Arithmetic properties encoded in undermonoids

Bangzheng Li; Felix Gotti

REVIEW 3 major objections 4 minor 28 references

Arithmetic properties encoded in undermonoids

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that a monoid whose every submonoid is atomic must satisfy the ascending chain condition on principal ideals, and that the weaker condition of atomicity of all undermonoids already forces it.

desk verdict Resolves a real conjecture with a clean, correct argument; minor exposition gaps in the group-case claims, but the paper deserves review and likely acceptance. read the letter →

arxiv 2412.11199 v1 pith:7CLL4ZSJ submitted 2024-12-15 math.AC

Felix Gotti , Bangzheng Li This is my paper · ORCID

classification math.AC MSC 13F1513A0520M1313F05

keywords undermonoidhereditaryatomicityunderatomicityascendingchainconditiononprincipalidealsatomicmonoidboundedfactorizationhalf-factoriallength-factorial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether a factorization property that holds for every “undermonoid” of a cancellative commutative monoid — a submonoid that generates the same group of formal differences — must hold for every submonoid. Its main answer is yes for atomicity: Theorem 3.3 equates underatomicity with hereditary atomicity. Building on that, Theorem 4.2 proves that every hereditarily atomic monoid satisfies the ascending chain condition on principal ideals (ACCP), settling a conjecture that had been open outside the torsion-free case. The same undermonoid-certification scheme works for the bounded factorization property, and for half-factoriality and length-factoriality it yields a complete classification: the only monoids all of whose undermonoids have either property are abelian groups with no element of infinite order. For reduced monoids, these results make hereditary atomicity and ACCP exactly the same condition.

What carries the argument

The load-bearing object is the undermonoid: $N\subseteq M$ with $\operatorname{gp}(N)=\operatorname{gp}(M)$. To force hereditary atomicity, the paper fixes an element $b$ that is not atomic in some submonoid and studies the poset $\mathcal{S}_b$ of submonoids $S$ for which $b\in S$ is not atomic, ordered by $S_1\preceq S_2$ when $S_1\subseteq S_2$ and $U(S_1)=S_1\cap U(S_2)$; Zorn’s lemma supplies a maximal $S$. The repeated structural move in the group case is the observation that if $u\in M$ either has some positive multiple in $S$ or admits no cancellation relation with $S$, then adjoining $2b+u$ to $S$ — taking $S'=S+\mathbb{N}_0(2b+u)$ — preserves the unit group and adds no new atoms except possibly $2b+u$ itself; maximality then forces $2b+u\in S$. Theorem 4.2 needs a second mechanism: Dickson’s lemma, which says every infinite subset of $\mathbb{N}_0^r$ contains an infinite strictly increasing coordinatewise chain, converts a non-stabilizing principal-ideal chain into a non-atomic submonoid.

What would settle it

A direct falsifier is a cancellative commutative monoid $M$ in which every submonoid is atomic but $M$ admits an infinite strictly ascending chain $m_0+M\subsetneq m_1+M\subsetneq\cdots$ of principal ideals; the paper’s Dickson-lemma argument claims such a chain necessarily produces a non-atomic submonoid, so any proposed chain can be tested against that construction.

Watch

Extended reading notes

Core claim

The central discovery is that a much smaller family of submonoids controls hereditary atomicity. A submonoid $N$ of $M$ is an undermonoid when $\operatorname{gp}(N)=\operatorname{gp}(M)$, i.e. when $N$ generates the same abelian group as $M$. Theorem 3.3 proves that if every undermonoid of $M$ is atomic, then every submonoid of $M$ is atomic. From there the paper proves Theorem 4.2: every hereditarily atomic monoid satisfies the ascending chain condition on principal ideals. The proof assumes an infinite strictly ascending chain $m_0+M\subsetneq m_1+M\subsetneq\cdots$ and uses Dickson’s lemma to select increasing blocks of the differences $a_n=m_n-m_{n-1}$, producing a submonoid generated by certain block sums $b_n$ and remainders $m'_n$ in which none of the $b_n$ can be atoms, contradicting atomicity of all submonoids. The paper then proves the analogous undermonoid-certification result for the bounded factorization property, and shows that the half-factorial and length-factorial versions hold only in abelian groups in which every element has finite order.

Load-bearing premise

The argument’s load-bearing premise is the group-case step that adjoining $2b+u$ to a maximal submonoid $S$ in which $b$ is not atomic preserves the invertible elements and atoms of $S$; if that preservation ever fails, the contradictions in the group cases of Theorems 3.3, 5.4, 6.2, and 6.3 do not go through.

