On the ascent of almost and quasi-atomicity to monoid semidomains

Felix Gotti; Ishan Panpaliya; Victor Gonzalez

REVIEW 3 major objections 3 minor 1 cited by

On the ascent of almost and quasi-atomicity to monoid semidomains

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that quasi-atomicity does not ascend to polynomial extensions and that neither almost atomicity nor quasi-atomicity ascends to monoid domains over finite fields.

desk verdict Plausible and worth exploring, but as written the two headline negative results are not proven: Theorem 4.1 rests on a false UFD assertion, and Theorem 5.6 has an unhandled unit case. read the letter →

arxiv 2501.04990 v1 pith:4DUZVAM5 submitted 2025-01-09 math.AC

Victor Gonzalez , Felix Gotti , Ishan Panpaliya This is my paper · ORCID

classification math.AC MSC 13F1513A0520M2506F0511Y0513G05

keywords almostatomicityquasi-atomicityatomicdomainmonoidsemidomainPuiseuxpolynomialextensionfinitefield

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

Two weakenings of atomicity—almost atomicity and quasi-atomicity—say that every element becomes factorable after multiplication by an atomic element, or by any element at all. This paper asks whether these properties are preserved when one passes from a semidomain to its polynomial extension or to a monoid semidomain. It establishes a positive result under extra structure: if the exponent monoid satisfies the ascending chain condition on principal ideals and the coefficient sets of indecomposable polynomials have maximal common divisors, then both weak forms of atomicity ascend. The paper then shows that the extra conditions are necessary: the quasi-atomic ring $\mathbb{Z}+\mathbb{Z}x+x^2\mathbb{R}[x]$ has a polynomial extension that is not quasi-atomic, and for every prime $p$ there is a rank-one torsion-free atomic monoid $M_p$ for which $\mathbb{F}_p[M_p]$ is not quasi-atomic. Thus two natural positive questions about weaker atomicity are answered in the negative, extending the known failure of atomicity itself.

What carries the argument

The load-bearing mechanism is a substitution trick for monoid domains. Given a Puiseux monoid $M$ containing $\mathbb{N}_0[1/p]$ and a low-degree polynomial $f_d(x)$ over $\mathbb{F}_p$ whose dilation $f_d(x^{d^n})$ is irreducible for every $n$, a purported factorization of $f_d(x)$ in $\mathbb{F}_p[M]$ is evaluated at $x \mapsto x^{1/(pd)^n}$ for large $n$. This sends all exponents into $\mathbb{N}_0$, so the factorization lands in the UFD $\mathbb{F}_p[x]$; irreducibility of $f_d(x^{d^n})$ forces one substituted atom to be divisible by it, and pulling the divisibility back through the substitution shows the original atom was reducible. A secondary mechanism is the MCD/indecomposable-length argument in Theorem 3.2: an ACCP exponent monoid is used to select a longest factorization into nonconstant polynomials, and extraction of coefficient maximal common divisors turns indecomposables into atoms.

What would settle it

Exhibit a reduced monoid $M$ satisfying the ACCP and a polynomial $f \in S[M]$ whose decompositions into nonconstant polynomials over $S$ have unbounded length; such an example would break the bounded-length step in Theorem 3.2. For the finite-field construction, a direct check in the $p=2$ case—deciding whether $x^2+x+1$ in $\mathbb{F}_2[M_{2,3}]$ has a nonzero multiple that factors into irreducibles—would settle the corresponding claim of Theorem 5.6.

Watch

Extended reading notes

Core claim

The central claim is that the two weakest commonly studied factorization properties do not transfer automatically. Under the hypotheses of Theorem 3.2—a reduced monoid $M$ satisfying the ACCP and a coefficient semidomain whose indecomposable-polynomial coefficient sets admit maximal common divisors—$S[M]$ is almost atomic when $S$ is, and quasi-atomic when $S$ is. The paper proves this by factoring an arbitrary polynomial into indecomposables of maximal length, pulling out coefficient MCDs, and using divisor-closedness of the monomial submonoid. It then shows sharpness: the ring $R = \mathbb{Z}+\mathbb{Z}x+x^2\mathbb{R}[x]$ is quasi-atomic, yet $R[y]$ is not; the proof forces the linear polynomial $\alpha x^2 y + \beta x^2$ to have no quasi-atomic multiple when $\alpha,\beta$ are algebraically independent irrationals. For finite fields, Theorem 5.6 constructs, for each prime $p$, an atomic Puiseux monoid $M_p$ with $\mathbb{F}_p[M_p]$ not quasi-atomic, via a substitution argument that transfers any purported factorization into the UFD $\mathbb{F}_p[x]$ and contradicts irreducibility of a chosen binomial or trinomial.

