The number of generators for an ideal and its dual in a numerical semigroup

Abstract

Let S be the numerical semigroup generated by , where the s_i are positive integers with . Let I be a relative ideal of S Then the dual of I is . Let μ· denote the size of a minimal generating set. A question motivated by the study of torsion in tensor products is under what conditions on S and I do we have the strict inequality μ(I + (S – I)) < μ(I)μ(S –I) Specifically, what is the smallest multiplicity of S, that is, the smallest value of s₁ for which equality can hold for some non-principal relative ideal I. We will examine this question in the case μ(I) = μ(S – I) = 2 and see that the smallest multiplicity is ten.