Editorial extensions

If this is right

Hereditary atomicity and underatomicity coincide: to certify that every submonoid is atomic, it suffices to check submonoids that generate the same Grothendieck group.
Every hereditarily atomic monoid satisfies ACCP; in reduced monoids the two conditions are equivalent, and every ACCP failure produces a non-atomic submonoid.
The hereditary bounded factorization property is likewise certified by undermonoids.
The monoids whose submonoids (or undermonoids) are all half-factorial are exactly the abelian groups with no infinite-order elements, and the same classification holds for length-factoriality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same maximal-element strategy is a natural template for other hereditary factorization properties; the paper’s open question about finite factorization monoids is the sharp test case, since adjoining $2b+u$ could create infinitely many factorizations of one element without creating new atoms.
The contrapositive of Theorem 4.2 is a constructive-looking statement: any non-stabilizing principal-ideal chain comes with an explicit non-atomic submonoid, which may be useful for transferring hereditary atomicity questions to structures built from monoids, such as monoid algebras.
The classification of half-factorial and length-factorial undermonoids as torsion groups suggests a purely group-theoretic rigidity: any infinite-order element embeds a copy of $\mathbb{N}_0$, and the atoms $2c,3c$ (or $3c,4c,5c$) then force two distinct factorizations, so these properties cannot coexist with infinite order.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: this paper resolves the Gotti–Vulakh conjecture (hereditarily atomic implies ACCP) and introduces the undermonoid notion, proving it is equivalent to hereditary atomicity. That is a genuinely substantial result. The main proof in Theorem 4.2 uses Dickson's lemma in an elegant way and, as far as I can see, it is correct. Theorem 3.3, underatomicity iff hereditary atomicity, is the conceptual core; the non-group case is a neat ideal trick, and the group case is a Zorn's lemma argument that was the most delicate part. The same group-case machinery is reused for BFM, HFM, and LFM, and it checks out, though not without effort.

The soft spots are mostly presentational. The proofs of the group-case claims in Theorems 3.3, 5.4, 6.2, and 6.3 are compressed with 'mimicking' references. One specific step, showing that 2b+u lies in the maximal S, is terse: from k≥2 it doesn't immediately follow that s+(k-1)(2b+u) is not a unit. The stress-test note fills this with a short subargument, so it is a gap in exposition, not a real flaw. Similarly, Lemma 5.3 and part of Lemma 6.1 are abbreviated, but the arguments are routine. There is a typo in Theorem 4.2 ('A(M'_n)' should be 'A(M')') that should be fixed. None of this undermines the main theorems.

What the paper does well: it introduces a clean concept that sharpens the distinction between checking submonoids and undermonoids, and it settles a conjecture that had been open for torsion-free monoids. The classification of monoids all of whose undermonoids are HFM or LFM (torsion abelian groups) is a satisfying endpoint. The writing is clear, and the examples at the end, separating the monoid-theoretic results from domain-theoretic analogues, are useful. One small caveat: the paper does not resolve the analogous FFM question (Question 5.5), and the authors say so honestly.

Who should read this: anyone working on atomicity, ACCP, or factorization theory in commutative monoids. It deserves a serious referee. My recommendation is to send it to review and ask for a revision that expands the compressed group-case arguments and fixes the typo. I would probably cite it in my own work on hereditary properties of monoids.

Referee Report

3 major / 4 minor

Summary. The paper studies, for cancellative commutative monoids, whether a factorization property holding for all undermonoids (submonoids with the same Grothendieck group) forces it to hold for all submonoids. The main results are: Theorem 3.3, underatomicity is equivalent to hereditary atomicity; Theorem 4.2, every hereditarily atomic monoid satisfies the ACCP, proving a conjecture of Gotti and Vulakh; Theorem 5.4, the analogous implication for the bounded factorization property; and Theorems 6.2 and 6.3, which characterize monoids whose undermonoids (equivalently, submonoids) are half-factorial or length-factorial as precisely the torsion abelian groups. The paper also gives examples separating monoid-level hereditary properties from ring-level hereditary properties of integral domains.

