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When is a Puiseux monoid atomic?

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Pith's one-line read In a Puiseux monoid with nonempty conductor, zero not being a limit point is equivalent to bounded factorization and ACCP.

desk verdict A genuinely useful survey-plus-research paper with clean characterizations; the central results hold, but the abstract overclaims a seminormal characterization and Theorem 4.5 needs a coefficient normalization. read the letter →

arxiv 1908.09227 v2 pith:RB7OTA3N submitted 2019-08-24 math.AC

Scott T. Chapman , Felix Gotti , Marly Gotti This is my paper · ORCID
classification math.AC MSC 20M1306F0520M14
keywords PuiseuxmonoidsatomicityfactorizationtheoryconductorACCPboundedfinitehalf-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

Puiseux monoids are additive submonoids of the nonnegative rational numbers, and the paper asks when every nonzero element can be written as a sum of irreducible elements. No general characterization of atomicity is known, so the authors chart the boundary of what is possible. The central result, Theorem 4.9, says that inside the class of nontrivial Puiseux monoids with nonempty conductor, the topological condition that $0$ is not a limit point of the nonzero elements is equivalent to being a bounded factorization monoid and to satisfying the ascending chain condition on principal ideals. This provides a sharp dividing line: the equivalence fails outside the nonempty-conductor class, and even inside it, atomicity and finite factorization remain strictly weaker. The paper also classifies factorial, half-factorial, and other-half-factorial Puiseux monoids and constructs infinite families realizing each level of the factorization hierarchy.

What carries the argument

The load-bearing object is the conductor $\mathrm{c}(M)$ of a Puiseux monoid $M$, the set of elements $x$ in the difference group for which $x+\widetilde M\subseteq M$, where $\widetilde M$ is the root closure (the elements of the difference group some positive multiple of which lies in $M$). Proposition 3.12 describes $\mathrm{c}(M)$ concretely: it is all of $M$ if $M$ is root-closed, empty if $M$ is not root-closed and $\widetilde M\setminus M$ is unbounded, and the tail $M_{\ge\sigma}$ otherwise. This tail description converts the metric information that $0$ is a limit point of $M^\bullet$ into an algebraic obstruction: when the conductor is nonempty, arbitrarily small atoms would form a summable sequence whose partial sums push the monoid into the conductor tail and generate a nonstabilizing chain of principal ideals. Theorem 4.7 supplies the opposite direction: if $0$ is not a limit point, some $\varepsilon>0$ bounds every nonzero element below, so every $x\in M$ has at most $\lfloor x/\varepsilon\rfloor$ atoms in any factorization, making $M$ a bounded factorization monoid.

What would settle it

A direct falsifier would be a Puiseux monoid with nonempty conductor, $0$ as a limit point of its nonzero elements, and the ascending chain condition on principal ideals; Theorem 4.9 asserts no such monoid exists. The paper's Example 4.11 exhibits the closest structure, the monoid $M=\langle 1/p \mid p\in\mathbb{P}\rangle\cup\mathbb{Q}_{\ge1}$, and shows it fails the ACCP by constructing the nonstabilizing chain of principal ideals $(x-s_n+M)_{n\in\mathbb{N}}$.

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Extended reading notes

Core claim

The central discovery is Theorem 4.9: for a nontrivial Puiseux monoid $M$ with nonempty conductor, the following are equivalent: (1) $0$ is not a limit point of $M^\bullet$; (2) $M$ is a bounded factorization monoid; (3) $M$ satisfies the ascending chain condition on principal ideals. The proof uses the conductor description from Proposition 3.12, which says that for a non-root-closed Puiseux monoid with $\sigma=\sup(\widetilde M\setminus M)<\infty$, the conductor is the tail $M_{\ge\sigma}$. With this tail structure, if atoms accumulated at $0$, their partial sums would generate a strictly ascending chain of principal ideals, contradicting the ACCP. The paper further shows that a nontrivial atomic Puiseux monoid is a unique factorization monoid or a half-factorial monoid exactly when it is isomorphic to $(\mathbb{N}_0,+)$, and that it is an other-half-factorial monoid exactly when it has at most two atoms. These characterizations, together with examples showing the failure of converses, delimit where the standard factorization hierarchy UFM $\Rightarrow$ FFM $\Rightarrow$ BFM $\Rightarrow$ ACCP $\Rightarrow$ atomic can be reversed in the class of Puiseux monoids.

Load-bearing premise

The equivalence in Theorem 4.9 depends on the conductor being nonempty, which means the root closure of the monoid has only a bounded gap from the monoid itself; without this hypothesis, a Puiseux monoid can satisfy the ACCP or the bounded factorization property while $0$ is a limit point of its nonzero elements.

