🔢 mathematics

Structure and Symmetry of Sally Type Semigroup Rings

This paper investigates the structure and symmetry of Sally type semigroup rings by characterizing when they are Gorenstein or symmetric, proving that specific semigroups formed by deleting consecutive integers are Gorenstein if and only if the deletion starts at a particular index, and constructing explicit minimal free resolutions for these cases.

Original authors: Srishti Singh, Hema Srinivasan

Published 2026-01-29

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Srishti Singh, Hema Srinivasan

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long, continuous row of numbered lockers, starting from a specific number $e$ and going up to $2e - 1$ . Let's call this the "Full Row."

In the world of mathematics, specifically a field called algebraic geometry, these lockers represent a "numerical semigroup." A semigroup is just a fancy way of saying a collection of numbers where if you add any two numbers from the collection, you get another number that is also in the collection.

The "Full Row" (lockers $e$ through $2e-1$ ) is a very special, perfectly organized collection. The authors of this paper are interested in what happens when we start taking lockers out of this row.

The "Sally Type" Experiment

The paper focuses on a specific type of experiment:

We start with the Full Row.
We remove a specific number of consecutive lockers (let's say $k$ lockers) from the middle.
We are left with a new collection of numbers.

The authors call these new collections "Sally type semigroups." They are named after a mathematician named Julia Sally, who discovered that these specific types of collections have a hidden, beautiful property called symmetry.

What is "Symmetry" in this context?

Think of symmetry like a perfectly balanced seesaw or a mirror image.

In a symmetric semigroup, the "gaps" (the missing numbers) and the "filled spots" are arranged in a way that is perfectly balanced. If you know the largest missing number (called the Frobenius number), you can predict exactly how many gaps there are.
In a non-symmetric semigroup, the arrangement is lopsided or chaotic.

The main goal of this paper is to answer a simple question: If we remove $k$ consecutive lockers from the Full Row, when does the remaining collection stay perfectly balanced (symmetric)?

The Rules of the Game

The authors found that the answer depends entirely on where you remove the lockers and how many you remove. They break the problem down into two main scenarios:

Scenario A: Removing a Small Number of Lockers ( $k$ is small)

Imagine the Full Row is a long line of people. If you ask a small group of people standing next to each other to step out:

The Rule: The line remains perfectly balanced only if the group you asked to step out starts exactly at the position equal to the number of people who stepped out.
- Example: If you remove 3 people, they must be the 3rd, 4th, and 5th people in the line. If you remove the 1st, 2nd, and 3rd, or the 4th, 5th, and 6th, the balance is broken.
The Exception: There is one special case where the group size is exactly half the total length of the row, and you remove the very first person. This is the only time the balance holds for that specific size.

Scenario B: Removing a Large Number of Lockers ( $k$ is large)

If you remove a large chunk of the row (more than half the length), the situation gets much more chaotic.

The Chaos: Usually, removing a large chunk breaks the symmetry completely.
The Rare Exceptions: However, the authors found that symmetry can survive if the numbers align in a very specific mathematical ratio. It's like a puzzle where the pieces only fit if the size of the gap and the size of the remaining row satisfy a precise equation.
- If the gap is too big or in the wrong spot, the symmetry is lost.
- If the gap is in a "Goldilocks" zone (not too big, not too small, and in the right spot relative to the total size), the symmetry is preserved.

The "Blueprint" (Structure and Resolutions)

Beyond just finding out when the symmetry exists, the authors also built a "blueprint" for these symmetric collections.

In mathematics, these collections are described by equations (ideals). The authors discovered that for these symmetric cases, the equations have a very specific, elegant structure. They can be built using determinants (a specific way of calculating values from a grid of numbers, like a Sudoku puzzle).

They showed that the "defining ideal" (the set of rules that makes the collection what it is) is essentially the sum of two specific types of determinant grids.
They also calculated the "Betti numbers." Think of these as a count of the "building blocks" or "scaffolding" needed to construct the mathematical object. The authors provided a formula to count exactly how many blocks are needed for any symmetric Sally type semigroup.

Summary of Findings

The "Consecutive" Rule: The paper focuses on removing consecutive numbers (like 5, 6, 7). If you remove non-consecutive numbers (like 5 and 7, skipping 6), the symmetry is usually broken, as shown in their final examples.
The "Sweet Spot": For small removals, the symmetry only happens if the gap starts at a specific index equal to the gap size.
The "Rare Alignment": For large removals, symmetry is rare and only happens if the gap size and position satisfy a complex ratio involving the total length of the row.
The Blueprint: When symmetry does exist, the authors provided the exact mathematical formula (the minimal free resolution) to describe it, proving that these symmetric structures are built from two specific types of determinant grids.

In short, the paper maps out the precise conditions under which a "broken" row of numbers can still maintain a perfect, mirror-like balance, and it provides the architectural plans for building those balanced structures.

