{f}((x_{1},\dots,x_{n}))=(f(x_{1}),\dots,f(x_{n})).$$

One can easily prove that $(f(x_{1}),\dots,f(x_{n}))$ is indeed a maximal chain.

What are necessary or sufficient conditions for $M_{f}$ to be surjective? That is, what conditions imply that all maximal chains in $Q$ have a preimage with a maximal chain running through it?

In my investigation I already have that $P$ and $Q$ are CW posets is this sufficent?

Any help is much appreciated! Thanks


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