Moebius function of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup and let \(\le_S\) be the relation order defined by \(S\). Given \(s \in S\), a chain of length \(l \ge 0\) between \(0\) and \(s\) is a subset of the set of divisors of \(s\),
\[ D(s) = \{s' \in S ~ | ~ s' \le_S s\}, \] of length \(l+1\), \(a_0, a_1, \ldots, a_l\) with \(a_i \ne a_j\) if \(i \ne j\), such that
\[ 0 = a_0 \le_S a_1 \le_S \cdots \le_S a_{l-1} \le_S a_l = s. \] Then, it is defined the Moebius function of \(S\) as the function \(\mu_S: S \to \mathbb{Z}\) such that for all \(s \in S\),
\[ \mu_S(s) = \sum_{l \ge 0} (-1)^l c_l(s), \] where \(c_l(s)\) denotes the number of chains of length \(l\) between \(0\) and \(s\). Since \(D(s)\) is finite for all \(s \in S\), the sum in \(\mu_s\) is always finite.
Examples
\(\circ\) Let \(S = \langle 4, 10, 11 \rangle\), let us compute \(\mu_S\) for \(s \in \{0, 4, 8\}\).
- \(D(0) = \{0\}\) and the unique chain is of length \(l = 0\), \(\{0\}\), then
\[ \mu_S(0) = (-1)^0 = 1. \]
- \(D(4) = \{0, 4\}\) and the unique chain is of length \(l = 1\), \(\{0,4\}\),
\[ \mu_S(4) = -1. \]
- \(D(8) = \{0, 4, 8\}\): there are the chains \(\{0, 8\}\) and \(\{0, 4, 8\}\),
\[ \mu_S(8) = (-1)^1 + (-1)^2 = 0. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 14, 33, 64, 76 \rangle\), in GAP:
gap> S := NumericalSemigroup(13, 14, 33, 64, 76);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the function MoebiusFunction returns the Moebius function of \(S\).
gap> f := MoebiusFunction(S);
function( n ) ... end
gap> 100 in S;
true
gap> f(100);
0Given a numerical semigroup \(S\) and an element \(n \in S\), the function MoebiusFunctionAssociatedToNumericalSemigroup returns \(\mu_S(n)\).
gap> MoebiusFunctionAssociatedToNumericalSemigroup(S, 100);
0References
https://gap-packages.github.io/ numericalsgps.