Today I asked in class whether one could extract more (topological) information from . I had the feeling that there had to be a connection between this and the Betti numbers of the order complex (those who have dealt with Euler characteristics before might have thought of this already): http://en.wikipedia.org/wiki/Betti_number.
I have not thought about it for too long, but I am wondering about what the topology could tell us about the combinatorics of the poset . . .