Counting the numerical semigroups with a specific special gap

Abstract

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