Oversemigroup

Definition

Let \(S\) be a numerical semigroup. It is said that \(T\) is a oversemigroup of \(S\) if \(T\) is a numerical semigroup and \(S \subseteq T\). The set of oversemigroups of \(S\) is denoted by \(\mathcal{O}(S)\).

The set \(\mathcal{O}(S)\) is non-empty and finite for every numerical semigroup, since \(S \in \mathcal{O}(S)\) and \(|\mathbb{N} \setminus S|\) is finite.

Examples

\(\circ\) Let \(S = \langle 4, 5, 6 \rangle = \{ 0, 4, 5, 6, 8, \rightarrow\}\). The gaps of \(S\) are \(G(S) = \{1, 2, 3, 7\}\), so to compute the oversemigroups we have to add gaps to \(S\) and check if it is in fact a numerical semigroup.

  • 0 gaps: we have \(S \subseteq S\) so \(S\) is trivially a oversemigroup of \(S\).

  • 1 gap: the candidates are \(T_{1,1} = S \cup \{1\}, T_{1,2} = S \cup \{2\}, T_{1,3} = S \cup \{3\}, T_{1,4} = S \cup \{7\}\). It is easy to check that \(T_4\) is the only numerical semigroup.

  • 2 gaps: in this case the oversemigroups are \(T_{2,1} = S \cup \{2, 7\}, T_{2,2} = S \cup \{3, 7\}\).

  • 3 gaps: the only oversemigroup is \(T_{3,1} = S \cup \{2, 3, 7\}\).

  • 4 gaps: we have the oversemigroup \(T_{4,1} = \mathbb{N}\).

To sum up, \(\mathcal{O}(S) = \{S, S \cup \{7\}, S \cup \{2, 7\}, S \cup \{3, 7\}, S \cup \{2, 3, 7\}, \mathbb{N}\}\).

Examples with GAP

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

\(\diamond\) Let \(S = \langle 2, 5 \rangle\), in GAP:

gap> S := NumericalSemigroup(2, 5);
<Numerical semigroup with 2 generators>

The functions Oversemigroups and OverSemigroupsNumericalSemigroup returns the set of oversemigroups.

gap> OverSemigroups(S);
[ <The numerical semigroup N>, <Numerical semigroup with 2 generators>, <Numerical semigroup with 2 generators> ]
gap> OverSemigroups(S) = OverSemigroupsNumericalSemigroup(S);
true

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 oversemigroups of \(S\) by its left elements and its conductor,

gap> List(OverSemigroups(S), l -> SmallElements(l));
[ [ 0 ], [ 0, 2 ], [ 0, 2, 4 ] ]

or by its gaps,

gap> List(OverSemigroups(S), l -> Gaps(l));
[ [  ], [ 1 ], [ 1, 3 ] ]

It is concluded that \(\mathcal{O}(S) = \{S, T, \mathbb{N}\}\), where \(T = \{0, 2, \rightarrow\}\).

\(\diamond\) Let \(S = \langle 4,5,6 \rangle\), in GAP:

gap> S := NumericalSemigroup(4,5,6);
<Numerical semigroup with 3 generators>

Given a numerical semigroup \(S\), the function DotOverSemigroupsNumericalSemigroup returns the Hasse diagram (in dot) of oversemigroups of \(S\).

gap> h := DotOverSemigroupsNumericalSemigroup(S);;
gap> Print(DotOverSemigroupsNumericalSemigroup(S));
digraph  NSGraph{rankdir = TB; edge[dir=back];
1 [label="〈 1 〉", style=filled];
2 [label="〈 2, 3 〉", style=filled];
3 [label="〈 2, 5 〉", style=filled];
4 [label="〈 3, 4, 5 〉", style=filled];
5 [label="〈 4, 5, 6, 7 〉"];
6 [label="〈 4, 5, 6 〉", style=filled];
1 -> 2;
2 -> 3;
2 -> 4;
3 -> 5;
4 -> 5;
5 -> 6;
}

The obtained graph is as follows.

NSGraph 1 〈 1 〉 2 〈 2, 3 〉 1->2 3 〈 2, 5 〉 2->3 4 〈 3, 4, 5 〉 2->4 5 〈 4, 5, 6, 7 〉 3->5 4->5 6 〈 4, 5, 6 〉 5->6

The nodes with grey background are irreducible 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.