The complexity of a numerical semigroup

Abstract

Let S and Δ be numerical semigroups.A numerical semigroup S is an I(Δ)-semigroup if S \ {0} is an ideal of Δ.We denote by J (Δ) = {S | S is an I(Δ)-semigroup}, and we say that Δ is an ideal extension of S if S ∈ J (Δ). In this work,we present an algorithm to build all the ideal extensions of a numerical semigroup.We recursively denote by J_0(N) = N, J_1(N) = J (N) and J_{ k+1}(N) = J (J_k(N)) forall k ∈ N. The complexity of a numerical semigroup S is the minimum of the set {k ∈ N | S ∈ J k(N)}. In addition,we introduce an algorithm to compute all the numerical semigroups with fixed multiplicity and complexity.