Load-bearing premise

The positive ascent theorem rests on the claim that in a reduced monoid satisfying the ACCP, any fixed polynomial in $S[M]$ has a bounded number of nonconstant factors; this claim is not proved and does not follow from ACCP for arbitrary commutative monoids.

Editorial extensions

If this is right

If the hypotheses of Theorem 3.2 hold, almost atomicity and quasi-atomicity survive the passage to monoid semidomains, so the positive ascent results cover polynomial extensions with an MCD condition.
Quasi-atomicity is not preserved by adjoining a variable: the quasi-atomic ring $\mathbb{Z}+\mathbb{Z}x+x^2\mathbb{R}[x]$ has a non-quasi-atomic polynomial extension.
For every prime $p$, there are rank-one torsion-free atomic monoids whose monoid algebras over $\mathbb{F}_p$ fail even quasi-atomicity, so neither weak ascent property holds for monoid domains over finite fields.
The examples separate the hierarchy: $\mathbb{Z}[x]+\mathbb{Q}[x]x^2$ is almost atomic but not atomic, while $\mathbb{Z}[x]+\mathbb{R}[x]x^2$ is quasi-atomic but not almost atomic.
Whether almost atomicity ascends to ordinary polynomial extensions remains open, as the paper explicitly asks.

Reading between the lines

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

An extension not pursued here is to test whether the substitution method works over coefficient fields other than $\mathbb{F}_p$, for instance over $\mathbb{Q}$ or number fields, provided one can find polynomials whose $d$-th power dilations are irreducible.
The boundedness step in Theorem 3.2 uses a claim about ACCP monoids that is not generally valid; if a counterexample exists, the positive ascent theorem would hold under a stronger hypothesis such as bounded factorization rather than ACCP.
The non-ascent example suggests a recipe for producing more quasi-atomic rings whose polynomial extensions fail: concentrate the non-atomicity in monomials of degree at least 2 and then form a linear polynomial whose coefficients are algebraically independent monomials.
A concrete computational test of the $p=2$ case would decide whether $x^2+x+1$ in $\mathbb{F}_2[M_{2,3}]$ admits a quasi-atomic multiple; a positive answer would force a revision of the finite-field construction.

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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

For my money the paper is worth engaging with, but not as written. The two headline negative results—quasi-atomicity not ascending to polynomial extensions (Theorem 4.1) and neither almost nor quasi-atomicity ascending to monoid domains over finite fields (Theorem 5.6)—are both plausible and interesting, but each has a proof gap that the authors need to close. The ascent theorem for monoid semidomains (Theorem 3.2) is also under-supported.

What is genuinely useful: the paper identifies the right questions and provides a coherent strategy. The construction in Section 3 of almost/quasi-atomic domains is nice and correct as far as I checked. The finite-field monoid construction in Section 5 is a natural extension of Coykendall–Gotti, and the reduction to irreducibility of binomials/trinomials via Lidl–Niederreiter is clever. If the gaps close, the significance is real: these would settle natural analogues of Roitman's theorem.

Now the soft spots, in order of severity.

First, Theorem 4.1. The proof asserts that R[x,y] is a UFD, where R = Z + Zx + x^2 R[x] (here R means the real polynomial ring). That is false. The element h = sqrt(2)x^2 lies in R and equals 2 * (sqrt(2)/2 x^2); both factors are nonunits, and the same move repeats, so h is not atomic and R fails ACCP. Hence R[x,y] is not atomic, let alone a UFD. The UFD claim is load-bearing: it is used to obtain equation (4.4) and isolate a degree-one factor. Without it the proof does not go through. Maybe the authors meant the real polynomial ring in x,y; then the factor r in (4.4) would be an arbitrary real, not necessarily an element of R, and subsequent steps still need repair.

Second, Theorem 5.6. The claim that B(x) is not a unit is not justified when g(x) is a constant and t = 1; in that case B(x) is a unit and the factorization is trivial. The argument needs to handle this case separately.

Third, Theorem 3.2's proof assumes that ACCP of M bounds the length of decompositions of a polynomial into nonconstant factors. That requires a short argument and may fail in general commutative monoids; in the rank-one linearly ordered setting it might be true, but it is not written.

These are not manufactured quibbles; the false UFD assertion in particular is a serious flaw in the current write-up. The paper should not be accepted as is, but it deserves a serious referee and a revision. If the authors fix these gaps, I would expect the results to hold and the paper to be a solid contribution.

Who is this for? People working in factorization theory and monoid domains. A reading group could have productive discussion on the constructions and on what would be needed to repair them. But I would not cite the theorem statements as established until the proofs are corrected.

