Numerical Semigroups of Sally Type

TL;DR

This paper studies numerical semigroups of Sally type, providing formulas for their Frobenius number, conditions for Gorenstein property, and algorithms for computing algebraic invariants of their semigroup rings.

Contribution

It introduces the concept of Sally type semigroups, derives formulas for key invariants, and develops computational tools for their algebraic analysis.

Findings

Formula for Frobenius number of Sally type semigroups

Necessary and sufficient conditions for Gorenstein property

Algorithm and GAP code for Betti number computation

Abstract

Judith Sally proved in 1980 that the associated graded ring of one-dimensional Gorenstein local rings of multiplicity $e$ and embedding dimension $e-2$ are Cohen-Macaulay. She showed that the defining ideal of the associated graded ring of such rings is generated by ${e-2 \choose 2}$ elements. Numerical semigroup rings are a big class of one-dimensional Cohen-Macaulay rings. In 2014, Herzog and Stamate proved that the numerical semigroup $<e,e+1,e+4,\ldots,2e-1 >$ defines a Gorenstein semigroup ring satisfying Sally's conditions above and such semigroups are called Gorenstein Sally Semigroups. We call a numerical semigroup $S$ as Sally type if $<S >= < e,e+1,\ldots,e+m-1, e+m+1,\ldots, e+n-1,e+n+1, \ldots 2e-1>$ for some $2 \leq m <n \leq e-2$. In this paper, we give a formula for its Frobenius number along with a necessary and sufficient condition for it to be Gorenstein. We compute the minimal number of generators for the defining ideal of the semigroup ring $k[S]$. Additionally, we present an algorithm and a GAP code used in applying Hochster's combinatorial formula to compute the first Betti number of $k[S]$.