Let \(S\) be a numerical semigroup, that is, a submonoid of \(\mathbb N\) with finite complement in \(\mathbb N\) (the set of nonnegative integers). The order associated to \(S\), is defined by \(a\leq_Sb\) if \(b-a\in S\). This paper studies the Möbius function associated to the poset \((\mathbb Z,\leq_S)\), when \(S\) is generated by an arithmetic sequence.
Given \(a,b\in\mathbb Z\), a chain connecting \(a\) and \(b\) of length \(l\) is a sequence of integers \(a_0,\ldots,a_l\) such that \(a_0=a\), \(a_l=b\) and \(a_{i-1}\leq_Sa_i\) for all \(i\in\{1,\ldots,l\}\). Denote by \(c_l(a,b)\) the number of chains of length \(l\) connecting \(a\) and \(b\). The Möbius function associated to \(S\) is then defined as \[ \mu_S(x,y)=\sum_{l\in\mathbb N}(-1)^lc_l(x,y). \] Observe that the number of chains joining \(a\) and \(b\) with \(a\leq b\) is the same as the number of chains joining \(0\) and \(b-a\). Hence \(\mu_S(x,y)=\mu_S(0,y-x)\). For this reason, \(\mu_S\) is considered as a function in one variable \(\mu_S(x)=\mu_S(0,x)\).
The authors give an alternative prove of a result by J. A. Deddens describing \(\mu_S(x)\) for \(S\) a numerical semigroup generated by two positive integers [J. Comb. Theory, Ser. A 26, 189-192 (1979; Zbl 0414.05005)]. They also give recursive formulas for the case \(S\) is generated by an arithmetic sequence. Special attention is paid to the case generated by \(\{2q,2q+d,2q+2d\}\), where more explicit results are obtained and illustrative tables are given.
One of the main tools used is the following recursive formula: \[ \mu_S(x)=-\sum_{y\in\mathrm{Ap}(S,a)\setminus\{0\}}\mu_S(x-y) \] for \(x\neq a\) and \(a\) the least positive integer in the semigroup (the multiplicity of \(S\)). The set \(\mathrm{Ap}(S,a)\) is the Apéry set of \(a\) in \(S\), that is, \(\{s\in S\mid s-a\not\in S\}\), which is known to have cardinality \(a\).