Editorial extensions

If this is right

  • In the nonempty-conductor class, checking whether a Puiseux monoid is a BFM or satisfies the ACCP reduces to checking whether $0$ is a limit point of $M^\bullet$.
  • The monoid $\langle 1/p \mid p\in\mathbb{P}\rangle$ satisfies the ACCP but is not a BFM, so among Puiseux monoids the implication ACCP $\Rightarrow$ BFM fails.
  • Every increasing Puiseux monoid, one generated by an increasing sequence of rationals, is a finite factorization monoid (Theorem 4.19).
  • A nontrivial atomic Puiseux monoid is a UFM or an HFM exactly when it is isomorphic to $(\mathbb{N}_0,+)$, and it is an OHFM exactly when it has at most two atoms.
  • Even with nonempty conductor, atomicity and the finite factorization property remain strictly weaker than the BFM/ACCP condition: Example 4.10 is a BFM that is not an FFM, and Example 4.11 is atomic but does not satisfy the ACCP.

Reading between the lines

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

  • A consequence the authors leave implicit is that the conductor-based dichotomy suggests a testable organizing principle: additive submonoids of $\mathbb{Q}$ whose root closure has bounded gaps are governed by limit points of their atoms, while monoids with large gaps in the root closure can hide unusual factorization behavior.
  • The proof recipe, taking atoms $a_n<2^{-n}$ and using their partial sums to build a nonstabilizing chain, should transfer to other rank-one positive monoids with a conductor; one could test it on submonoids of $\mathbb{R}_{\ge0}$ with the same tail structure.
  • Because the full characterization of atomic Puiseux monoids remains open, a natural next step is to look for a dichotomy in terms of the set of denominators $\mathrm{d}(M^\bullet)$, since the ACCP and BFM examples in the paper differ sharply in how their denominators grow.
  • One could probe whether the equivalence in Theorem 4.9 survives under a weaker condition than a nonempty conductor, such as the root closure being finitely generated as a module over $M$, or whether some bounded-gap condition is truly necessary.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. This paper studies atomicity and factorization-theoretic properties of Puiseux monoids, i.e. additive submonoids of Q_{\ge 0}. It surveys and extends known results about such monoids: it describes the root closure, complete integral closure, and conductor of a Puiseux monoid (Section 3), and it investigates the atomic hierarchy UFM \Rightarrow FFM \Rightarrow BFM \Rightarrow ACCP \Rightarrow atomic. The main results are: Theorem 4.5, asserting that every submonoid of \langle 1/p \mid p \in P\rangle satisfies the ACCP; Theorem 4.7, giving the BFM property when 0 is not a limit point of M^{\bullet}; Theorem 4.9, establishing that for Puiseux monoids with nonempty conductor, having 0 not as a limit point is equivalent to being a BFM and to satisfying the ACCP; Theorem 4.19, showing that increasing Puiseux monoids are FFMs; and Propositions 4.22 and 4.25, characterizing UFM, HFM, and OHFM Puiseux monoids. The paper also provides explicit infinite families of examples showing that the reverse implications in the hierarchy fail.

Significance. If the proof issue in Theorem 4.5 is repaired, this is a genuinely useful contribution. The conductor-based equivalence in Theorem 4.9 gives a clean boundary for when the atomic hierarchy collapses inside the nonempty-conductor class, and the paper provides parameter-free, elementary proofs of all its central claims. The counterexamples are concrete and reproducible, and the survey component, including Proposition 3.5 and Proposition 3.12 on closures and conductors, is valuable. The paper is also strengthened by stating several open characterizations honestly rather than claiming results that are not proved. The main reservations are the typeset coefficient bound in Theorem 4.5 and the abstract's promise of a seminormal characterization that does not appear in the body.

major comments (2)
  1. [§4.2, Theorem 4.5, Eq. (4.4)] The coefficient bound is typeset as \alpha_j \in [0, p_j]. With the closed interval, the representation (4.4) is not unique: for any prime p_j, 1 = p_j(1/p_j) and 1 = 1 + 0(1/p_j) are both admissible. The uniqueness argument applies the p_j-adic valuation to (\alpha_j - \beta_j)/p_j and infers \alpha_j = \beta_j from p_j \mid (\alpha_j - \beta_j); this inference requires |\alpha_j - \beta_j| < p_j, which fails under the stated bound. Since the well-definedness of n(x) and s(x), and hence the ACCP conclusion, depends on uniqueness, the proof as written is incomplete. The standard repair is to require 0 \le \alpha_j \le p_j - 1, absorbing each p_j(1/p_j) into the integer part; please make that normalization explicit. Corollary 4.6 and Example 4.11 use this theorem, so the sharpness of the conductor hypothesis in Theorem 4.9 rests on this repair.
  2. [Abstract] The abstract promises characterizations of when M is seminormal, but no definition of seminormal appears in the paper and no characterization of seminormal Puiseux monoids is stated or proved. The body characterizes root closure, the complete integral closure, and the Prüfer property (e.g., Proposition 3.5 and Corollary 3.7), but in commutative monoid theory seminormal is a separate property, not covered by those statements. Either add the missing definition and characterization, or remove the word 'seminormal' from the abstract.
minor comments (3)
  1. [Proof of Theorem 4.5] In the paragraph after Eq. (4.4), the notation 'x' |_MP x' and 'x' \in MP' appears to be a typo for the ambient monoid M; please correct the subscripts.
  2. [Proof of Corollary 3.9] The claim that for every d \in d(M^{\bullet}) there is a d' \in d(M^{\bullet}) properly dividing d is false when d = 1. The argument is repairable: for d > 1 take d' = 1, and for d = 1 note that infinite d(M^{\bullet}) contains some e > 1, so n = e(n/e) is a nontrivial decomposition of n in \tilde{M}.
  3. [Proposition 3.5(2), lcm-closure argument] The existence of N, c_1, c_2 \in \mathbb{N}_0 in the lcm-closure proof is asserted without justification; a sentence explaining that a Bézout combination can be shifted by adding a large multiple of (A/g, B/g) would make the argument fully explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are proved from definitions and standard valuation arguments, and the self-citations are either independently proved in the paper or used only for non-central examples.

