Saturated numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is said that \(S\) is saturated if for all \(s, s_1, \ldots, s_r \in S\) such that \(s_i \le s\) for all \(i \in \{1, 2, \ldots, r\}\) and for all \(z_1, \ldots, z_r \in \mathbb{Z}\) so that \(z_1 s_1 + \cdots + z_r s_r \ge 0\), it holds \(s + z_1 s_1 + \cdots + z_r s_r \in S\).
Examples
\(\circ\) Let \(S = \langle 7, 11, 15, 16, 17, 19, 20 \rangle\) and let \(s = s_1 = 11\) and \(s_2 = 7\). For \(z_1 = 2, z_2 = -3\), we have \(z_1 s_1 + z_2 s_2 = 2 \cdot 11 - 3 \cdot 7 = 1 \ge 0\), but \(11 + 1 = 12 \not \in S\). Therefore, \(S\) is not saturated. It can be proven that \(S\) is Arf.
\(\circ\) Every saturated numerical semigroup \(S\) is Arf. Indeed, given \(x \ge y \ge z\), we have \(y - z \ge 0\). Since \(S\) is saturated, \(x + y - z \in S\) and \(S\) is Arf.
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 14, 17, 46, 49 \rangle\), in GAP:
gap> S := NumericalSemigroup(14, 17, 46, 49);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the function IsSaturated returns true or false depending on whether the numerical semigroup is saturated or not.
gap> IsSaturated(S);
false\(\diamond\) Given an integer \(f\), the function SaturatedNumericalSemigroupsWithFrobeniusNumber returns a list with all the saturated numerical semigroups with Frobenius number \(f\).
gap> Length(SaturatedNumericalSemigroupsWithFrobeniusNumber(23));
84References
https://gap-packages.github.io/ numericalsgps.