On finitary power monoids of linearly orderable monoids

Bangzheng Li; Felix Gotti; Jiya Dani; Leo Hong; Shimon Schlessinger

REVIEW 5 major objections 5 minor 27 references

On finitary power monoids of linearly orderable monoids

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For any linearly orderable monoid M, the finitary power monoid P_fin(M) is atomic exactly when M is atomic and every finite subset of M has a maximal common divisor.

desk verdict New ascent and non-ascent results for power monoids of linearly orderable monoids, but the main characterization theorem has a real proof gap that needs to be fixed. read the letter →

arxiv 2501.03407 v1 pith:PX6FVUY7 submitted 2025-01-06 math.AC

Jiya Dani , Felix Gotti , Leo Hong , Bangzheng Li , Shimon Schlessinger This is my paper · ORCID

classification math.AC MSC 13A0513F1513A1513G05

keywords finitarypowermonoidatomiclinearlyorderablemaximalcommondivisorPuiseuxalmostFurstenbergpropertyTIDF

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

A finitary power monoid is built from a commutative monoid by taking all nonempty finite subsets and adding them elementwise. This paper asks which arithmetic properties survive that subset construction when the original monoid can be totally ordered compatibly with addition. Its central result is a complete characterization: for a linearly orderable monoid M, the power monoid P_fin(M) is atomic precisely when M is atomic and every nonempty finite subset of M has a maximal common divisor. This settles the ascent of atomicity on the whole class of linearly orderable monoids, extending an earlier result for Puiseux monoids. The paper also proves several weaker properties do not ascend, while certain chain conditions and Furstenberg-type properties do.

What carries the argument

The engine is the sumset operation $S+T=\{s+t:s\in S,\ t\in T\}$ together with order-sensitive identities: when M is linearly ordered, $\min(S+T)=\min S+\min T$ and $\max(S+T)=\max S+\max T$ (Lemma 2.2), and cardinalities satisfy $|S+T|\ge |S|+|T|-1$ with strict growth when one summand has size at least two (Lemma 3.1). These facts make divisibility in the power monoid track minima and cardinalities, which is what makes atoms of P_fin(M) analyzable. The other load-bearing object is the maximal common divisor (MCD): Proposition 4.1 shows M is an MCD-monoid exactly when P_fin(M) is, and Lemma 4.2 builds indecomposable sets of the form $S\cup\{4\max S\}$ or $S\cup\{4\min S\}$ that are used to extract MCDs from atomic decompositions.

What would settle it

To test Theorem 4.3, one could try to exhibit a linearly orderable monoid M that is atomic and an MCD-monoid but whose power monoid contains a finite subset that is not a finite sum of atoms; no such example exists if the theorem is correct. Concretely, for the monoid (5.6) of Theorem 5.5, one could compute whether a nonempty finite subset S exists such that $S+\{(2/5,20/3),(3/7,10)\}$ is a finite sum of atoms of P_fin(M); a positive answer would contradict the paper's non-quasi-atomicity conclusion.

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Extended reading notes

Core claim

Theorem 4.3 establishes that for any linearly orderable monoid M, P_fin(M) is atomic if and only if M is atomic and every nonempty finite subset of M has a maximal common divisor. The forward direction shows atomicity of the power monoid forces M to be atomic, because singletons form a divisor-closed submonoid, and also forces the maximal common divisor property by an argument using specially constructed indecomposable sets. The reverse direction transfers the MCD property from M to P_fin(M) and then shows every finite subset decomposes into atoms. Along the way the paper constructs an atomic Puiseux monoid whose power monoid is not atomic, giving an alternative proof that atomicity does not ascend in general, and it constructs a rank-2 almost atomic monoid whose power monoid is not even quasi-atomic.

Load-bearing premise

In the rank-2 construction behind Theorem 5.5, the proof assumes that every element of the monoid (5.6) that is neither 0 nor an atom is divisible by an element of D, and that the sum of two atoms stays in D+M; if that structural claim fails, the forced-coordinate argument showing that P_fin(M) is not quasi-atomic collapses.

Editorial extensions

If this is right

For any linearly orderable M, atomicity of P_fin(M) can be read off from two properties of M itself, completely solving the ascent problem for atomicity on this class.
The earlier Puiseux-monoid criterion becomes a special case of Theorem 4.3, which covers all cancellative torsion-free monoids regardless of rank.
When M is an atomic MCD-monoid, P_fin(M) is not only atomic but also an MCD-monoid, so the MCD property itself ascends.
The quasi-ACCP and almost ACCP ascend from M to P_fin(M), as do the Furstenberg, quasi-Furstenberg, almost Furstenberg, and nearly Furstenberg properties on linearly orderable monoids.
On positive Archimedean monoids, both the finite factorization property and the TIDF property ascend; in contrast, near atomicity, almost atomicity, and quasi-atomicity do not ascend in general.