full rationale

The main derivation chain is self-contained. Theorem 4.5 establishes an explicit normal form for elements of the monoid generated by {1/p : p in P} and proves uniqueness by p-adic valuations; the normal form is constructed in the proof rather than assumed from the conclusion, and the ACCP conclusion is derived from the associated measures n(x) and s(x). The later equivalence in Theorem 4.9 is proved directly: (1) implies (2) by Theorem 4.7, (2) implies (3) by standard ACCP theory, and (3) implies (1) uses Proposition 3.12 to control the tail of the root closure. Proposition 3.12 is cited to the authors' preprint [25], but it is also stated and proved in the present paper from the definitions of root closure and conductor, so the self-citation is not load-bearing. Example 4.8 cites [37, Cor. 5.6] for atomicity of a monotone Puiseux monoid; this is a self-citation, but it is a parameter-free external theorem about monotone Puiseux monoids used only to exhibit a counterexample to the converse of Theorem 4.7, not to establish the paper's main characterization. The paper explicitly flags the scope limitation of Theorem 4.9 by noting that without the nonempty-conductor condition the implications fail, and it supplies Corollary 4.6 and Example 4.8 as boundary cases; this explicit demarcation is the opposite of circularity. The typeset coefficient interval in Theorem 4.5 may contain a repairable correctness gap, since the closed interval [0, p_j] permits duplicative representations, but that is a proof-repair issue, not a reduction of the theorem to its inputs. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work to force a choice, and no known result is merely renamed. Therefore the paper receives a circularity score of 0.

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

This is a pure mathematics paper; no free parameters or invented entities appear. The proofs are parameter-free derivations from stated definitions and standard background facts. The listed axioms are the main unproved background inputs used by the central arguments.

assumptions (5)
  • standard math Rank-1 torsion-free abelian groups are isomorphic to subgroups of (Q,+).
    Invoked in Proposition 3.3 via Fuchs [21, Section 85] to represent any rank-1 torsion-free monoid as a Puiseux monoid.
  • standard math The p-adic valuations v_p on Q are discrete and satisfy v_p(x+y) >= min(v_p(x), v_p(y)).
    Used in Theorem 4.5 to prove uniqueness of representations and in Proposition 4.3 to detect atoms.
  • standard math Every submonoid of (N0,+) is finitely generated.
    Used in Proposition 3.14 to choose a finite minimal generating set for M cap N0.
  • standard math A monoid satisfying ACCP is atomic, and finitely generated monoids are atomic.
    Background from Geroldinger-Halter-Koch [26], used throughout Section 4.
  • domain assumption Puiseux monoids are reduced, cancellative, and torsion-free submonoids of Q>=0.
    Inherited from the ambient additive group Q; this is the standing domain assumption of the paper.

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Pith. "Pith review of When is a Puiseux monoid atomic?." pith.science (2026). https://pith.science/paper/RB7OTA3N

@misc{pith2026190809227,
  author       = {Pith},
  title        = {Pith review of: When is a Puiseux monoid atomic?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RB7OTA3N}},
  note         = {Machine review of arXiv:1908.09227}
}
abstract

A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If $M$ is a Puiseux monoid, then the question of whether each non-invertible element of $M$ can be written as a sum of irreducible elements (that is, $M$ is atomic) is surprisingly difficult. Although various techniques have been developed over the past few years to identify subclasses of Puiseux monoids that are atomic, no general characterization of such monoids is known. Here we survey some of the most relevant aspects related to the atomicity of Puiseux monoids. We provide characterizations of when $M$ is finitely generated, factorial, half-factorial, other-half-factorial, Pr\"ufer, seminormal, root-closed, and completely integrally closed. In addition to the atomicity, characterizations are also not known for when $M$ satisfies the ACCP, the bounded factorization property, or the finite factorization property. In each of these cases, we construct an infinite class of Puiseux monoids satisfying the corresponding property.

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