Möbius function of semigroup posets through Hilbert series

Jonathan Chappelon (1) , Ignacio García-Marco (1) , Luis Pedro Montejano (1) , Jorge Luis Ramírez Alfonsín (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

In this paper, we investigate the Möbius function $\mu_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. For, we introduce and develop a new approach to study $\mu_{\mathcal{S}}$ by using the Hilbert series associated to $\mathcal{S}$. The latter allow 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 finite locally poset to be isomorphic to a semigroup poset. With this in hand, we are 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.

Mots clés: Möbius function; locally finite poset; semigroup; Hilbert series; denumerant

Domaines: Théorie des groupes [math.GR]; Combinatoire [math.CO]; Théorie des nombres [math.NT]

Jonathan Chappelon, Ignacio García-Marco, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. Möbius function of semigroup posets through Hilbert series. 2014. ⟨hal-00981841v1⟩