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arXiv · 1404.5710

M\"obius function of semigroup posets through Hilbert series

Jonathan Chappelon·Ignacio García-Marco·Luis Pedro Montejano·Jorge Luis Ramírez Alfonsín
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Abstract

In this paper, we investigate the M{\"o}bius 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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BibTeXRIS

Jonathan Chappelon, Ignacio García-Marco, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. 2014-04-23. M\"obius function of semigroup posets through Hilbert series. https://doi.org/10.1016/j.jcta.2015.07.006

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