Reading between the lines

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

The MCD characterization suggests a transfer principle for factorization algorithms: to decide whether a finite subset of a linearly orderable monoid is atomic in the power monoid, one could focus on solving MCD problems in the base monoid rather than searching over subset decompositions.
The constructions of Sections 4 and 5 can be read as a recipe: adjoining elements that destroy MCDs in the base monoid destroys atomicity, and even near or quasi-atomicity, in the power monoid; this recipe may produce similar counterexamples in other ordered monoids, such as higher-rank additive submonoids of R^n.
Question 5.6 leaves open whether there is a rank-1 torsion-free almost atomic or quasi-atomic monoid whose power monoid fails the corresponding property; a natural next attempt is to project the rank-2 example (5.6) onto a single Archimedean coordinate while preserving its D-divisibility structure.

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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

Here's my take. The paper has real content: Theorem 3.3 (ascent of quasi- and almost-ACCP) is proved cleanly, Proposition 4.1 gives a self-contained equivalence between MCD properties of M and its power monoid, and the constructions in Sections 5 and 6 are genuine counterexamples that answer open questions. Example 4.5 is a nice improvement over [13, Example 3.3]. The main characterization, Theorem 4.3, is the right statement and would be a significant strengthening of the Puiseux result.

The soft spot is the proof of Theorem 4.3. The forward direction (a)⇒(c) is dispatched with 'mutatis mutandis' to [13, Theorem 3.2], which is about Puiseux monoids. That is not enough. The paper does not identify which parts of the Puiseux argument use rank 1, nonnegativity, or Archimedeanity, and arbitrary linearly orderable monoids have none of those in general. Since this is the existence half of the characterization, the central result is not actually proved in this version. The reverse direction also has an asserted step ('after replacing S by a suitable subset') that is likely repairable, because zero-sum pairs in a linearly orderable monoid are inverses and hence units, but it still needs a line.

There are smaller issues. Theorem 5.5 relies on a structural claim about the monoid (5.6) — every non-atomic element is divisible by something in D — which is stated but not isolated and proved in the lemmas. The proof of Theorem 5.3 is dense and the final step is hand-wavy. Theorem 6.5 asserts A(M)={a,b} and universal divisibility with 'one can readily verify', and there is a z+y/z-y typo. None of these are fatal on their own, but together they make verification harder than it should be.

Citation pattern looks fine; self-citations are to prior independent results. Who is this for? Specialists in factorization theory, especially people working on power monoids or generalized ACCP/atomicity conditions. They will get value from the theorems and the examples. I would not rely on Theorem 4.3 in its current form, but the rest of the paper stands on its own.

Recommendation: send to a serious referee. The referee should be asked to supply a complete proof of Theorem 4.3(a)⇒(c) or clarify exactly which parts of [13] carry over, and to fill the gaps in 5.5 and 6.5. This is the kind of paper that can be fixed with a solid revision.

Referee Report

5 major / 5 minor

Summary. The paper studies when atomic and divisibility properties ascend from a linearly orderable commutative monoid M to its finitary power monoid P_fin(M). Section 3 proves ascent of the quasi-ACCP and almost ACCP. Section 4 gives a proposed characterization: P_fin(M) is atomic iff M is an atomic MCD-monoid, with an auxiliary construction of an atomic Puiseux monoid that is not MCD. Section 5 constructs counterexamples showing that near atomicity does not ascend and that an almost atomic rank-2 monoid can have a power monoid that is not quasi-atomic. Section 6 proves ascent of Furstenberg-type properties and of the FFM/TIDF properties for positive Archimedean monoids, and gives a proposed linearly orderable TIDF monoid whose power monoid is not IDF.

