On the M\"obius function of the locally finite poset associated with a numerical semigroup

TL;DR

This paper investigates the M"obius function of a locally finite poset derived from an arithmetic numerical semigroup, providing insights into its combinatorial and algebraic structure.

Contribution

It characterizes the M"obius function for the poset induced by arithmetic semigroups, a specific class of numerical semigroups, expanding understanding of their combinatorial properties.

Findings

Explicit formulas for the M"obius function in the arithmetic case

Connections between the M"obius function and semigroup properties

Enhanced understanding of the poset structure associated with numerical semigroups

Abstract

Let $S$ be a numerical semigroup and let $\left(\mathbb{Z},\leqslant\_S\right)$ be the (locally finite) poset induced by $S$ on the set of integers $\mathbb{Z}$ defined by $x \leqslant\_S y$ if and only if $y-x\in S$ for all integers $x$ and $y$. In this paper, we investigate the M{\"o}bius function associated to $\left(\mathbb{Z},\leqslant\_S\right)$ when $S$ is an arithmetic semigroup.