Technical Summary: Structure and Symmetry of Sally Type Semigroup Rings

Problem Statement
This paper investigates the structural properties of "Sally type" numerical semigroups. These are numerical semigroups with multiplicity $e$ and width $e-1$ , minimally generated by a subset of the interval $[e, 2e-1]$ . The study focuses on semigroups obtained by removing $k$ consecutive integers from the interval semigroup $\langle e, e+1, \dots, 2e-1 \rangle$ . The primary objectives are to:

Determine the defining ideals of the associated semigroup rings, specifically characterizing their structure as sums of determinantal ideals.
Establish necessary and sufficient conditions for these semigroups to be symmetric (equivalently, for their semigroup rings to be Gorenstein).
Construct explicit minimal free resolutions and compute Betti numbers for the symmetric cases.

The work extends previous research by Herzog and Stamate, who introduced the concept of Sally semigroups to construct Gorenstein ideals, and builds upon results regarding the $k=1$ and $k=2$ cases found in prior literature.

Methodology
The authors employ a combination of combinatorial semigroup theory and commutative algebra techniques:

Determinantal Structures: The defining ideals are analyzed using ideals of $2 \times 2$ minors of specific matrices. The authors utilize a criterion by W. Gastinger to prove that a binomial ideal generated by these minors is indeed the defining ideal of the semigroup ring. This involves calculating the dimension of the quotient ring modulo the ideal and a specific variable.
Gap Analysis: To determine symmetry, the authors compute the Frobenius number ( $F$ ) and the genus ( $g$ , the number of gaps) for the semigroups. They utilize the symmetry condition $2g = F + 1$ .
Case Division: The analysis is split into two regimes based on the number of removed generators $k$ $k$ relative to the multiplicity $e$ $e$ :
- Case 1: $k \le e/2$ .
- Case 2: $k > e/2$ .
  Within these regimes, further distinctions are made based on the starting position $j$ of the removed consecutive integers and whether $j=1$ .
Resolution Construction: For symmetric cases, the minimal free resolutions are constructed using the mapping cylinder of a map between dual Eagon-Northcott complexes. This approach leverages the determinantal structure of the ideals.

Key Contributions and Results

Determinantal Structure for Embedding Dimension $e-2$ :
For the case where two consecutive generators are removed ( $k=2$ ), the authors prove that the defining ideal is the sum of two determinantal ideals, $I_2(A) + I_2(B)$ , where $A$ and $B$ are specific matrices constructed from the variables of the polynomial ring. This generalizes the known structure for interval semigroups.
Symmetry Classification for $k \le e/2$ :
The paper provides a complete classification for symmetry in this range. A Sally type semigroup $S_e^k(j)$ (removing $k$ consecutive integers starting at $j$ ) is symmetric if and only if:
- $j = k$ , OR
- $j = 1$ and $e = 2k$ (a specific boundary case).
  In all other cases within this range, the semigroup is not symmetric.
Symmetry Classification for $k > e/2$ :
The situation for $k > e/2$ is more complex. The authors derive explicit formulas for the Frobenius number and genus depending on the parameters $e, k, j$ .
- For $j \neq 1$ : Symmetry occurs only if there exists an integer $n$ such that $\frac{n-1}{n} \le \frac{k-j}{e-2} < \frac{n}{n+1}$ and the condition $n(k-j) = (n-1)(e-2)$ holds. The paper proves that no symmetric semigroups exist at "odd levels" ( $2n-1$ ) of the gap structure.
- For $j = 1$ : Symmetry occurs if and only if $(n-1)k = (n-2)e - (n-3)$ , where $n$ is determined by the ratio $\frac{k}{e-1}$ .
- Uniqueness: In the symmetric cases where $k > e/2$ , there is at most one such symmetric semigroup for a given embedding dimension.
Minimal Free Resolutions and Betti Numbers:
For all identified symmetric cases (both $k \le e/2$ and $k > e/2$ ), the authors construct the minimal free resolution of the semigroup ring.
- The resolution is described as the mapping cylinder of a map $\psi: E^*(nk) \to E(nk)$ , where $E(nk)$ is the Eagon-Northcott resolution of a determinantal ideal.
- Explicit formulas for the Betti numbers $\beta_t$ are provided. Notably, the Betti numbers for the symmetric cases in the $k > e/2$ regime are shown to be identical to those in the $k \le e/2$ regime when the parameters align to produce symmetry.

Significance
The paper provides a comprehensive structural characterization of Sally type semigroup rings, moving beyond specific small cases to a general theory for any $k$ . By identifying the precise conditions under which these semigroups are symmetric, the work clarifies the landscape of Gorenstein numerical semigroups of maximal embedding dimension minus $k$ . The construction of minimal free resolutions for these symmetric cases offers concrete algebraic tools for studying their homological properties. The results unify the understanding of these semigroups, showing that despite the chaotic nature of the $k > e/2$ case, symmetry is a rare and highly constrained phenomenon that follows a deterministic pattern based on the arithmetic relationship between $e, k,$ and $j$ .