2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.