Interpreting the Möbius function of a poset

Question

I have just learned about incidence algebras and Möbius inversion. I know that the Möbius function is the inverse of the zeta function, and that it appears in the important Möbius inversion formula. But does it have any interpretation outside these two contexts? Does the Möbius function itself have any meaning? In other words, what can I learn about a poset by looking only at the Möbius function and its values?

Answer 1

Here is one answer: Every interval in the poset has an associated abstract simplicial complex consisting of all chains in the interval that do not contain the maximum or the minimum. This complex gives rise to some topological space, which is often an interesting space for common posets (e.g. a sphere). The Mobius function evaluated on the interval gives you the reduced Euler characteristic of this complex because it is the number of chains of odd length minus the number of chains of even length (which is another way to interpret it). Here we consider the empty set to be a chain of even length.