Möbius function of semigroup posets through Hilbert series

In this paper, we investigate the Möbius function μSμS associated to a (locally finite) poset arising from a semigroup SS of ZmZm. We introduce and develop a new approach to study μSμS by using the Hilbert series of SS. The latter enables us to provide formulas for μSμS when SS belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.