The complexity of a numerical semigroup

Abstract

Let S and Delta be numerical semigroups. A numerical semigroup S is an I(Delta)-semigroup if S \ {0} is an ideal of Delta. We denote by J (Delta) = {S vertical bar S is an I(Delta)-semigroupg, and we say that Delta is an ideal extension of S if S is an element of J (Delta). 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)) for all k is an element of N: The complexity of a numerical semigroup S is the minimum of the set {k is an element of N vertical bar S is an element of J(k) (N)}. In addition, we introduce an algorithm to compute all the numerical semigroups with fixed multiplicity and complexity.