Numerical Semigroups of Sally Type II

TL;DR

This paper investigates the algebraic properties of numerical semigroups of Sally type, focusing on their minimal resolutions, Betti numbers, and conjectures relating symmetric and non-symmetric cases.

Contribution

It constructs minimal resolutions for symmetric Sally type semigroup rings and proposes conjectures linking Betti numbers of symmetric and non-symmetric cases.

Findings

Constructed minimal resolutions for symmetric Sally type semigroup rings.

Computed Betti numbers for these semigroups.

Proposed conjectures relating Betti numbers across different Sally types.

Abstract

In this paper we study numerical semigroups of Sally type of multiplicity $e$ and embedding dimension $\nu \ge e-2$. We construct the minimal resolutions for these semigroup rings when they are symmetric and compute their Betti numbers. We also construct a minimal resolution for another special class of such semigroups of type $\nu-1$. Finally, we propose some conjectures for the Betti numbers of families of non-symmetric Sally type semigroups in the above cases in relation to those of the corresponding Gorenstein cases of Sally type semigroups.