Almost symmetric numerical semigroup

Definition

Let \(S\) be a numerical semigroup, \(F(S)\) the Frobenius number, \(g(S)\) the genus and \(t(S)\) the type of \(S\). It is said that \(S\) is an almost symmetric numerical semigroup if it satisfies the equality \(2g(S) = F(S) + t(S)\).

Examples

\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow\}\). With few calculations, the set of gaps is \(G(S) = \{1, 2, 3, 4, 6, 8, 11, 13\}\), the Frobenius number is \(F(S) = 13\) and the set of pseudo-Frobenius numbers is \(PF(S) = \{11, 13\}\). Therefore, \(g(S) = |G(S)| = 8, t(S) = |PF(S)| = 2\) and \(2g(S) = 16 \ne 15 = 13 + 2 = F(S) + t(S)\), to sum up, \(S\) is not an almost symmetric numerical semigroup.

\(\circ\) Let \(a,b \in \mathbb{N}\) with \(gcd(a,b) = 1\) and \(S = \langle a, b \rangle\). It is well known that for numerical semigroups of embedding dimension two, \(t(S) = 1\) and \(2g(S) = F(S) + 1 = F(S) + t(S)\). Therefore, every numerical semigroup with embedding dimension two is almost symmetric.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 12, 15, 19 \rangle\), in GAP:

gap> S := NumericalSemigroup(12, 15, 19);
<Numerical semigroup with 3 generators>

The functions IsAlmostSymmetric and IsAlmostSymmetricNumericalSemigroup return true or false depending on whether a numerical semigroup is almost symmetric or not.

gap> IsAlmostSymmetric(S);
true
gap> IsAlmostSymmetric(S) = IsAlmostSymmetricNumericalSemigroup(S);
true

Therefore, \(2g(S) = F(S) + t(S)\), let us check it. The functions Genus, FrobeniusNumber and Type return the genus, the Frobenius number and the type of a numerical semigroup, respectively.

gap> g := Genus(S);
36
gap> F := FrobeniusNumber(S);
71
gap> t := Type(S);
1
gap> 2*g = F + t;
true

\(\diamond\) Given a numerical semigroup, the function AlmostSymmetricNumericalSemigroupsFromIrreducible returns a list with all the numerical semigroups that can be constructed from by removing some of its generators.

gap> S := NumericalSemigroup(5, 8, 9, 11);
<Numerical semigroup with 4 generators>
gap> l := AlmostSymmetricNumericalSemigroupsFromIrreducible(S);
[ <Numerical semigroup with 4 generators>,
  <Numerical semigroup with 5 generators>,
  <Numerical semigroup with 5 generators> ]
gap> List(l, MinimalGenerators);
[ [ 5, 8, 9, 11 ], [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ]

\(\diamond\) If we are interested in how many almost symmetric numerical semigroups exist with a fixed Frobenius number \(f\) and type \(t\), the function AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType returns a list of all these almost symmetric numerical semigroups.

gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType(12,6));
4

There is an extension of the previous function, that is, AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber, which returns all the almost symmetric numerical semigroups with fixed Frobenius number \(f\) and type greater than or equal to \(t\).

gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12));
15

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.