Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

*Question*: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ Note that $\mu(\hat{0},\hat{1}) = \pm1$.



It is obviously satisfied by the boolean lattices. Now a boolean lattice is the face lattice of a simplex and John's comment below suggests that it could be satisfied by the face lattice of an arbitrary (convex?) polytope. Is it true? Can we extend to any [Eulerian lattice][1]? Anything else?


  [1]: https://en.wikipedia.org/wiki/Eulerian_poset