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On quotients of numerical semigroups for almost arithmetic progressions

This paper presents a method to reduce the computation of the Apéry set for quotients of numerical semigroups to a simple minimization problem when the divisor divides the smallest generator, thereby deriving closed formulas for the Frobenius number in cases involving almost arithmetic progressions and partially resolving an open problem posed by Adeniran et al.

Original authors: Feihu Liu

Published 2026-04-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Feihu Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 giant, infinite warehouse filled with boxes. You can only build stacks of boxes using specific "base sizes" (let's call them $a_1, a_2, \dots$ ). You can combine these base sizes however you like, but you can't break them apart.

In mathematics, this collection of all possible stack heights you can build is called a Numerical Semigroup.

The Problem: The "Missing Heights"

Sometimes, there are certain heights you simply cannot build. For example, if your base sizes are 3 and 5, you can build 3, 5, 6, 8, 9, 10... but you can never build a stack of height 1, 2, 4, or 7.

The Frobenius Number is the tallest "missing" stack you can't build. (In the 3 and 5 example, it's 7).
The Genus is just the count of all the missing heights.

Mathematicians have known how to calculate these missing heights for a long time when you have just two base sizes. But when you have three or more, or when the base sizes follow a specific pattern, it becomes a nightmare to find a simple formula.

The New Twist: The "Quotient" Filter

This paper introduces a new way of looking at the problem. Imagine you have a special filter (let's call it $p$ ).

You take your original warehouse of stacks.
You apply the filter: "Only keep the stacks that are exactly divisible by $p$ ."
Then, you shrink everything down by dividing by $p$ .

Mathematically, this creates a new set of stacks (a new semigroup). The big question is: What are the new "missing heights" in this shrunken, filtered world?

Usually, calculating this is incredibly hard. It's like trying to guess the shape of a shadow without knowing the object casting it.

The Author's Solution: The "Ladder" Trick

The author, Feihu Liu, found a clever shortcut, but only works when the filter number $p$ divides the very first base size ( $a_1$ ).

Think of the original base sizes as a ladder. The first rung is $a_1$ . The other rungs are $a_2, a_3$ , etc.
The author realized that if you divide the whole ladder by $p$ , the structure of the "missing heights" doesn't change randomly. Instead, it behaves like a scaled-down version of the original problem, with a few predictable adjustments.

He developed a method to turn this complex puzzle into a simple "minimization game."

The Game: You have a target number. You need to reach that number using the fewest steps possible, where each step has a specific cost.
The Magic: Once you solve this simple game, you can instantly write down the formulas for the "missing heights" (Frobenius number) and the "count of missing heights" (Genus) for the new filtered world.

What Did They Actually Solve?

The paper focuses on a specific, tricky type of pattern called "Almost Arithmetic Progressions."

Normal Arithmetic Progression: 10, 12, 14, 16 (adding 2 every time).
Almost Arithmetic Progression: 10, 22, 24, 26... (The first number is special, then they follow a pattern).

The author solved the "missing height" puzzle for several variations of these patterns:

Standard Almost Progressions: The first number is different, the rest follow a pattern.
Progressions with Gaps: The first few numbers in the pattern are missing.
Progressions with Odd Terms: Only specific steps in the pattern are included.

Why Does This Matter?

Before this paper, if someone asked, "What is the tallest impossible stack if I use these weird, almost-patterned numbers and filter them by 5?" the answer was usually "We don't know, it's too hard to calculate."

This paper provides a recipe book. It says: "If your numbers look like this, and you filter by that, here is the exact formula to find the answer."

The Big Picture

Think of the paper as a master carpenter who figured out how to build a specific, complex type of chair (the quotient semigroup) using a new, simpler tool (the minimization problem).

The Open Problem: Mathematicians had been stuck on a specific chair design (proposed by Adeniran et al.) for a while.
The Breakthrough: This paper didn't solve every chair design, but it solved the most common and difficult ones where the filter divides the first leg of the chair.

In short, the paper takes a scary, abstract math problem about "impossible numbers" and turns it into a manageable puzzle with clear, closed-form answers for many real-world scenarios.

1. Problem Statement

The paper addresses the computation of invariants for quotients of numerical semigroups.

Context: Let $\langle A \rangle$ be a numerical semigroup generated by a set of relatively prime positive integers $A = \{a_1, a_2, \dots, a_n\}$ . For a positive integer $p$ , the quotient semigroup is defined as $\langle A \rangle_p = \{x \in \mathbb{N} \mid px \in \langle A \rangle\}$ .
The Challenge: While $\langle A \rangle_p$ is known to be a numerical semigroup, calculating its fundamental invariants—the Frobenius number ( $F$ , the largest integer not in the semigroup) and the genus ( $g$ , the count of non-negative integers not in the semigroup)—is generally difficult and often lacks closed-form solutions.
Specific Focus: The paper focuses on the case where $p$ is a positive divisor of the smallest generator $a_1$ (denoted as $a$ ).
Open Problem: The work aims to partially solve an open problem proposed by Adeniran et al. regarding the formulas for $F(\langle A \rangle_p)$ and $g(\langle A \rangle_p)$ when $A$ consists of almost arithmetic progressions (AAPs) and $p \mid a$ .

2. Methodology

The author employs a reduction strategy based on the Apéry set of the numerical semigroup.