Significance. If the main characterization (Theorem 4.3) is correct, it would generalize the Puiseux-monoid result of [13] to all linearly orderable monoids, which is a substantial contribution. The paper also supplies a useful toolkit: Lemma 2.2, Lemma 3.1, and the MCD-based arguments are clean and likely reusable. The counterexample constructions in Sections 4 and 5 address natural open ascent questions and are therefore of interest. Several proofs are self-contained and carefully structured. However, the paper's significance is currently limited by two load-bearing gaps: the forward direction of Theorem 4.3 is delegated to a mutatis mutandis argument from a Puiseux-monoid paper, and the counterexample in Theorem 6.5 contains a concrete construction error that invalidates the example as written. The remaining results appear plausible, but the overall contribution is not yet established.

major comments (5)

[Theorem 4.3, proof of (a)⇒(c)] The proof that P_fin(M) is atomic when M is an atomic MCD-monoid is dispatched as 'follows the line of the proof of [13, Theorem 3.2] mutatis mutandis.' Since [13, Theorem 3.2] is stated for Puiseux monoids, which are rank-1 submonoids of Q_{≥0}, and the present paper explicitly allows non-Archimedean and non-positive linearly orderable monoids, the adaptation is not routine. The implication (a)⇒(c) is the substantive half of the characterization, and the paper must either prove it in full or state precisely which steps of [13] carry over and why they do not use rank 1, nonnegativity, or Archimedeanity.
[Section 5.2, Theorem 5.5 and the paragraph before Lemma 5.4] The assertion that 'every element that is neither the identity element nor an atom is divisible by an element of D' is used in the proof of Theorem 5.5 to infer that an atom A_i of P_fin(M) contains either (0,0) or an atom of M. Lemma 5.4, as proved, only establishes that M is almost atomic, that A(M)=A∪B, and that A(M)+A(M)⊆D+M; it does not prove the quoted structural assertion. The divisibility step requires a common divisor of all elements of A_i by a single element of D, not just individual divisibility. This gap affects the conclusion that π(q_i)∈{0,1/5,1/7}, on which the contradiction in Theorem 5.5 rests.
[Theorem 6.5, construction of M] The construction of M is internally inconsistent. Since X_n = N x_n + gp(A) and Y_n = N(x_n−y) + gp(B), the subgroup gp(⟨∪_n (X_n ∪ Y_n)⟩) contains both x_n and x_n−y for each n, hence contains y, and also contains −a and −b. As Z = Nz + gp(⟨∪_n (X_n ∪ Y_n)⟩), the monoid M contains −a and −b as well as a and b, so a and b are units. Consequently A(M) is empty (indeed M is a group), contradicting the claim that A(M)={a,b} and that each element is divisible by a or b. The later argument that {x_n, x_n−y} are atoms of P_fin(M) also collapses because in a group every finite set is invertible and there are no atoms.
[Theorem 4.3, proof of (b)⇒(a)] The reduction 'after replacing S by a suitable subset, we can further assume that s+t≠0 for all s,t∈S' is not justified. In a linearly ordered monoid, s+t=0 implies s and t are mutual inverses and hence units; the reduction may remove elements, and it is not shown that the MCD of the reduced subset yields an MCD of the original S. This step is needed to apply Lemma 4.2 and should be proved.
[Theorem 5.3, proof of the Claim] The statement 'these values are actually the exact multiplicities for every factorization' is asserted without proof, but it is essential for the conclusion that the coefficients are minima in each summand and for the final use of Lemma 5.2. The proof should spell out why the bounded p-adic valuations force exact multiplicities in every factorization.

minor comments (5)

[Theorem 6.5, proof of (2)] The text says 'atoms of P dividing {z, z + y}' but the subsequent divisibility statement concerns {z, z − y}; the plus sign appears to be a typo.
[Theorem 6.2(4)] The final sentence 'Hence we conclude that P is nearly atomic' should read 'nearly Furstenberg.'
[Lemma 5.4(1)] The inclusion A(M) ⊆ A∪B∪D is stated, but the reason that no element of D is an atom is not given; it would be clearer to note explicitly that each (0,1/2^n) is the sum of two equal elements of D.
[Theorem 4.3 and Section 2.6] The use of M for both the monoid and the set of singletons {{m}:m∈M} is confusing; a different symbol for the singleton submonoid would improve readability.
[Example 4.5] In the proof that {1,4/3} has no MCD, the symbol n is reused for an index and for the exponent in 1/2^n; the argument is ultimately understandable but would benefit from a change of variables.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main derivations are self-contained, though a few proof steps are delegated or asserted without full detail.

