M\"obius function of semigroup posets through Hilbert series

In this paper, we investigate the M{"o}bius function $\mu \mathrm{\_}\mathcal{S}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of ${\mathbb{Z}}^m$. We introduce and develop a new approach to study $\mu \mathrm{\_}\mathcal{S}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $\mu \mathrm{\_}\mathcal{S}$ when $\mathcal{S}$ belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.