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

Let $S$ be a numerical semigroup and let $(\mathbb{Z},⩽\mathrm{\_}S)$ be the (locally finite) poset induced by $S$ on the set of integers $\mathbb{Z}$ defined by $x⩽\mathrm{\_}Sy$ 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 $(\mathbb{Z},⩽\mathrm{\_}S)$ when $S$ is an arithmetic semigroup.