M\"obius function of semigroup posets through Hilbert series

TL;DR

This paper introduces a novel method using Hilbert series to analyze the Möbius function of semigroup posets, providing formulas for specific semigroup families and characterizing when a poset is isomorphic to a semigroup poset.

Contribution

The paper develops a new approach to study the Möbius function of semigroup posets via Hilbert series and offers formulas and characterizations for these structures.

Findings

Formulas for the Möbius function in certain semigroup families

A characterization of when a poset is isomorphic to a semigroup poset

A new approach connecting Hilbert series with Möbius functions

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.