{"id":"ee7e884f-42c1-4959-86d5-6b6a5bc35b29","arxiv_id":"1908.09227","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Puiseux monoids, UFM equals HFM equals N0, OHFM is equivalent to having at most two atoms, and the BFM/ACCP/limit-point trichotomy holds when the conductor is nonempty.","lead":"Puiseux monoids are additive number systems built from nonnegative fractions, and this paper asks when every element can be written as a finite sum of irreducible atoms. It supplies clean characterizations for several atom-related properties and infinite example families that separate the standard factorization hierarchy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.5's coefficient bound allows nonunique representations; the ACCP class and the sharpness of Theorem 4.9's conductor hypothesis depend on the fix.","rationale":"I focused on Theorem 4.9, the strongest claim identified by the reader. The proof of (3) => (1) is internally sound: ACCP gives atomicity; 0 as a limit point gives atoms a_n < 1/2^n; the partial sums s_n lie in M and are bounded by 1; choosing x above 1 + sup(~M \\ M), the conductor hypothesis forces x - s_n in M; and the resulting chain of principal ideals is strictly ascending, contradicting ACCP. The root-closed case needs a small convention for sup(empty set), but this is harmless. I found no flaw in the core equivalence. The load-bearing issue I did identify is in Theorem 4.5, which is used to prove Corollary 4.6 and is invoked in Example 4.11. Those two results are the concrete evidence that the nonempty-conductor assumption in Theorem 4.9 cannot be dropped. As typeset, the uniqueness proof in Theorem 4.5 is invalid because the interval [0, p_j] allows the ambiguity 1 = p_j*(1/p_j) = 1 + 0. The valuation argument only forces alpha_j - beta_j to be a multiple of p_j, not to be zero. This is exactly the coefficient-bound issue the reader flagged, and it is fixable, but it is not merely cosmetic: the measures n(x) and s(x) used to prove ACCP must be attached to a canonical representation. The reader's weakest_assumption was the role of the conductor and the dependence on Proposition 3.12; my concern concerns the same sharpness examples and is therefore partial agreement. Since the identified problem is real but mechanical, the existing CONDITIONAL verdict remains appropriate.","tokens_in":20598,"tokens_out":22694,"duration_ms":235920,"concrete_test":"Re-prove Theorem 4.5 with the corrected bound 0 <= alpha_j <= p_j - 1: first show existence by carrying p_j*(1/p_j) = 1 into the integer part whenever a coefficient reaches p_j, then rerun the p_j-adic valuation argument on (4.5). If uniqueness holds under this bound, verify that Corollary 4.6's length-unboundedness argument and Example 4.11's atomicity assertion still go through unchanged. If the revised proof succeeds, Theorem 4.9's statement and its sharpness examples survive modulo a typesetting correction; if it fails, the ACCP construction in Corollary 4.6 would need a different supporting argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.5 claims a representation x = n + sum alpha_j/p_j with alpha_j in [0, p_j], then proves uniqueness by applying the p_j-adic valuation to (alpha_j - beta_j)/p_j and concluding that p_j | alpha_j - beta_j forces alpha_j = beta_j. That conclusion is valid only when |alpha_j - beta_j| < p_j. With the stated closed interval, alpha_j = p_j and beta_j = 0 are both permitted, and the same element has two representations, e.g. 1 = p_j*(1/p_j) and 1 = 1 + 0. Thus the uniqueness step, and with it the well-definedness of the measures n(x) and s(x) used in the ACCP argument, fails as typeset. The gap is repairable by requiring 0 <= alpha_j <= p_j - 1 and absorbing p_j/p_j into the integer part, but the proof as written does not state this normalization. Corollary 4.6 and Example 4.11 explicitly use Theorem 4.5 to exhibit ACCP Puiseux monoids with 0 as a limit point, and these examples are precisely what shows that the nonempty-conductor hypothesis in Theorem 4.9 is necessary. Hence the sharpness of the boundary asserted in Theorem 4.9 rests on this coefficient normalization.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20836,"tokens_out":14665,"duration_ms":147668,"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":[{"comment":"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.","section":"§4.2, Theorem 4.5, Eq. (4.4)"},{"comment":"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.","section":"Abstract"}],"minor_comments":[{"comment":"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.","section":"Proof of Theorem 4.5"},{"comment":"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}.","section":"Proof of Corollary 3.9"},{"comment":"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.","section":"Proposition 3.5(2), lcm-closure argument"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, readable paper, more research contribution than pure survey. The new material that matters: the conductor trichotomy (Theorem 4.9), the HFM=UFM collapse (Proposition 4.22), and the OHFM iff at most two atoms (Proposition 4.25). Theorem 4.7 (no limit point at 0 implies BFM) is a clean observation, and the examples in Section 4 separating FFM/BFM/ACCP/atomic are well chosen and instructive. The proofs are mostly self-contained and I don't see hidden fitting or circularity. The increasing-monoid theorem 4.19 is acknowledged as a special case of [31], and the proof they give is a nice simplification; that's honest.