Referee Report

3 major / 3 minor

Summary. This paper studies the ascent of almost atomicity and quasi-atomicity to polynomial and monoid semidomains. Theorem 3.2 states that if S is almost atomic (resp. quasi-atomic) and M is a reduced monoid satisfying the ACCP, with an MCD condition on coefficient sets, then S[M] is almost atomic (resp. quasi-atomic). Section 4 claims a counterexample to the ascent of quasi-atomicity to polynomial extensions, using R = Z + Zx + x^2 R[x] (with real coefficients) and proving that R[y] is not quasi-atomic. Section 5 constructs, for each prime p, a rank-one torsion-free atomic Puiseux monoid M_p such that F_p[M_p] is not quasi-atomic, yielding non-ascent of almost and quasi-atomicity to monoid domains over finite fields. The paper also provides examples of almost atomic non-atomic domains and quasi-atomic non-almost-atomic domains.

Significance. If the main results were established, they would be a meaningful contribution: Theorem 5.6 would strengthen the known non-ascent of atomicity to monoid domains, and Theorem 4.1 would settle negatively the ascent of quasi-atomicity to polynomial extensions. The explicit constructions and the use of external irreducibility criteria (Lidl-Niederreiter, Blake et al.) are appropriate and nontrivial. However, several load-bearing steps in the proofs are not justified as written, so the paper does not currently establish its central claims.

major comments (3)

[Section 3, proof of Theorem 3.2] The sentence 'The fact that M satisfies the ACCP guarantees that the set L is bounded' is not a consequence of the ACCP. The ACCP stabilizes ascending chains of principal ideals; it does not bound the number of factors in a factorization of a fixed element in an arbitrary commutative monoid. Since the proof chooses ℓ as the largest element of L, this assertion is load-bearing and the argument collapses without it. The theorem may be salvageable with an additional bounded-factorization hypothesis, but as stated the proof is incomplete.
[Section 4, proof of Theorem 4.1] The assertion 'R[x,y] is an atomic domain (indeed, R[x,y] is a UFD)' is false for the ring R defined in (4.1). The element h = sqrt(2)x^2 belongs to R and satisfies h = 2 * (sqrt(2)/2 x^2), where both 2 and (sqrt(2)/2)x^2 are nonunits in R; iterating gives h = 2^n * (sqrt(2)/2^n)x^2 for every n, so R (and hence R[x,y]) fails the ACCP and is not atomic. This invalidates the factorization-rearrangement used to pass from (4.3) to (4.4) and to isolate a y-degree-one irreducible factor. The notation in (4.1) is also ambiguous because the right-hand side uses the same symbol for the real numbers and for the ring being defined.
[Section 5, final paragraph of the proof of Theorem 5.6] The claim that 'the equality B(x) = G(x^{(pd)^{-n}}) guarantees that B(x) is not a unit' is not justified. If t = 1 and g(x) is a nonzero constant, then G(x) is a unit and B(x) is a unit; the displayed equality alone does not exclude this case. The contradiction to the irreducibility of a_j requires B(x) to be a nonunit, so this is a load-bearing gap. The authors need to prove that for the chosen index j either t ≥ 2 or g(x) is nonconstant.

minor comments (3)

[Section 4, equation (4.1)] Please disambiguate the use of R for the ring under construction and blackboard R for the real numbers in the defining expression; the current notation is circular if read literally.
[Throughout] There are numerous typographical and extraction artifacts ('A TOMICITY', 'elem ent', 'semi domain', 'must exits'); a careful proofreading pass is needed.
[Section 3, proof of Lemma 3.4(1)] The statement that the set of lengths of decompositions into nonconstant polynomials is bounded should be justified explicitly by a degree argument; the bound is clear in the ordinary polynomial ring but should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all central results are obtained from explicit constructions and external irreducibility criteria, never by assuming the target property.

full rationale

The paper contains no circular derivation. Section 3 states a conditional ascent theorem and proves it from definitions and a maximal-length decomposition; despite a gap in the boundedness justification, the conclusion is not assumed among the hypotheses. Section 4's non-ascent proof proceeds by contradiction, assuming a factorization into irreducibles and deriving a contradiction via coefficient and order arguments; the fact that R[x,y] is claimed to be a UFD is a mathematical error if R is the ring in (4.1), but an erroneous premise is not a circular premise. Section 5 constructs explicit atomic Puiseux monoids (Proposition 5.2) and uses the external irreducibility criteria of Lidl-Niederreiter (Theorem 5.3) and Blake et al. (Theorem 5.5), plus the cited [10, Lemma 5.3] for the p=2 trinomial; none of these assume non-quasi-atomicity of the monoid domain under construction. The self-citations to [10] and [21] provide constructions and motivation, but the load-bearing mathematics is either proved in the paper or is an independent statement about finite-field polynomials. The open question in Section 4 and the remark preceding Example 3.3 are honest scope limitations. The correctness concerns raised by a reviewer, namely the unbounded-length assertion in the proof of Theorem 3.2 and the UFD assertion for R[x,y] in Theorem 4.1, are validity gaps rather than circularity, because they do not identify a place where an output is equivalent to an input by definition or by fit. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new fitted constants or postulated entities. Its main extra assumptions are the MCD condition in Theorem 3.2 and the external irreducibility criteria, both invoked explicitly.

