Symmetric numerical semigroup

Definition

A numerical semigroup \(S\) is said to be symmetric if it is irreducible and its Frobenius number is odd.

It can be proven that \(S\) is symmetric if, and only if,

\[ g(S) = \frac{F(S)+1}{2}, \] where \(g(S)\) denotes the genus of \(S\).

Examples

\(\circ\) Let \(S = \langle 4, 10, 17 \rangle\). This numerical semigroup is irreducible and if we compute the first elements of \(S\), we obtain that \(S = \{0, 4, 8, 10, 12, 14, 16, 17, 18, 20, 21, 22, 24, \rightarrow \}\). Hence, \(F(S) = 23\) and \(S\) is a symmetric numerical semigroup.

Examples with GAP

The following example is made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 4, 5, 7 \rangle\), in GAP:

gap> S := NumericalSemigroup(4, 5, 7);
<Numerical semigroup with 3 generators>

The functions IsSymmetric and IsSymmetricNumericalSemigroup return if \(S\) is a symmetric numerical semigroup.

gap> IsSymmetric(S);
false
gap> IsSymmetric(S) = IsSymmetricNumericalSemigroup(S);
true

The output is false, thus, \(S\) is not a symmetric numerical semigroup, we can check where it fails. First, the function IsIrreducible returns true or false depending on whether the numerical semigroup is irreducible or not.

gap> IsIrreducible(S);
true

Now, the function FrobeniusNumber compute the Frobenius number of \(S\).

gap> FrobeniusNumber(S);
6

As we can see, \(F(S)\) is not odd. The irreducible numerical semigroups with Frobenius number even are called pseudo-symmetric numerical semigroups.

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.