Betti element

Definition

Let \(S\) be a numerical semigroup, an element \(s \in S\) and \(\nabla_s\) the graph associated to \(s\). It is said that \(s \in S\) is a Betti element if \(\nabla_s\) is not connected.

It can be proven that in order to obtain a presentation of a numerical semigroup, we only have to take into account the Betti elements.

Examples

\(\circ\) Let \(S = \langle 15, 18, 35 \rangle\) and \(s = 90 \in S\). The set of factorizations of \(s\) is \(\mathbf{Z}(90) = \{(6,0,0), (0,5,0)\}\) and \((6,0,0), (0,5,0)\) are not R-related. Then, \(\nabla_{90}\) has two connected components and \(s\) is a Betti element.

\(\circ\) Let \(S = \langle 10, 11, 15 \rangle\) and \(s = 100 \in S\). We have \(\mathbf{Z}(100) = \{ (10, 0, 0), (3, 5, 1), (7, 0, 2), (0, 5, 3), (4, 0, 4), (1, 0, 6) \}\) but each pair is \(\mathcal{R}-\)related. Therefore, \(s\) is not a Betti element.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 12, 34, 50, 79, 107 \rangle\), in GAP:

gap> S := NumericalSemigroup(12, 34, 50, 79, 107);
<Numerical semigroup with 5 generators>

Given a numerical semigroup, its Betti elements can be computed with the functions BettiElements and BettiElementsOfNumericalSemigroup.

gap> BettiElements(S);
[ 84, 136, 141, 150, 158, 179, 181, 186, 207, 214 ]
gap> BettiElements(S) = BettiElementsOfNumericalSemigroup(S);
true

Let us check that \(s = 84\) is a Betti element. The function Factorizations computes the factorization of an element with respect to a numerical semigroup.

gap> Factorizations(S, 84);
[ [ 7, 0, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ] ]

On the other hand, the function RClassesOfSetOfFactorizations returns a list with the \(\mathcal{R}-\)classes. As each class represent a connected component of the graph \(\nabla_s\), the number of \(\mathcal{R}-\)classes is equal to the number of components of the graph.

gap> RClassesOfSetOfFactorizations(Factorizations(S, 84));
[ [ [ 7, 0, 0, 0, 0 ] ], [ [ 0, 1, 1, 0, 0 ] ] ]
gap> Length(RClassesOfSetOfFactorizations(Factorizations(S, 84)));
2

Therefore, \(s = 84\) is a Betti element.

\(\diamond\) Let \(S = \langle 25, 30, 62, 69, 108, 114, 146 \rangle\), in GAP:

gap> S := NumericalSemigroup(25, 30, 62, 69, 108, 114, 146);
<Numerical semigroup with 7 generators>

Given a numerical semigroup \(S\) with set of Betti elements \(Betti(S)\), the function HasseDiagramOfBettiElementsOfNumericalSemigroup returns the Hasse diagram of \(S\) by the relation order of S \(\le_S\), that is, \(u \le_S v \Longleftrightarrow v - u \in S\). The vertices are the Betti elements of \(S\). The function DotBinaryRelation returns a GraphViz dot that represents the given binary relation.

gap> H := HasseDiagramOfAperyListOfNumericalSemigroup(S);
<general mapping: Domain([ 0, 30, 60, 62, 69, 90, 92, 99, 108, 114, 120, 122, 129, 131, 138,
  146, 152, 159, 161, 168, 176, 182, 191, 198, 228 ]) -> Domain([ 0, 30, 60, 62, 69, 90, 92, 99,
  108, 114, 120, 122, 129, 131, 138, 146, 152, 159, 161, 168, 176, 182, 191, 198, 228 ]) >
gap>
gap> h := DotBinaryRelation(H);;
gap> Print(h);
digraph  NSGraph{rankdir = TB; edge[dir=back];
1 [label="0"];
2 [label="30"];
3 [label="60"];
4 [label="62"];
5 [label="69"];
6 [label="90"];
7 [label="92"];
8 [label="99"];
9 [label="108"];
10 [label="114"];
11 [label="120"];
12 [label="122"];
13 [label="129"];
14 [label="131"];
15 [label="138"];
16 [label="146"];
17 [label="152"];
18 [label="159"];
19 [label="161"];
20 [label="168"];
21 [label="176"];
22 [label="182"];
23 [label="191"];
24 [label="198"];
25 [label="228"];
2 -> 1;
4 -> 1;
5 -> 1;
9 -> 1;
10 -> 1;
16 -> 1;
3 -> 2;
7 -> 2;
8 -> 2;
15 -> 2;
21 -> 2;
6 -> 3;
12 -> 3;
13 -> 3;
20 -> 3;
7 -> 4;
14 -> 4;
21 -> 4;
8 -> 5;
14 -> 5;
15 -> 5;
11 -> 6;
17 -> 6;
18 -> 6;
24 -> 6;
12 -> 7;
19 -> 7;
13 -> 8;
19 -> 8;
20 -> 8;
15 -> 9;
21 -> 10;
25 -> 10;
22 -> 11;
25 -> 11;
17 -> 12;
23 -> 12;
18 -> 13;
23 -> 13;
24 -> 13;
19 -> 14;
20 -> 15;
21 -> 16;
22 -> 17;
25 -> 18;
23 -> 19;
24 -> 20;
25 -> 24;
}

The Hasse diagram is as follows.

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.