Möbius function of semigroup posets through Hilbert series

Abstract : In this paper, we investigate the Möbius function $\mu_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study $\mu_{\mathcal{S}}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $\mu_{\mathcal{S}}$ 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.
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Submitted on : Thursday, March 31, 2016 - 11:41:52 AM
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Jonathan Chappelon, Ignacio García-Marco, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. Möbius function of semigroup posets through Hilbert series. Journal of Combinatorial Theory, Series A, Elsevier, 2015, 136, pp.238-251. ⟨10.1016/j.jcta.2015.07.006⟩. ⟨hal-00981841v4⟩

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