Möbius function of semigroup posets through Hilbert series
Résumé
In this paper, we investigate the Möbius function $\mu_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. In order to do this, we introduce and develop a new approach to study $\mu_{\mathcal{S}}$ by using the Hilbert series associated to $\mathcal{S}$. The latter allows us to provide formulas for $\mu_{\mathcal{S}}$ when $\mathcal{S}$ is a semigroup with unique Betti element, and when $\mathcal{S}$ is a complete intersection numerical semigroup with three generators. We also give a characterization for a locally finite poset to be isomorphic to a semigroup poset. We are thus able to calculate the Möbius function of certain posets (for instance the classical arithmetic Möbius function) by computing the Möbius function of the corresponding semigroup poset.
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