Wilf semigroup

Definition

Let \(S\) be a numerical semigroup with embedding dimension \(e(S)\), \(n(S)\) left elements and conductor \(C(S)\). It is said that \(S\) is a Wilf semigroup if \(W(S) \ge 0\), where

\[ W(S) = e(S) n(S) - C(S). \] It is said that \(W(S)\) is the Wilf number of \(S\). Nowadays, it is not known a numerical semigroup such that \(W(S) < 0\), Wilf’s conjecture states that every numerical semigroup is a Wilf semigroup.

Examples

\(\circ\) Let \(S = \langle 4, 7, 10 \rangle = \{ 0, 4, 7, 8, 10, 11, 12, 14, \rightarrow\}\). We have \(e(S) = 3, n(S) = 7\) and \(C(S) = 14\). The Wilf number of \(S\) is \(W(S) = e(S) n(S) - C(S) = 7 \ge 0\) and \(S\) is a Wilf semigroup.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 17, 31, 66, 80, 91 \rangle\), in GAP:

gap> S := NumericalSemigroup(17, 31, 66, 80, 91);
<Numerical semigroup with 5 generators>

Given a numerical semigroup \(S\); the functions EmbeddingDimension, SmallElements and Condcutor returns the embedding dimension, the list of small elements (left elements with the conductor) and the conductor, respectively.

gap> e := EmbeddingDimension(S);
5
gap> n := Length(SmallElements(S)) - 1;
67
gap> C := Conductor(S);
170
gap> e*n - C;
165

The functions WilfNumber and WilfNumberOfNumericalSemigroup return the Wilf number given a numerical semigroup.

gap> WilfNumber(S);
165
gap> WilfNumberOfNumericalSemigroup(S);
165

therefore, \(W(S) = 165 \ge 0\) and \(S\) is Wilf.

References

Delgado, Manuel. 2019. “Conjecture of Wilf: A Survey.” https://arxiv.org/abs/1902.03461.

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.