Arf semigroup

Definition

Let \(S\) be a numerical semigroup. It is said that \(S\) is Arf if every integrally closed proper ideal of \(S\) is stable. It can be proven that a numerical semigroup \(S\) is Arf if, and only if, for any \(s,t \in S\) with \(s \le t\), it holds \(2t - s \in S\).

Examples

\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\), \(I = \{9, 10, 12, 14\}\) and \(E = I + S\). It is easy to prove that \(E = (9 + \mathbb{N}) \cap S = S(9)\). Then, \(E\) is integrally closed, but \(r(E) = 2\), since

\[ 2E = E + E = \{18, \rightarrow \} \ne \{18, 19, 21, 23, \rightarrow \} = m(E) + E, \]

and

\[ 3E = E + (E + E) = \{27, \rightarrow \} = m(E) + 2E. \]

Therefore, \(E\) is integrally closed but not stable. Thus, \(S\) is not an Arf semigroup. Let us also note that \(2 \cdot 9 - 7 = 11 \not \in S\), but \(7,9 \in S\).

\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0, 3, 5, \rightarrow \}\), let’s prove that for any \(s,t \in S\) with \(s \le t\), it holds \(2t - s \in S\). The conductor of \(S\) is \(C(S) = 5\), we divide into cases:

• If \(t < C(S)\), it must be \(t = 0\) or \(t = 3\) and the property holds easily.

• If \(t \ge C(S)\), let \(s \in S\) with \(s \le t\). Then, \(2t - s = t + (t-s) \ge t \ge C(S)\) and \(2t - s \in S\).

Therefore, \(S\) is an Arf semigroup. In fact, we have proven that \(S\) is Arf if, and only if, for any \(s \in S\) and \(t \in N(S)\) with \(s \le t\), where \(N(S)\) denotes the set of left elements of \(S\), it holds \(2t - s \in S\).

Examples with GAP

The following examples are made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 20, 24, 45, 66 \rangle\), in GAP:

gap> S := NumericalSemigroup(20, 24, 45, 66);
<Numerical semigroup with 4 generators>

The function IsArf returns true or false depending on whether a numerical semigroup is Arf or not.

gap> IsArf(S);
false

\(\diamond\) There are several functions related to how many Arf numerical semigroups satisfy a certain condition.

• Condition on Frobenius number: the function ArfNumericalSemigroupsWithFrobeniusNumber returns the Arf numerical semigroups with Frobenius number equals to a given number.

gap> Length(ArfNumericalSemigroupsWithFrobeniusNumber(10));
9

This function is extended with the function ArfNumericalSemigroupsWithFrobeniusNumberUpTo, which returns the Arf numerical semigroups with Frobenius number less than or equal to a given number.

gap> Length(ArfNumericalSemigroupsWithFrobeniusNumberUpTo(10));
46

• Condition on genus: the function ArfNumericalSemigroupsWithGenus returns the Arf numerical semigroups with genus equals to a given number.

gap> Length(ArfNumericalSemigroupsWithGenus(12));
31

This function is extended with the function ArfNumericalSemigroupsWithGenusUpTo, which returns the Arf numerical semigroups with genus less than or equal to a given number.

gap> Length(ArfNumericalSemigroupsWithGenusUpTo(12));
143

• Condition on Frobenius number and genus: The function ArfNumericalSemigroupsWithGenusAndFrobeniusNumber returns the Arf numerical semigroups with genus and Frobenius number both fixed. The first parameter is the genus and the next one is the Frobenius number.

gap> Length(ArfNumericalSemigroupsWithGenusAndFrobeniusNumber(12,15));
6

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.

Delgado, M., P. A. Garcia-Sanchez, and J. Morais. 2024. “NumericalSgps, a Package for Numerical Semigroups, Version 1.4.0.” https://gap-packages.github.io/ numericalsgps.