full rationale

The paper’s claimed results do not reduce by construction to their own inputs. The central characterization, Theorem 4.3, contains one substantive direction proved in the text: (b)⇒(a), showing that atomicity of P_fin(M) forces M to be an atomic MCD-monoid, using Lemma 4.2 and a decomposition of a carefully chosen finite set T into atoms. The converse direction (a)⇒(c) is not proved in the text; it is delegated with the sentence “The proof that P_fin(M) is an atomic monoid follows the line of the proof of [13, Theorem 3.2] mutatis mutandis.” This is an omitted proof and a reliance on an external preprint, and it is a real completeness concern for the strength of the characterization, but it is not circular: [13, Theorem 3.2] is a Puiseux-monoid theorem whose proof is being adapted, not an input whose conclusion is identical to the present theorem. Similarly, Corollary 6.4(1) delegates an FFM ascent argument “mutatis mutandis” to [13, Theorem 4.2], again a proof delegation rather than a circular reduction. The unproved reduction “after replacing S by a suitable subset” in Theorem 4.3(b)⇒(a) and the structural claim before Lemma 5.4 that every non-atom, non-zero element of M is divisible by an element of D are asserted details that would need justification, but neither makes a conclusion equivalent to a hypothesis. The self-citations used, [15] and [16], provide definitions and prior independent results and are not load-bearing in a circular way. Section 3 and Sections 5–6 contain direct proofs of the ascent and non-ascent results, and the counterexamples are constructed from scratch rather than by assuming the target theorem. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is a pure mathematics paper. It introduces no fitted parameters and no hypothetical entities. It relies on standard theorems (Levi, Hölder, Halter-Koch) and on the prior literature's estimates for power monoids. The constructed counterexample monoids are fully explicit and carry their own proofs, though a few structural claims are asserted tersely (e.g., A(M)={a,b} in Theorem 6.5 and the divisibility-by-D claim in Section 5.2), which the reader must verify.

assumptions (5)

domain assumption Every linearly orderable monoid is cancellative and torsion-free, and conversely (Theorem 2.1, Levi).
This classical theorem grounds the class under study and is used throughout, e.g., to embed M into an ordered group and to apply Hölder's theorem.
standard math Hölder's theorem: an Archimedean linearly ordered abelian group is order-isomorphic to a subgroup of (R, +).
Used in Proposition 6.3 and Corollary 6.4 to reduce positive Archimedean monoids to submonoids of R≥0.
standard math Halter-Koch's theorem [19, Theorem 2]: a monoid is an FFM iff it is atomic and IDF.
Used to conclude TIDF implies FFM for positive Archimedean monoids and to characterize FFMs.
standard math ZFC set theory and standard arithmetic of p-adic valuations on Q.
Underlies all constructions and valuation arguments in Examples 4.5, 5.2, and Theorem 5.3.
standard math The inequality |S+T| ≥ |S|+|T|-1 for finite subsets of a linearly ordered monoid ([9, Proposition 3.5]).
Frequently invoked in Lemma 3.1, Lemma 4.2, Lemma 6.1, and elsewhere; accepted from prior literature.

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Pith. "Pith review of On finitary power monoids of linearly orderable monoids." pith.science (2026). https://pith.science/paper/PX6FVUY7

@misc{pith2026250103407,
  author       = {Pith},
  title        = {Pith review of: On finitary power monoids of linearly orderable monoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PX6FVUY7}},
  note         = {Machine review of arXiv:2501.03407}
}

abstract

A commutative monoid $M$ is called a linearly orderable monoid if there exists a total order on $M$ that is compatible with the monoid operation. The finitary power monoid of a commutative monoid $M$ is the monoid consisting of all nonempty finite subsets of $M$ under the so-called sumset. In this paper, we investigate whether certain atomic and divisibility properties ascend from linearly orderable monoids to their corresponding finitary power monoids.

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Works this paper leans on

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[1] [13]

On the atomicity of power monoids of Puiseux monoids

V. Gonzalez, E. Li, H. Rabinovitz, P. Rodriguez, and M. T irador, On the atomicity of power monoids of puiseux monoids , International Journal of Algebra and its Applications (to appear). Preprint available on arXiv: https://arxiv.org/pdf/2401.12444

work page Pith review arXiv

[2] [1]

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

work page 1990

[3] [2]

D. D. Anderson and B. Mullins, Finite Factorization Domains , Proc. Amer. Math. Soc. 124 (1996) 389-396

work page 1996

[4] [3]

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

work page 2022

[5] [4]

N. R. Baeth, S. T. Chapman, and F. Gotti, Bi-atomic classes of positive semirings , Semigroup Forum 103 (2021) 1–23

work page 2021

[6] [5]

P. Y. Bienvenu and A. Geroldinger, On algebraic properties of power monoids of numerical monoi ds, Israel J. Math. (to appear). Preprint on arXiv: https://arxiv.org/p df/2205.00982.pdf

work page Pith review arXiv

[7] [6]

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

work page 2015

[8] [7]

P. L. Clark, The Euclidean Criterion for irreducibles , Amer. Math. Monthly 124 (2017) 198–216

work page 2017