An analogue of covering space theory for ranked posets (Q5947941)

A locally finite poset with least element and a rank function (denoted 0 and \(|x|)\) to which is assigned an associated function \(n\): \(P\times P\to\{0,1,2, \dots\}\) such that \(n(x,y)\neq 0\) iff \(x\leq y\), and for all \(x<y\) and non-negative \(k\), \(|x|\leq k\leq|y|\), \(n(x,y)= \sum_{|z|= k}n(x,z)n(z,y)\), is a weighted relation poset, obviously with nice conditions implying nice combinatorial properties already including those of larger classes known to be nice. A theory may be developed for these objects, including one of cover maps \(\widetilde P\to P\) (surjections), unique up to isomorphism, factoring through any other covering map \(P'\to P\), for which every principal order ideal is a chain, and the weight assigned to each covering relation of \(\widetilde P\) is 1, with \(\widetilde P\) often having a simple description in the case \(P\) itself is somehow a poset of ``natural'' combinatorial objects. Several interesting examples \(\widetilde P\to P\) are discussed, producing useful observations and proofs, either new or known or completely new of until now unknown propositions.