Presentation of a numerical semigroup

Definition

Let \(S\) be a numerical semigroup, let \(\phi\) be the factorization homomorphism of \(S\) and let \(\ker(\phi)\) be the kernel of \(\phi\) as a congruence. A set \(\rho \subseteq \ker(\phi)\) is a presentation of \(S\) if \(\rho\) is a generator system of \(\ker(\phi)\), that is, if \(\ker(\phi) = Cong(\rho)\). It is said that \(\rho\) is a minimal presentation if no proper subsets of \(\rho\) is a presentation.

If \(\rho\) is a minimal presentation, its elements are called minimal relators. Since there is not generally an unique minimal presentation, there may be minimal relators which do not belong in a particular minimal presentation.

Examples

\(\circ\) Let us consider \(a,b \in \mathbb{N} \setminus \{0,1\}\) with \(gcd(a,b) = 1\) and \(S = \langle a, b \rangle\). Let us prove that \(\{[(b,0), (0,a)]\}\) is a minimal presentation of \(S\). Given \([(\alpha_1, \beta_1), (\alpha_2, \beta_2)] \in \ker(\phi)\), and with loss of generality let us suppose \(\alpha_1 > \alpha_2\), we have

\[ \alpha_1a + \beta_1b = \phi(\alpha_1, \beta_1) = \phi(\alpha_2, \beta_2) = \alpha_2a + \beta_2 b, \]

or equivalently,

\[ (\alpha_1 - \alpha_2)a = (\beta_2 - \beta_1)b. \]

From this equality, since \(gcd(a,b) = 1\), it is deduced that exists \(q \in \mathbb{N}\) such that \(\alpha_1 = \alpha_2 + bq\) and \(\beta_2 = \beta_1 + aq\). If \(q = 0\) it is clear. Otherwise,

\[ [(\alpha_1, \beta_1), (\alpha_2, \beta_2)] = [(\alpha_2, \beta_1), (\alpha_2, \beta_1)] + q[(b,0), (0,a)]. \]

\[ = [(b,0), (0,a)] + (q-1)[(b,0), (0,a)] + [(\alpha_2, \beta_1), (\alpha_2, \beta_1)]. \]

Therefore, \([(\alpha_1, \beta_1), (\alpha_2, \beta_2)] \in Cong([(b, 0), (0, a)])\). Finally, is a minimal presentation since it only has one element.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 7, 15, 20, 31, 39 \rangle\), in GAP:

gap> S := NumericalSemigroup(7, 15, 20, 31, 39);
<Numerical semigroup with 3 generators>

In order to compute the minimal presentation of \(S\), the function MinimalPresentation returns a list with the minimal generators of the kernel of \(\phi\).

gap> MinimalPresentation(S);
[ [ [ 0, 0, 0, 0, 2 ], [ 1, 0, 2, 1, 0 ] ], [ [ 0, 0, 0, 1, 1 ], [ 0, 2, 2, 0, 0 ] ],
  [ [ 0, 0, 0, 2, 0 ], [ 1, 1, 2, 0, 0 ] ], [ [ 0, 0, 1, 0, 1 ], [ 2, 3, 0, 0, 0 ] ],
  [ [ 0, 0, 1, 1, 0 ], [ 3, 2, 0, 0, 0 ] ], [ [ 0, 0, 3, 0, 0 ], [ 0, 4, 0, 0, 0 ] ],
  [ [ 0, 1, 0, 0, 1 ], [ 2, 0, 2, 0, 0 ] ], [ [ 0, 1, 0, 1, 0 ], [ 1, 0, 0, 0, 1 ] ],
  [ [ 0, 1, 1, 0, 0 ], [ 5, 0, 0, 0, 0 ] ], [ [ 0, 3, 0, 0, 0 ], [ 2, 0, 0, 1, 0 ] ] ]

This minimal presentation says that \(S\) is a commutative monoid generated by three elements, \(a,b,c,d, e\) under the above relations.

If we want all the minimal relators in a numerical semigroup, the function AllMinimalRelationsOfNumericalSemigroup computes it.

gap> AllMinimalRelationsOfNumericalSemigroup(S);
[ [ [ 0, 2, 2, 0, 0 ], [ 0, 0, 0, 1, 1 ] ], [ [ 0, 4, 0, 0, 0 ], [ 0, 0, 3, 0, 0 ] ],
  [ [ 1, 0, 0, 0, 1 ], [ 0, 1, 0, 1, 0 ] ], [ [ 1, 0, 2, 1, 0 ], [ 0, 0, 0, 0, 2 ] ],
  [ [ 1, 1, 2, 0, 0 ], [ 0, 0, 0, 2, 0 ] ], [ [ 2, 0, 0, 1, 0 ], [ 0, 3, 0, 0, 0 ] ],
  [ [ 2, 0, 2, 0, 0 ], [ 0, 1, 0, 0, 1 ] ], [ [ 2, 1, 0, 1, 0 ], [ 0, 0, 3, 0, 0 ] ],
  [ [ 2, 3, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ] ], [ [ 3, 0, 0, 0, 1 ], [ 0, 0, 3, 0, 0 ] ],
  [ [ 3, 2, 0, 0, 0 ], [ 0, 0, 1, 1, 0 ] ], [ [ 4, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 1 ] ],
  [ [ 4, 2, 1, 0, 0 ], [ 0, 0, 0, 0, 2 ] ], [ [ 5, 0, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ] ],
  [ [ 5, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 1 ] ], [ [ 6, 0, 1, 0, 0 ], [ 0, 0, 0, 2, 0 ] ],
  [ [ 9, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 2 ] ], [ [ 10, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1 ] ]
 ]

On the other hand, the function IsMinimalRelationOfNumericalSemigroup returns true or false depending on whether a pair (a relation) is a minimal relator or not.

gap> IsMinimalRelationOfNumericalSemigroup([ [0, 1, 0, 0, 1], [2, 0, 2, 0, 0] ], S);
true
gap> IsMinimalRelationOfNumericalSemigroup([ [4, 0, 0, 5, 1], [12, 4, 0, 0, 2] ], S);
false

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.