pith. sign in

arxiv: 2312.06096 · v1 · submitted 2023-12-11 · 🧮 math.NT · math.CO

On quotients of numerical semigroups for almost arithmetic progressions

Feihu Liu This is my paper

Pith reviewed 2026-05-24 05:40 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords numerical semigroupsquotientsApéry setFrobenius numberalmost arithmetic progressionsminimization
0
0 comments X

The pith

When p divides a1, the Apéry set of a1/p in the quotient reduces to a minimization problem that yields closed formulas for the Frobenius number of almost arithmetic progression semigroups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for a numerical semigroup generated by relatively prime integers, if p divides the first generator a1, then the Apéry set of a1/p inside the quotient semigroup can be computed by solving a straightforward minimization. This reduction produces explicit closed formulas for the Frobenius number when the original semigroup belongs to the families of almost arithmetic progressions or almost arithmetic progressions with initial gaps. A reader would care because these quotients are otherwise difficult to analyze directly, and the formulas give concrete access to their invariants while also addressing part of an open question on such quotients.

Core claim

When p is a positive divisor of a1, the computation of the Apéry set of a1/p in <A>/p reduces to a simple minimization problem. This allows closed formulas for the Frobenius number of the quotient when <A> is an almost arithmetic progression or an almost arithmetic progression with initial gaps, and it partially solves an open problem on quotients of numerical semigroups.

What carries the argument

Reduction of the Apéry set of a1/p in the quotient to a minimization problem over the original generators.

If this is right

  • The Frobenius number of <A>/p admits a closed formula when <A> is an almost arithmetic progression.
  • The same holds for almost arithmetic progressions with initial gaps.
  • The minimization reduction gives a practical route to other invariants of the quotient beyond the Frobenius number.
  • The approach partially resolves the open problem stated by Adeniran et al. on quotients of numerical semigroups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The minimization technique could be tested on other structured families of numerical semigroups to see whether similar closed formulas appear.
  • The explicit formulas may allow faster enumeration or classification of quotient semigroups in computational experiments.
  • Connections between the minimization step and linear Diophantine equations might be examined to generalize the method.

Load-bearing premise

The generators are relatively prime positive integers and p divides the first generator a1.

What would settle it

Pick a concrete almost arithmetic progression with generators a1 to an where p divides a1, compute the Frobenius number of the quotient both via the derived closed formula and via direct listing of elements up to the expected bound, and check whether the two values agree.

read the original abstract

Let $\langle A\rangle$ be the numerical semigroup generated by relatively prime positive integers $\{a_1,a_2,...,a_n\}$. The quotient of $\langle A\rangle$ with respect to a positive integer $p$ is defined by $\frac{\langle A\rangle}{p}=\{x\in \mathbb{N} \mid px\in \langle A\rangle\}$. The quotient $\frac{\langle A\rangle}{p}$ is known to be a semigroup but is hard to study. When $p$ is a positive divisor of $a_1$, we reduce the computation of the Ap\'ery set of $\frac{a_1}{p}$ in $\frac{\langle A\rangle}{p}$ to a simple minimization problem. This allow us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These includes the cases when $\langle A\rangle$ is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps, etc. In particular, we partially solve an open problem proposed by A. Adeniran et al.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the quotient semigroup <A>/p for a numerical semigroup <A> generated by coprime positive integers a1,...,an. When p divides a1, it reduces computation of the Apéry set of a1/p inside the quotient to a minimization problem over the original generators. This reduction is then used to derive closed-form expressions for the Frobenius number of the quotient when <A> belongs to the families of almost arithmetic progressions, almost arithmetic progressions with initial gaps, and related variants. The work partially resolves an open problem posed by Adeniran et al.

Significance. If the reduction to minimization and the resulting closed formulas hold, the paper supplies explicit, computable expressions for the Frobenius number of quotients in structured families where general methods yield only algorithmic descriptions. This is a concrete contribution to the study of numerical-semigroup quotients, which are otherwise difficult to analyze explicitly.

minor comments (3)
  1. The abstract asserts that the minimization yields closed formulas but does not indicate the precise form of the objective function being minimized; adding a one-sentence description of this function in the abstract would improve readability.
  2. Section 2 (or the introductory definitions) would benefit from a short numerical example illustrating the quotient construction before the general reduction is stated.
  3. The paper should include at least one explicit verification that the derived Frobenius formula matches direct computation of the Apéry set for a small almost-arithmetic-progression example.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on quotients of numerical semigroups and the recommendation for minor revision. The referee's summary accurately captures the main contributions: the reduction of the Apéry set computation to a minimization problem when p divides a1, the resulting closed-form Frobenius formulas for almost arithmetic progressions and variants, and the partial resolution of the open problem from Adeniran et al.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the quotient semigroup and Apéry set in the standard way from the numerical semigroup generated by coprime integers, then asserts a reduction of the Apéry-set computation (when p divides a1) to an explicit minimization that exploits the almost-arithmetic-progression structure to produce closed-form Frobenius numbers. This is a direct algebraic/combinatorial argument under the usual coprimality hypothesis; no step equates a claimed result to a fitted parameter, renames an input, or rests on a self-citation chain whose own justification is internal to the paper. The cited open problem is external. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of numerical semigroups generated by relatively prime integers and the definition of their quotients; no free parameters, invented entities, or non-standard axioms are introduced in the abstract.

