Perfect Pairs of Ideals and Duals in Numerical Semigroups

Abstract

This article considers numerical semigroups S that have a nonprincipal relative ideal I such that μ _S (I)μ _S (S − I) = μ _S (I + (S − I)). We show the existence of an infinite family of such pairs (S, I) in which I + (S − I) = S\{0}. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them.

Key Words:

• Dual
• Minimal generating set
• Numerical semigroup
• Relative ideal

Mathematics Subject Classification:

• 20M14

Notes

Communicated by I. Swanson.