Complete intersection numerical semigroup

Definition

Let \(S\) be a numerical semigroup. It is said that \(S\) is a complete intersection if the cardinality of any minimal presentation of \(S\) is equal to \(e(S) - 1\), where \(e(S)\) denotes the embedding dimension of \(S\).

It can be proven that every numerical semigroup is finite presented, that every minimal presentation has the same cardinal and that every presentations \(\rho\) of \(S\) has cardinal \(|\rho| \ge e(S) - 1\).

Examples

\(\circ\) Let us consider \(a,b \in \mathbb{N} \setminus \{0,1\}\) with \(gcd(a,b) = 1\) and \(S = \langle a, b \rangle\). A minimal presentation of \(S\) is \(\rho = \{(bx_1, ax_2)\}\), whose cardinal is \(1 = e(S) - 1\). Then, any numerical semigroup with embedding dimension two is a complete intersection.

\(\circ\) Given a numerical semigroup with embedding dimension three, the concepts of symmetric, free and complete intersection are equivalent.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 22, 24, 51, 115, 122 \rangle\), in GAP:

gap> S := NumericalSemigroup(22, 24, 51, 115, 122);
<Numerical semigroup with 5 generators>

Given a numerical semigroup, the function IsCompleteIntersection returns true or false depending on whether a numerical semigroup is a complete intersection or not.

gap> IsCompleteIntersection(S);
false

The function MinimalPresentation returns a minimal presentation of a given numerical semigroup. On the other hand, the function EmbeddingDimension returns the embedding dimension.

brk_3> Length(MinimalPresentation(S));
11
brk_3> EmbeddingDimension(S)-1;
4

\(\diamond\) Given an integer \(z \in \mathbb{Z}\), the function CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber returns a list with all numerical semigroups that are a complete intersection with Frobenius number \(z\).

gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(21));
5

The code List(List_NS, l -> Function(l)); returns a list where it has applied Function to every element in \(List\_NS\), for example, we can see the numerical semigroups with complete intersection and Frobenius number \(f = 21\) by its minimal generators.

gap> Li := CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(21);
[ <Numerical semigroup with 2 generators>,
  <Numerical semigroup with 3 generators>,
  <Numerical semigroup with 3 generators>,
  <Numerical semigroup with 3 generators>,
  <Numerical semigroup with 3 generators> ]
gap> List(Li, l -> MinimalGenerators(l));
[ [ 2, 23 ], [ 4, 6, 19 ], [ 4, 10, 15 ], [ 4, 11, 14 ],
  [ 6, 8, 11 ] ]

References

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.

Rosales, J. C., and P. A. Garcı́a-Sánchez. 2009. Numerical Semigroups. Springer.