axioms (2)
  • standard math A numerical semigroup is generated by a finite set of relatively prime positive integers.
    Invoked in the opening sentence of the abstract as the starting object.
  • standard math The quotient of a numerical semigroup by p is itself a semigroup.
    Stated as known in the abstract.

pith-pipeline@v0.9.0 · 5710 in / 1276 out tokens · 18333 ms · 2026-05-24T05:40:03.087571+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    Adeniran, S

    A. Adeniran, S. Butler, C. Defant, Y. Gao, P. E. Harris, C. Hettle, Q. Liang, H. Nam, and A. Volk, On the genus of a quotient of a numerical semigroup , Semigroup Forum. 98 (2019), 690–700

    work page 2019

  2. [2]

    Brauer, On a problem of partitions , Amer

    A. Brauer, On a problem of partitions , Amer. J. Math. 64 (1942), 299–312

    work page 1942

  3. [3]

    Brauer, and J

    A. Brauer, and J. E. Shockley, On a problem of Frobenius , J. Reine Angew Math. 211 (1962), 215–220

    work page 1962

  4. [4]

    Cabanillas, Quotients of numerical semigroups generated by two numbers , arXiv:1904.082402v2

    E. Cabanillas, Quotients of numerical semigroups generated by two numbers , arXiv:1904.082402v2. (2019)

    work page arXiv 1904

  5. [5]

    Curtis, On formulas for the Frobenius number of a numerical semigroup , Math

    F. Curtis, On formulas for the Frobenius number of a numerical semigroup , Math. Scand. 67 (1990), 190–192

    work page 1990

  6. [6]

    numericalsgps

    M. Delgado, P. A. Garc´ ıa-S´ anchez, and J. Morais, “numericalsgps”: a gap package on numerical semigroups , Version 1.2.0 dev (2019), (Refereed GAP package), http://gap- packages.github.io/numericalsgps

    work page 2019

  7. [7]

    Delgado, P

    M. Delgado, P. A. Garc´ ıa-S´ anchez, and J. C. Rosales,Numerical semigroups problem list , CIM Bulletin. 33 (2013), 15–26

    work page 2013

  8. [8]

    Komatsu, Sylvester sum on the Frobenius set in arithmetic progression with initial gaps, arXiv:2210.17019v1, 2022

    T. Komatsu, Sylvester sum on the Frobenius set in arithmetic progression with initial gaps , arXiv:2210.17019v1, (2022)

    work page arXiv 2022

  9. [9]

    A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)

    F. Liu, and G. Xin, A combinatorial approach to Frobenius numbers of some special sequences (Complete Version), arXiv:2303.07149, (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  10. [10]

    J. L. Ram´ ırez Alfons´ ın,The Diophantine Frobenius Problem , Oxford Lecture Series in Mathe- matics and Its Applications, vol. 30, Oxford University Press, 2005

    work page 2005

  11. [11]

    J. B. Roberts, Note on linear forms , Proc. Amer. Math. Soc. 7 (1956), 465–469

    work page 1956

  12. [12]

    A. M. Robles-P´ erez, and J. C. Rosales,Equivalent proportionally modular Diophantine inequali- ties, Archiv der Mathematik. 90 (2008), 24–30

    work page 2008

  13. [13]

    ¨O. J. R¨ odseth,On a linear diophantine problem of Frobenius , J. Reine Angew Math. 301 (1978), 171–178

    work page 1978

  14. [14]

    J. C. Rosales, Fundamental gaps of numerical semigroups generated by two elements , Linear Algebra Appl. 405 (2005), 200–208

    work page 2005

  15. [15]

    J. C. Rosales, P. A. Garc´ ıa-S´ anchez, J. I. Garc´ ıa-Garc´ ıa, and J. M. Urbano-Blanco,Proportionally modular Diophantine inequalities, J. Number Theory. 103 (2003), 281–294

    work page 2003

  16. [16]

    J. C. Rosales, and P. A. Garc´ ıa-S´ anchez,Numerical Semigroups, Developments in Mathematics. Vol.20, Springer, New York (2009)

    work page 2009

  17. [17]

    E. S. Selmer, On the linear Diophantine problem of Frobenius , J. Reine Angew. Math. 293/294 (1977), 1–17

    work page 1977

  18. [18]

    Strazzanti, Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup, Internat

    F. Strazzanti, Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup, Internat. J. Algebra Comput. 25(06) (2015), 1043–1053. ON QUOTIENTS OF NUMERICAL SEMIGROUPS FOR ALMOST ARITHMETIC PROGRESSIONS 11

    work page 2015

  19. [19]

    J. J. Sylvester, On sub-invariants, i.e., semi-invariants to binary quanties of an unlimited order , Amer. J. Math. 5 (1882), 119–136

    work page

  20. [20]

    J. J. Sylvester, Mathematical questions with their solutions , Eductional Times. 41 (1884), 21. 1School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China Email address : 1liufeihu7476@163.com

    work page