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arxiv: 2605.20974 · v1 · pith:GU6CDXEVnew · submitted 2026-05-20 · 🧮 math.GR

Finite factorization is detected by undermonoids

Yutong Zhang , Yaoran Yang This is my paper

Pith reviewed 2026-05-21 01:59 UTC · model grok-4.3

classification 🧮 math.GR
keywords cancellative commutative monoidfinite factorization monoidundermonoidsubmonoidGrothendieck groupfactorization propertyhereditary
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The pith

For every cancellative commutative monoid, every submonoid is a finite factorization monoid if and only if every undermonoid is.

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

The paper proves that the finite factorization property is detected exactly by the undermonoids of a cancellative commutative monoid. If every undermonoid has the property that every element has only finitely many factorizations of each length, then the same holds for every submonoid. This equivalence answers a question of Gotti and Li and shows that it suffices to check the property on the full submonoids that generate the entire Grothendieck group. The argument fixes an element with infinitely many factorizations of length ell and enlarges any bad submonoid to a bad undermonoid via a divisor complement ideal or a careful perturbation.

Core claim

For a cancellative commutative monoid M, the conditions that every submonoid is an FFM and that every undermonoid is an FFM are equivalent. The proof isolates a fixed length ell and an infinite set of length-ell factorizations of one element b. In the non-group case a divisor-complement ideal I = {m in M : m does not divide b} enlarges the bad submonoid to a bad undermonoid while preserving the factorizations. In the group case a maximality argument is combined with the two-sided perturbation S maps to S + N0(2b + u) that creates no new units and does not split atoms, yielding an undermonoid with infinitely many factorizations of b.

What carries the argument

Divisor-complement ideal or two-sided perturbation that enlarges a submonoid with infinite factorizations into an undermonoid with the same infinite factorizations of a fixed element b.

Load-bearing premise

The enlargement of the bad submonoid via the divisor-complement ideal or the perturbation preserves the chosen infinite set of length-ell factorizations without introducing new units or splitting atoms.

What would settle it

Finding a cancellative commutative monoid M together with a submonoid S that is not an FFM while every undermonoid of M is an FFM.

read the original abstract

Let $M$ be a cancellative commutative monoid and call a submonoid $S$ of $M$ an undermonoid if $\G(S)=\G(M)$ inside the Grothendieck group of $M$. Gotti and Li asked whether the finite factorization property is hereditary once it is known on all undermonoids: if every undermonoid of $M$ is a finite factorization monoid, must every submonoid of $M$ be a finite factorization monoid? We give an affirmative answer. Equivalently, for every cancellative commutative monoid $M$, the following two conditions coincide: every submonoid of $M$ is an FFM, and every undermonoid of $M$ is an FFM. The proof isolates a fixed length $\ell$ and an infinite set of length-$\ell$ factorizations of one element $b$. In the non-group case, a divisor-complement ideal $I=\{m\in M:m\nmid_M b\}$ enlarges the bad submonoid to a bad undermonoid while preserving the chosen length-$\ell$ factorizations. In the group case, a maximality argument over submonoids for which these factorizations survive is combined with a two-sided perturbation $S\mapsto S+\Nzero(2b+u)$. The key point is that the perturbation creates no new units and does not split any atom occurring in the fixed factorizations. This yields an undermonoid with infinitely many factorizations of $b$, contradicting the hypothesis.

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 / 2 minor

Summary. The manuscript proves that for a cancellative commutative monoid M, every submonoid is a finite factorization monoid (FFM) if and only if every undermonoid is an FFM, where an undermonoid S satisfies G(S)=G(M) in the Grothendieck group of M. This affirmatively resolves a question of Gotti and Li by constructing, from any bad submonoid with infinitely many length-ℓ factorizations of an element b, a corresponding bad undermonoid. The non-group case enlarges via the divisor-complement ideal I={m∈M : m∤_M b}; the group case combines a maximality argument on factorization-preserving submonoids with the two-sided perturbation S↦S+ℕ₀(2b+u).

Significance. The result shows that the finite factorization property is detected precisely by the undermonoids. The argument is a direct, parameter-free construction from the definitions of cancellative commutative monoids, Grothendieck groups, and finite factorization; it explicitly verifies that the perturbation creates no new units and does not split the relevant atoms, with no reduction to previously fitted quantities or circularity.

minor comments (2)
  1. [§2] §2 (non-group case): the claim that I enlarges any bad submonoid to a bad undermonoid while preserving the chosen length-ℓ factorizations of b is central; a brief verification that G(I∪S)=G(M) holds under the standing cancellativity assumption would strengthen the exposition.
  2. [Abstract] Abstract and §3 (group case): the notation ℕ₀(2b+u) is clear in context but would benefit from an explicit sentence confirming it denotes the submonoid generated by 2b+u.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of the manuscript, including the accurate summary of the main result and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's proof proceeds by direct construction from the definitions of cancellative commutative monoids, Grothendieck groups, and finite factorization monoids. It isolates a fixed length ℓ with infinitely many factorizations of an element b, then applies the divisor-complement ideal I = {m ∈ M : m ∤_M b} in the non-group case to enlarge a bad submonoid to an undermonoid while preserving those factorizations, and uses a maximality argument plus the perturbation S ↦ S + ℕ₀(2b + u) in the group case, verifying no new units or split atoms. These steps rely only on cancellativity and commutativity with no reduction to fitted inputs, self-citations, or ansatzes; the derivation is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard definition of cancellative commutative monoids and the Grothendieck group construction; no free parameters, invented entities, or ad-hoc axioms beyond ordinary monoid theory are indicated.

axioms (1)
  • domain assumption M is a cancellative commutative monoid
    Explicitly stated as the ambient setting for the theorem.

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