Apéry Set Reduction:
- The Apéry set of an element $m$ in a semigroup $S$ , denoted $Ap(S, m)$, is the set of the smallest elements in $S$ for each residue class modulo $m$ .
- The Frobenius number and genus can be directly calculated from the Apéry set (via Lemma 2.1):
  - $F(S) = \max(Ap(S, m)) - m$
  - $g(S) = \frac{1}{m} \sum_{w \in Ap(S, m)} w - \frac{m-1}{2}$
- The core innovation is reducing the computation of the Apéry set elements of the quotient $\langle A \rangle_p$ to a simple minimization problem, denoted as $OB(M)$.
- For a generator set $A = (a, ha + dB)$, the elements of the Apéry set of $\langle A \rangle_p$ are expressed as:
  $N_{dr, p} = \min \left\{ OB(ma + rp) \cdot \frac{ha}{p} + \left(\frac{ma}{p} + r\right)d \;\Big|\; m \in \mathbb{N} \right\}$
  where $OB(M)$ is the minimum number of generators required to sum to $M$ using a specific subset of the generators.
Monotonicity Analysis:
- The author analyzes the function $N_{dr, p}(m)$ to determine if it is increasing with respect to $m$ . If it is increasing, the minimum occurs at $m=0$ , allowing for explicit closed formulas.
- This approach modifies a reduction method previously proposed by the author in a prior work [9].

3. Key Contributions and Results

The paper provides closed-form formulas for the Frobenius number and genus for several specific types of almost arithmetic progressions under the condition $p \mid a$ .

A. General Scaling Property (Theorem 1.2)

The paper establishes a relationship between the invariants of a semigroup generated by $(a, dB)$ and one generated by $(a, B)$ .

If $A = (a, db_1, \dots, db_k)$ $A = (a, d b_{1}, \dots, d b_{k})$ , then:
- $F(\langle A \rangle_p) = d \cdot F(\langle (a, b_1, \dots, b_k) \rangle_p) + \frac{a(d-1)}{p}$
- $g(\langle A \rangle_p) = d \cdot g(\langle (a, b_1, \dots, b_k) \rangle_p) + \frac{(a-p)(d-1)}{2p}$
This allows the reduction of problems involving scaled generators to simpler base cases.

B. Almost Arithmetic Progressions (Theorem 1.3)

For $A = (a, ha+d, ha+2d, \dots, ha+kd)$ with $p \mid a$ :

Frobenius Number:
$F(\langle A \rangle_p) = \left\lceil \frac{a-p}{k} \right\rceil \cdot \frac{ha}{p} + \frac{a(d-1)}{p} - d$
Genus:
$g(\langle A \rangle_p) = h \cdot \sum_{r=1}^{\frac{a}{p}-1} \left\lceil \frac{rp}{k} \right\rceil + \frac{(a-p)(d-1)}{2p}$
Significance: When $h=1$ , this provides a partial solution to the open problem posed by Adeniran et al. for standard arithmetic progressions.

C. AAPs with Initial Gaps (Theorem 1.4)

For $A = (a, ha+(K+1)d, \dots, ha+kd)$ (skipping the first $K$ terms of the progression):

The formulas depend on the relationship between $p$ and the gap size $K$ .
Case $p > K$ : The formulas resemble the standard AAP case.
Case $p \le K$ : The formulas become more complex, involving a split summation and a floor function term $\lfloor K/p \rfloor$ $⌊ K / p ⌋$ .
- The Frobenius number involves a maximum of two potential candidates depending on the residue classes.

D. Specific Small Cases (Theorems 1.5 & 1.6)

Three-term AAPs with gaps (Theorem 1.5): For $A = (a, ha-d, ha+d)$, the paper derives explicit formulas involving a parameter $s = \lfloor \frac{ha-d}{2hp} \rfloor$ .
AAPs with Odd Terms (Theorem 1.6): For $A = (a, ha+d, ha+3d, \dots, ha+(2k+1)d)$ $A = (a, ha + d, ha + 3 d, \dots, ha + (2 k + 1) d)$ .
- If the remainder $t'$ (from $a-p$ divided by $2k+1$ ) is even, a closed formula is provided.
- If $t'$ is odd, the paper notes that the case is complicated and a closed formula is difficult to obtain, highlighting a boundary of the current method.

4. Significance and Impact

Partial Solution to Open Problems: The work successfully resolves the specific case of the Adeniran et al. open problem where the divisor $p$ divides the first generator $a$ . This was previously an unsolved area for general $n \ge 3$ .
Unified Framework: By utilizing the Apéry set reduction to the $OB(M)$ minimization problem, the paper provides a unified method to handle various "almost" arithmetic structures (standard, with gaps, with odd terms).
Computational Efficiency: The derived formulas are closed-form (or involve simple finite sums), making the calculation of Frobenius numbers and genera for these specific semigroups computationally efficient compared to brute-force enumeration.
Verification: The author validates the theoretical results using numerical examples and cross-references them with the numericalsgps GAP package, ensuring the correctness of the derived formulas.

5. Conclusion

Feihu Liu's paper advances the theory of numerical semigroups by providing explicit formulas for the invariants of quotient semigroups generated by almost arithmetic progressions. The key breakthrough is the reduction of the Apéry set computation to a minimization problem, which allows for the derivation of closed formulas in cases where $p$ divides the smallest generator. While the method covers a wide range of structured sequences, it also identifies specific parity conditions (as seen in Theorem 1.6) where closed formulas remain elusive, pointing to directions for future research.