IDEALS AND DUALS GENERATED BY TWO ELEMENTS IN NON-SYMMETRIC NUMERICAL SEMIGROUPS

Abstract

Let S be a numerical semigroup and let I be a relative ideal of S. Let S − I denote the dual of I and let μ _S (·) represent the size of a minimal generating set. We investigate the inequality μ _S (I)μ _S (S − I) ≥ μ _S (I + (S − I)) under the assumptions that μ(I) = μ _S (S − I) = 2 and S is non-symmetric. Specifically, we will identify some criterion involving the associated sets H(S) and S′ which imply the strict inequality μ _S (I)μ _S (S − I) > μ _S (I + (S − I)).

Acknowledgments