The generators, relations and type of the Backelin semigroup

Abstract

We present an explicit minimal set of generators for the defining ideal of the family of Backelin semigroups and find its Betti numbers. In particular, we compute the type of the semigroup as well, correcting a claim in the literature. Additionally, we show that the Betti numbers of the corresponding numerical semigroup ring coincide to those of its tangent cone.

Keywords:

• Betti numbers
• defining ideal
• numerical semigroup
• semigroup type
• tangent cone

2020 Mathematics Subject Classification:

• 20M14
• 20M25
• 13F99

Acknowledgment

This project was initiated as part of the RIMMES program at Georgia State University and continued as the MS thesis project of the first author. The authors thank Dumitru Stamate for a number of conversations on the subject that helped improve the paper, and in particular for mentioning Backelin’s semigroup as an interesting subject of study. In addition, the authors thank the anonymous referee and the coresponding editor for several important suggestions, which included the use of the results in [11] to greatly simplify our work.