assumptions (4)

standard math F_p[x] is a UFD for every prime p.
Used throughout Section 5 to factor standard polynomial images and isolate irreducible factors.
standard math Binomial irreducibility criterion of Lidl-Niederreiter (Theorem 5.3) and trinomial criterion of Blake et al. (Theorem 5.5).
Invoked to guarantee that f_d(x^{d^n}) is irreducible for every n, which is load-bearing for the contradiction in Theorem 5.6.
ad hoc to paper The MCD condition on coefficient sets in Theorem 3.2.
A nonstandard hypothesis introduced to make the ascent proof work; it is not derived from the other assumptions.
domain assumption Semidomains embed as subsemirings of integral domains, and monoid semidomains of torsion-free monoids are semidomains.
Used in Section 2 to justify that S[M] is a semidomain and that monomials form a divisor-closed submonoid.

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Pith. "Pith review of On the ascent of almost and quasi-atomicity to monoid semidomains." pith.science (2026). https://pith.science/paper/4DUZVAM5

@misc{pith2026250104990,
  author       = {Pith},
  title        = {Pith review of: On the ascent of almost and quasi-atomicity to monoid semidomains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DUZVAM5}},
  note         = {Machine review of arXiv:2501.04990}
}

read the original abstract

A commutative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral (semi)domain is atomic if its multiplicative monoid is atomic. Notions weaker than atomicity have been introduced and studied during the past decade, including almost atomicity and quasi-atomicity, which were coined and first investigated by Boynton and Coykendall in their study of graphs of divisibility of integral domains. The ascent of atomicity to polynomial extensions was settled by Roitman back in 1993 while the ascent of atomicity to monoid domains was settled by Coykendall and the second author in 2019 (in both cases the answer was negative). The main purpose of this paper is to study the ascent of almost atomicity and quasi-atomicity to polynomial extensions and monoid domains. Under certain reasonable conditions, we establish the ascent of both properties to polynomial extensions (over semidomains). Then we construct an explicit example illustrating that, with no extra conditions, quasi-atomicity does not ascend to polynomial extensions. Finally, we show that, in general, neither almost atomicity nor quasi-atomicity ascend to monoid domains, improving upon a construction first provided by Coykendall and the second author for the non-ascent of atomicity.

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Works this paper leans on

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Pith tools

Reviewed August 10, 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

work page 1990

[2] [2]

D. F. Anderson and F. Gotti: Bounded and ﬁnite factorization domains . In: Rings, Monoids, and Module Theory (Eds. A. Badawi and J. Coykendall) Springer Proceedings in M athematics & Statistics, Vol. 382, Singapore, 2022

work page 2022

[3] [3]

I. F. Blake, X. Gao, R. C. Mullin, S. A. Vanstone, and T. Yag hoobian, Applications of ﬁnite ﬁelds , Springer Science + Business Media, New York 1993

work page 1993

[4] [4]

Boynton and J

J. Boynton and J. Coykendall, On the graph divisibility of an integral domain , Canad. Math. Bull. 58 (2015) 449–458

work page 2015

[5] [5]

A. Bu, F. Gotti, B. Li, and A. Zhao, One-dimensional monoid algebras and ascending chains of pr incipal ideals . Submitted. Preprint on arXiv: https://arxiv.org/abs/240 9.00580

work page

[6] [6]

S. T. Chapman, J. Jang, J. Mao, and S. Mao, On the set of Betti elements of a Puiseux monoid , Bull. Aust. Math. Soc. (to appear). DOI: https://doi.org/10.1017/S0004972 724000352

work page doi:10.1017/s0004972

[7] [7]

Chouinard, Krull semigroups and divisor class groups , Canad

L. Chouinard, Krull semigroups and divisor class groups , Canad. J. Math. 33 (1981) 1459–1468

work page 1981

[8] [8]

P. M. Cohn, Bezout rings and and their subrings , Proc. Cambridge Philos. Soc. 64 (1968) 251–264

work page 1968