\n\nSoft spots, in proportion:\n\n- Theorem 4.5 has a genuine typesetting/normalization gap. The interval 0 <= alpha_j <= p_j allows two representations of the same element (p_j*(1/p_j) = 1 + 0*(1/p_j)), so the uniqueness argument 'p_j divides alpha_j - beta_j implies alpha_j = beta_j' is false as written. The fix is standard: take 0 <= alpha_j <= p_j - 1 and absorb p_j/p_j into the integer part. Then the ACCP argument works. This is minor but should be fixed; Corollary 4.6 and Example 4.11 use the theorem, so the proof needs to be clean.\n\n- The abstract promises a characterization of seminormal Puiseux monoids. I could not find seminormal even defined in the body, let alone characterized. Either add the missing material or trim the abstract. This is an editorial mismatch, not a mathematical failure.\n\n- Theorem 4.9 relies on the conductor description cited from the authors' preprint [25]. That is a legitimate building block, and they prove the trichotomy in the paper; the conductor hypothesis is genuinely necessary, as the examples show. No concern.\n\nBottom line: the paper deserves a serious referee. The math is sound in its central claims, the examples are useful, and the exposition is clear. I would send it out and ask for the normalization fix and abstract alignment, not more.","headline":"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.","tokens_in":21422,"tokens_out":3561,"would_cite":true,"duration_ms":34927,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M13","06F05","20M14"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a Puiseux monoid with nonempty conductor, zero not being a limit point is equivalent to bounded factorization and ACCP.","keywords":["Puiseux monoids","atomicity","factorization theory","conductor","ACCP","bounded factorization monoids","finite factorization monoids","half-factorial monoids"],"falsifier":"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 $(x-s_n+M)_{n\\in\\mathbb{N}}$.","tokens_in":20383,"feed_emoji":"➕","tokens_out":19046,"duration_ms":154178,"temperature":0.7,"pith_summary":"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.","feed_headline":"Zero not a limit point guarantees ACCP and bounded factorization","feed_subtitle":"With nonempty conductor, the reverse holds too: the condition is equivalent to ACCP and bounded factorization.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the conductor description used in the proof of Theorem 4.9: for a non-root-closed Puiseux monoid with finite $\\sigma=\\sup(\\widetilde M\\setminus M)$, the conductor is the tail $M_{\\ge\\sigma}$.","marker":"[25]"},{"why":"Provides the standard implication from bounded factorization to ACCP, specifically Corollary 1.3.3, used to derive (2) implies (3) in Theorem 4.9.","marker":"[26]"},{"why":"Gives the atomicity criterion for the monotone Puiseux monoid in Example 4.8, which shows a bounded factorization monoid can have $0$ as a limit point, closing the converse gap of Theorem 4.7.","marker":"[37]"}],"fun_headline_variants":["Nonempty conductor: zero non-limit point iff ACCP and BFM in Puiseux monoids","Puiseux monoids with conductor: zero non-limit point equivalent to ACCP and BFM","Atomic Puiseux monoid: UFM or HFM iff isomorphic to N0","Atomic Puiseux monoid: other-half-factorial iff at most two atoms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Nonempty conductor: zero non-limit point iff ACCP and BFM in Puiseux monoids","Puiseux monoids with conductor: zero non-limit point equivalent to ACCP and BFM","Atomic Puiseux monoid: UFM or HFM iff isomorphic to N0","Atomic Puiseux monoid: other-half-factorial iff at most two atoms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002101,"raw_usage":{"total_tokens":8197,"prompt_tokens":1009,"completion_tokens":7188,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":7091}},"tokens_in":625,"tokens_out":7188,"duration_ms":53290,"temperature":1.0,"reasoning_tokens":7091,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:19:48.079205+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $(x-s_n+M)_{n\\in\\mathbb{N}}$.","supporting_citations":[{"cited_title":"On strongly primary monoids, with a focus on Puiseux monoids","cited_arxiv_id":"1910.10270","evidence_quote":"Supplies the conductor description used in the proof of Theorem 4.9: for a non-root-closed Puiseux monoid with finite $\\sigma=\\sup(\\widetilde M\\setminus M)$, the conductor is the tail $M_{\\ge\\sigma}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the standard implication from bounded factorization to ACCP, specifically Corollary 1.3.3, used to derive (2) implies (3) in Theorem 4.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the atomicity criterion for the monotone Puiseux monoid in Example 4.8, which shows a bounded factorization monoid can have $0$ as a limit point, closing the converse gap of Theorem 4.7."}],"review_version":1}