Significance. If the results stand, the paper closes a conjecture in full generality and gives a clean structural answer for HFM and LFM: the only monoids all of whose submonoids are half-factorial or length-factorial are torsion groups. The proof of Theorem 4.2 is a genuine strength: the Dickson's lemma argument in Lemma 4.1 and the block-construction in the Claim are written out in detail, and the non-group case of Theorem 3.3 is elegant. The paper is also honest about its limits, explicitly leaving the FFM analogue open in Question 5.5. The main concerns are local: a misstated definition in Section 5 and a compressed maximality-extension step in the group cases, both of which are repairable without changing the overall strategy.

major comments (3)

[Section 5, Definition 5.2] The definition of 'boundedly atomic' is incorrect as written. It says that b is boundedly atomic if there exists n such that b can be written as a sum of at most n non-invertible elements. Under this existential reading, every non-invertible element satisfies the condition with n = 1, so the assertion that a monoid is a BFM iff every element is boundedly atomic is false. The proofs in Lemma 5.3 and Theorem 5.4 use the intended universal reading: there is a uniform bound on the length of every decomposition of b into non-invertible elements, equivalently that b is atomic and sup L_M(b) is finite. This definition must be corrected and the surrounding arguments should be adjusted to match.
[Theorem 3.3, proof of Claim 2] The step concluding that k = 1 and s is a unit of S from a' = s + k(2b + u) is missing an argument. The non-unitness of 2b + u only rules out k >= 2 if the other summand s + (k-1)(2b + u) is not a unit, and that is not proved. The gap is fillable: if x = s + (k-1)(2b + u) were a unit of S', then x is a unit of S, so -x is in S and (k-1)u + [s + (2k-2)b - x] = 0, contradicting condition (ii); if condition (i) holds instead, multiplying the unit relation by n0 forces a positive multiple of b to be a unit, contradicting b not in U(S). This sublemma should be written out, since the same compressed step is reused in the group cases of Theorems 5.4, 6.2, and 6.3.
[Theorem 5.4, Case 2] The assertion that 'the fact that b is not boundedly atomic in S immediately implies that b is not boundedly atomic in S′' is not immediate and is not proved. Given a decomposition b = sum_i (s_i + k_i(2b + u)) in S' with K = sum_i k_i >= 1, one obtains K u + q = 0 for q = sum_i s_i + (2K-1)b in S, so condition (ii) fails and condition (i) gives n0 u in S; multiplying through by n0 then forces a positive multiple of b to be a unit, contradicting b not in U(S). Hence every such decomposition has all k_i = 0 and is actually a decomposition in S. This step needs to be stated explicitly for the proof to be complete.

minor comments (4)

[Theorem 4.2, proof] In the sentence 'm'_n /in A(M'_n)', the notation should be 'A(M′)', not 'A(M′_n)'; the intended statement is that m'_n is not an atom of the submonoid M'.
[Theorem 6.3, proof] In the paragraph on pairwise non-associate atoms, the variable s is used inconsistently in place of c, e.g. '3s /in A(S)' and '4s /in A(S)' should refer to 3c and 4c.
[Theorem 4.2, proof of the Claim] The sentence explaining why the sequence (k_n) cannot be decreasing should say 'cannot be non-increasing' or 'cannot have no strict increase', since a merely non-increasing sequence of natural numbers could stabilize, which would also contradict the strict increase of the x_n.
[Example 6.5] The notation for the two factorizations of p^2 q^2 is clear, but it may help to note explicitly that p^2 and q^2 are atoms in the Hilbert monoid because the set Q includes products p q with p and q not necessarily distinct.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central theorems are derived by Zorn's lemma and Dickson's lemma, and self-citations are background only.

full rationale

The paper's central derivation chain is self-contained. Theorem 3.3 proves underatomicity iff hereditary atomicity by a Zorn's lemma maximal-element argument; the group-case step showing 2b+u belongs to the maximal S is a standard maximal-counterexample extension argument, not a presupposition of the conclusion. Theorem 4.2 independently proves hereditary atomicity implies ACCP via Dickson's lemma and the construction of a submonoid M' whose atomicity forces the contradictory representation m0 in the monoid generated by the b_n; it does not rely on Theorem 3.3 or on the conjecture from [22] that it proves. The bounded-factorization and half-/length-factoriality results follow the same maximal-element scheme and reduce the undermonoid hypotheses to the desired submonoid conclusions rather than importing them. Self-citations ([12], [16], [17], [22]) provide definitions, background, a previously settled torsion-free case, or an illustrative example; none carries the weight of the main proofs, and the cited statements are not used to define the target properties into existence. No fitted parameter, normalization choice, renamed empirical pattern, or equation that equals its own input by construction appears. The only minor blemish is a typographical 'A(M'_n)' where 'A(M')' is meant, which does not affect the derivation. Hence the paper receives a low score reflecting only the presence of non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The results rest on standard set-theoretic and monoid-theoretic tools: Zorn's lemma, Dickson's lemma, and the embedding properties of Grothendieck groups for cancellative commutative monoids. Standard factorization-theory implications (ACCP implies atomic, BFM implies ACCP) are cited from the literature. No ad hoc axioms or fitted parameters are introduced.

