Counting the numerical semigroups with a specific special gap

Abstract

Let S be a numerical semigroup. An element $x\in \mathbb{N}\backslash S$ is a special gap of S if $S\cup \left\{x\right\}$ is also a numerical semigroup. If a is a positive integer, we denote by $\mathcal{A}(a)$ the set of all numerical semigroups for which a is a special gap. We say that an element of $\mathcal{A}(a)$ is $\mathcal{A}(a)$-irreducible if it cannot be expressed as the intersection of two numerical semigroups of $\mathcal{A}(a),$ properly containing it. The main aim of this paper is to describe three algorithmic procedures: the first one calculates the elements of $\mathcal{A}(a),$ the second one determines whether or not a numerical semigroup is $\mathcal{A}(a)$-irreducible and the third one computes all the $\mathcal{A}(a)$-irreducibles numerical semigroups.

Keywords:

• $\mathcal{A}$(a)-irreducible numerical semigroup
• ANI-semigroup
• atomic numerical semigroup
• Frobenius number
• gap
• genus
• irreducible numerical semigroup

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20M14
• Secondary: 11Y16

Acknowledgements

The authors would like to thank the referees for their useful comments and suggestions that helped to improve this work.

Additional information

Funding

M. A. Moreno-Frías was partially supported by MTM2017-84890-P and by Junta de Andalucía Group FQM-298. J. C. Rosales was partially supported by MTM2017-84890-P and by Junta de Andalucía Group FQM-343.