Minimal Generating Sets for Relative Ideals in Numerical Semigroups of Multiplicity Eight

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 assumption that S has multiplicity 8. We will show that if I is non-principal, then the strict inequality μ _S (I)μ _S (S − I) > μ _S (I + (S − I)) always holds.

Key Words:

• Numerical semigroup
• Relative ideal
• Minimal generating set

Mathematics Subject Classification:

• 20M14

Acknowledgments

We wish to thank Frank Golf and Major Scott Sears for providing some of the initial inspiration for many of the proofs in this paper.

A Note on Technology: The research for this paper could not have been accomplished without technological support. We would like to recognize and thank Dr. Juan Ignacio Garcia-Garcia and Dr. Pedro Garcia-Sanchez at the University of Granada for writing a wonderful “semigroup tester” program which helped to run a large quantity of specific examples for this investigation. We would also like to note that Microsoft Excel was invaluable in managing the large number of cases under our consideration.

Notes

^#Communicated by I. Swanson.