assumptions (6)

standard math Zorn's lemma is used to obtain maximal elements in the posets S_b defined in Lemmas 3.2, 5.3, and 6.1.
Invoked in Lemma 3.2(2), Lemma 5.3(2), and Lemma 6.1(2) to prove the posets have maximal elements, which is the structural engine of Theorems 3.3, 5.4, 6.2, and 6.3.
standard math Dickson's lemma: every subposet of (N0^r, <=) has finitely many minimal elements.
Used in Lemma 4.1 to construct arbitrarily long strictly increasing sequences in infinite subsets of N0^r, which underpins the proof of Theorem 4.2.
standard math The Grothendieck group of a cancellative commutative monoid is the universal group, N embeds in gp(M), and gp(N) is identified with the subgroup of gp(M) generated by N.
Used throughout to define undermonoids and to argue that M' = N union I has gp(M') = gp(M) in Theorems 3.3, 5.4, 6.2, and 6.3.
standard math In a cancellative monoid, if a finite sum of elements equals the identity, then each summand is invertible; also U(N) is contained in U(M) for every submonoid N of M.
Used repeatedly in the group-case arguments of Theorems 3.3, 5.4, 6.2, and 6.3 to derive contradictions from relations such as (k+1)u + (s + (2k+2)b) = 0.
domain assumption Every monoid satisfying the ACCP is atomic, from Geroldinger and Halter-Koch [14, Proposition 1.1.4].
Cited in Section 2 and used in Corollary 4.3, part (c) implies (a), to close the equivalence with hereditary atomicity.
domain assumption Every bounded factorization monoid satisfies the ACCP, from Geroldinger and Halter-Koch [14, Corollary 1.3.3].
Cited in Section 2 to justify the naive expectation in Section 5 that hereditary atomicity might imply the BFM property, which Example 5.1 refutes.

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Pith. "Pith review of Arithmetic properties encoded in undermonoids." pith.science (2026). https://pith.science/paper/7CLL4ZSJ

@misc{pith2026241211199,
  author       = {Pith},
  title        = {Pith review of: Arithmetic properties encoded in undermonoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CLL4ZSJ}},
  note         = {Machine review of arXiv:2412.11199}
}

abstract

Let $M$ be a cancellative and commutative monoid. A submonoid $N$ of $M$ is called an undermonoid if the Grothendieck groups of $M$ and $N$ coincide. For a given property $\mathfrak{p}$, we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of $M$ satisfy $\mathfrak{p}$ to conclude that all submonoids of $M$ satisfy $\mathfrak{p}$? In this paper, we give a positive answer to this question for the property of being atomic, and then we prove that if $M$ is hereditarily atomic (i.e., every submonoid of $M$ is atomic), then $M$ must satisfy the ACCP, proving a recent conjecture posed by Vulakh and the first author. We also give positive answers to our main question for the following well-studied factorization properties: the bounded factorization property, half-factoriality, and length-factoriality. Finally, we determine all the monoids whose submonoids/undermonoids are half-factorial (or length-factorial).

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorizations in integral domains , J. Pure Appl. Algebra 69 (1990) 1–19

1990

[2] [2]

D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains II , J. Algebra 152 (1992) 78–93

work page 1992

[3] [3]

D. F. Anderson and F. Gotti, Bounded and ﬁnite factorization domains . In: Rings, Monoids, and Module Theory (Eds. A. Badawi and J. Coykendall) pp. 7–57. Springer Procee dings in Mathematics & Statistics, Vol. 382, Singapore, 2022

work page 2022

[4] [4]

J. G. Boynton and J. Coykendall, An example of an atomic pullback without the ACCP , J. Pure Appl. Algebra 223 (2019) 619–625

work page 2019

[5] [5]

A. Bu, J. Vulakh, and A. Zhao, Length-factoriality and pure irreducibility , Comm. Algebra 51 (2023) 3745–3755

work page 2023

[6] [6]

Carlitz, A characterization of algebraic number ﬁelds with class num ber two , Proc

L. Carlitz, A characterization of algebraic number ﬁelds with class num ber two , Proc. Amer. Math. Soc. 11 (1960) 391–392

work page 1960

[7] [7]

S. T. Chapman, A tale of two monoids: a friendly introduction to nonunique f actorizations, Math. Mag. 87 (2014) 163–173

work page 2014

[8] [8]

S. T. Chapman, J. Coykendall, F. Gotti, and W. W. Smith, Length-factoriality in commutative monoids and integral domains, J. Algebra 578 (2021) 186–212

work page 2021