Numerical Semigroups with a Fixed Fundamental Gap

Abstract

A gap a of a numerical semigroup S is fundamental if {2a, 3a} ⊆ S. In this work, we will study the set B(a) = {S | S is a numerical semigroup and a is a fundamental gap of S}. In particular, we will give an algorithm to compute all the elements of B(a) with a given genus. The intersection of two elements of B(a) is again one element of B(a). A B(a)- irreducible numerical semigroup is an element of B(a) that cannot be expressed as an intersection of two elements of B(a) containing it properly. In this paper, we will study the B(a)-irreducible numerical semigroups. In this sense we will give an algorithm to calculate all of them. Finally, we will study the submonoids of (N,+) that can be expressed as an intersection (finite or infinite) of elements belonging to B(a).