Judith Sally proved in 1980 that the associated graded ring of a one-dimensional Gorenstein local ring of multiplicity and embedding dimension \(e-2\) is Cohen-Macaulay. Numerical semigroup rings provide a large and important class of one-dimensional Cohen-Macaulay rings. In 2014, Herzog and Stamate showed that the numerical semigroup \(\langle e,e+1,e+4,e+5,\dots,2e+1\rangle\)  defines a Gorenstein semigroup ring satisfying Sally’s conditions. We call a numerical semigroup Sally type if its multiplicity is one more than its width; equivalently, it can be obtained from the interval semigroup \(\langle e,e+1,\dots, 2e-1\rangle\) by removing some generators.

In this talk, we first focus on the case where two generators are removed: \(S= \{e, e+1,.., 2e-1\} \backslash \{e+m,e+n\}\), and study the following properties: Frobenius numbers, Cohen-Macaulay type, and computing minimal generating sets for their defining ideals using Hochster’s formula. If time permits, we will use Apéry sets to understand when “similar” numerical semigroups have the same Betti numbers.