2Dan: aha, thanks! Looks like that's not enough, but to get any further I should actually formulate what are my "higher transitivities" in 3) - otherwise I can't get further than abstract simplicial complexes.

2Andrej (3): Now let's say that C_n(N(1),... ,N(k)) = True if at least on of k options holds, where option #k sounds like this: There exists a nest N'(i) in N(i) for every i not equal to k, where |N'(i)∩I| > |N(i)∩J| if i<k and |N'(i)∩I| < |N(i)∩J| if i>k. Then C_n's should assemble into the structure that I'm interested in, where converse of axiom 2 fails (example: C_3(2|1|34,2|14|3, 24|13) = False but True for all pairs).

2Andrej (2): By a theorem of Laplante-Anfossi, summands of Saneblidze-Umble diagonal on permutahedra can be described as pairs of nestings (N,M) such that for every (I,J) in D(n) at least one of the following holds: either there exists a nest N' in N such that |N'∩I|>|N'∩J| or there exists a nest M' in M such that |M'∩I|<|M'∩J|. We then say C_2(N,M)=True. (Note that it's a suitable generalization of Bruhat order on vertices).

2Andrej (1): In my example of interest, the higher poset will be a generalization of Bruhat poset of permutations. My set P will consist of all ordered partitions of n elements (aka faces of permutahedron, aka nestings = filtrations on the set {1,2,...,n}). Let D(n) be the set of pairs (I,J) where I and J are non-intersecting subsets of {1,2,...,n} of the same cardinality with min(I + J) in I.

Oh thank you very much! This makes a certain proof a lot simpler than I though.

Oh, thanks for telling! Maybe I should call them "short" or "tight".

Ah, good point! I will edit my question: I meant that the condition "one source, one sink" is brutally required from all the faces as well.

Thanks you very much!

Oh. I don't even know the definition of Moebius functions for posets... Does this look like an ok source? arxiv.org/pdf/1803.06664.pdf

Markl arxiv.org/abs/math/0401007 ?