Generic numerical semigroup

Definition

Let \(S\) be a numerical semigroup. It is said that \(S\) is a generic numerical semigroup if in every minimal relation all minimal generators occur.

It can be proven that every generic numerical semigroup is uniquely presented.

Examples

\(\circ\) Let \(S = \langle 3, 7, 11 \rangle\). In order to compute all minimal relators, we have to consider the Betti elements of \(S\), which are \(Betti(S) = \{14, 18, 22\}\). Their set of factorizations are

\[ \mathbf{Z}(14) = \{(0,2,0), (1,0,1)\}, ~~ \mathbf{Z}(18) = \{(6,0,0), (0,1,1)\}, ~~ \mathbf{Z}(22) = \{(5,1,0), (0,0,2)\}, \]

from which it follows that the set of minimal relations of \(S\) is

\[ \rho = \{[ (1, 0, 1), (0, 2, 0) ], [ (5, 1, 0), (0, 0, 2) ], [ (6, 0, 0), (0, 1, 1) ] \}, \]

and every minimal generator appears in each minimal relation. Therefore, \(S\) is generic.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 31, 49, 74, 118, 128, 150 \rangle\), in GAP:

gap> S := NumericalSemigroup(31, 49, 74, 118, 128, 150);
<Numerical semigroup with 6 generators>

Given a numerical semigroup \(S\), the functions IsGeneric and IsGenericNumericalSemigroup return true if \(S\) is a generic numerical semigroup and false otherwise.

gap> IsGeneric(S);
false

The function AllMinimalRelationsOfNumericalSemigroup returns a list with all minimal relations of a given numerical semigroup.

gap> AllMinimalRelationsOfNumericalSemigroup(S);
[ [ [ 0, 0, 2, 2, 0, 0 ], [ 0, 0, 0, 0, 3, 0 ] ],
  [ [ 0, 2, 0, 0, 2, 0 ], [ 0, 0, 0, 3, 0, 0 ] ],
  [ [ 0, 2, 1, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 2 ] ],
  [ [ 0, 2, 2, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 0 ] ],
  [ [ 1, 0, 1, 1, 0, 1 ], [ 0, 5, 0, 0, 1, 0 ] ],
  [ [ 1, 5, 0, 0, 0, 0 ], [ 0, 0, 2, 0, 1, 0 ] ],
  [ [ 2, 0, 0, 1, 0, 1 ], [ 0, 0, 1, 0, 2, 0 ] ],
  [ [ 2, 0, 0, 2, 0, 0 ], [ 0, 0, 2, 0, 0, 1 ] ],
  [ [ 2, 0, 1, 0, 2, 0 ], [ 0, 8, 0, 0, 0, 0 ] ],
  [ [ 2, 2, 0, 0, 0, 1 ], [ 0, 0, 1, 2, 0, 0 ] ],
  [ [ 2, 2, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 1 ] ],
  [ [ 2, 2, 1, 0, 0, 1 ], [ 0, 0, 0, 0, 3, 0 ] ],
  [ [ 2, 4, 0, 0, 1, 0 ], [ 0, 0, 0, 2, 0, 1 ] ],
  [ [ 3, 0, 0, 0, 1, 0 ], [ 0, 3, 1, 0, 0, 0 ] ],
  [ [ 3, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 1, 0, 0 ] ],
  [ [ 4, 0, 0, 1, 0, 1 ], [ 0, 8, 0, 0, 0, 0 ] ],
  [ [ 4, 2, 0, 0, 0, 0 ], [ 0, 0, 3, 0, 0, 0 ] ],
  [ [ 5, 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 2, 0 ] ],
  [ [ 5, 0, 0, 1, 0, 0 ], [ 0, 1, 1, 0, 0, 1 ] ],
  [ [ 5, 1, 0, 0, 0, 1 ], [ 0, 0, 0, 3, 0, 0 ] ],
  [ [ 5, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1 ] ],
  [ [ 7, 1, 0, 0, 0, 0 ], [ 0, 0, 2, 1, 0, 0 ] ],
  [ [ 7, 1, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 3, 0 ] ],
  [ [ 8, 0, 0, 0, 0, 0 ], [ 0, 2, 0, 0, 0, 1 ] ],
  [ [ 10, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 2, 0, 0 ] ],
  [ [ 10, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 3, 0 ] ] ]

For example, \([ (0, 0, 2, 2, 0, 0), (0, 0, 0, 0, 3, 0) ]\) is a minimal relation but not all minimal generators occur.

References

Blanco, V., P. A. García-Sánchez, and A. Geroldinger. 2011. “Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids.” Illinois Journal of Mathematics 55 (4): 1385–1414. https://doi.org/10.1215/ijm/1373636689.

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.