{"id":"5b4230ab-f65b-4f1e-b8f3-f06fda06d0b5","arxiv_id":"2412.11199","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A monoid is hereditarily atomic exactly when all its undermonoids are atomic, this forces the ACCP, and torsion abelian groups are precisely the monoids all of whose undermonoids are half- or length-factorial.","lead":"The paper introduces 'undermonoids', submonoids that generate the same Grothendieck group, and proves that checking several factorization properties on undermonoids alone is enough to conclude they hold for every submonoid. It settles a conjecture connecting hereditary atomicity to the ascending chain condition on principal ideals, and classifies monoids all of whose undermonoids are half- or length-factorial as torsion abelian groups.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the group-case maximality-extension hinge in Theorems 3.3, 5.4, 6.2, and 6.3 is compressed but can be completed, and the Dickson-lemma argument for Theorem 4.2 is internally sound.","rationale":"The reader's weakest-assumption identification points to exactly the right structural hinge: the group-case maximality-extension argument. I examined that hinge closely and found that the printed proof is compressed but not incorrect. The critical inference in Claim 2 can be justified by a short sublemma using the disjunctive hypothesis on u and the already-proved unit preservation U(S') = U(S). The same pattern appears in Theorems 6.2 and 6.3, and the same fill applies. Theorem 5.4's analogous claim is actually simpler because non-bounded-atomicity is inherited by S' directly. I also checked the main ACCP theorem, Theorem 4.2, including the base case of the claim, the Dickson's lemma induction, the injectivity of f_s, the strict-increasing-sequence extraction, and the final construction of M'. Every step is internally consistent; the only issues are expository, such as the terse Claim 2 and a minor typo. Because the central claims appear correct and the reader's flagged concern can be resolved without changing the argument's substance, the verdict should remain ACCEPT with no adjustment.","tokens_in":20583,"tokens_out":34368,"duration_ms":329548,"concrete_test":"Independently re-derive Claim 2 of Theorem 3.3 with the missing sublemma spelled out: if k>=2 and x = s+(k-1)(2b+u) is a unit of S', then x is in U(S), and the failure of condition (ii) together with condition (i) forces a positive multiple of b to be a unit, contradicting b not in U(S). If this sublemma cannot be completed, the maximality-extension step collapses and the group case of Theorem 3.3 is in doubt; if it can be completed, the central theorems stand. As a secondary check, recompute Equation (4.2) in Theorem 4.2 to confirm that every summand lies in M before concluding that the trailing block is a unit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The most load-bearing place is the group-case maximality-extension step in Theorem 3.3, reused in Theorems 5.4, 6.2, and 6.3: for a maximal non-atomic submonoid S and u satisfying condition (i) or (ii), one must show that 2b+u belongs to S. The printed proof of Claim 2 is terse: from a' = s + k(2b+u) being an atom of S' it asserts directly that k=1 and s is a unit of S'. If k>=2, the decomposition a' = (s+(k-1)(2b+u)) + (2b+u) only rules out k>=2 when s+(k-1)(2b+u) is not a unit. The missing subargument can be supplied: if x = s+(k-1)(2b+u) were a unit, then x is in U(S), so -x is in S and the equality (k-1)u + s+(2k-2)b + (-x) = 0 violates condition (ii); since u satisfies (i) or (ii), condition (i) gives n0u in S, and multiplying by n0 forces a positive multiple of b to be a unit, contradicting b not in U(S). With this sublemma, Claims 2 and 3 go through and the final contradiction with v outside gp(S) is valid. The analogous HFM/LFM versions have the same fillable gap, and Theorem 5.4 avoids the delicate atom-classification issue entirely. Theorem 4.2's Claim, Lemma 4.1, and the final submonoid construction were checked step-by-step and are consistent; the only blemish is a typographical 'A(M'_n)' where 'A(M')' is meant.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20988,"tokens_out":21909,"duration_ms":200110,"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":[{"comment":"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.","section":"Section 5, Definition 5.2"},{"comment":"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.","section":"Theorem 3.3, proof of Claim 2"},{"comment":"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.","section":"Theorem 5.4, Case 2"}],"minor_comments":[{"comment":"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'.","section":"Theorem 4.2, proof"},{"comment":"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.","section":"Theorem 6.3, proof"},{"comment":"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.","section":"Theorem 4.2, proof of the Claim"},{"comment":"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.","section":"Example 6.5"}],"recommendation":"major_revision","confidential_remarks":"The central claims appear sound and the strategy is convincing. The two load-bearing issues are the misstated Definition 5.2 and the compressed maximality-extension step in the group-case proofs; both are repairable within the scope of the paper. I do not see a novelty or circularity concern. I would support acceptance once these points are fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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. \n\nThe 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. \n\nWhat 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. \n\nWho 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.","headline":"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.","tokens_in":21521,"tokens_out":1895,"would_cite":true,"duration_ms":16246,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F15","13A05","20M13","13F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["undermonoid","hereditary atomicity","underatomicity","ascending chain condition on principal ideals","atomic monoid","bounded factorization monoid","half-factorial monoid","length-factorial monoid"],"falsifier":"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.","tokens_in":20378,"feed_emoji":"🔢","tokens_out":15539,"duration_ms":123190,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every hereditarily atomic monoid satisfies the ACCP","feed_subtitle":"Checking only submonoids with the same Grothendieck group yields atomicity everywhere — and proves ACCP.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces hereditary atomicity in integral domains and supplies the conjecture that Theorem 4.2 answers.","marker":"[12]"},{"why":"Settles the torsion-free case of the ACCP conjecture and poses the general monoid conjecture answered here.","marker":"[22]"},{"why":"Provides the standard monoid and factorization background, including the fact that ACCP implies atomicity used in the corollaries.","marker":"[14]"},{"why":"Introduces the bounded factorization property whose hereditary/under version is Theorem 5.4.","marker":"[1]"},{"why":"Introduces half-factoriality, the property classified in Theorem 6.2.","marker":"[27]"},{"why":"Introduces length-factoriality (as other-half-factoriality), the property classified in Theorem 6.3.","marker":"[13]"}],"fun_headline_variants":["Atomic undermonoids imply the ACCP","Undermonoids suffice for atomicity and ACCP","Undermonoid check decides atomicity and BFP","Half-factorial and length-factorial via undermonoids","Hereditarily atomic? Undermonoids prove the ACCP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Atomic undermonoids imply the ACCP","Undermonoids suffice for atomicity and ACCP","Undermonoid check decides atomicity and BFP","Half-factorial and length-factorial via undermonoids","Hereditarily atomic? Undermonoids prove the ACCP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000573,"raw_usage":{"total_tokens":2735,"prompt_tokens":1003,"completion_tokens":1732,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":1651}},"tokens_in":619,"tokens_out":1732,"duration_ms":13959,"temperature":1.0,"reasoning_tokens":1651,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:11:50.189526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Zaks, Half-factorial domains , Bull","cited_arxiv_id":null,"evidence_quote":"Introduces half-factoriality, the property classified in Theorem 6.2."},{"cited_title":"Coykendall, F","cited_arxiv_id":null,"evidence_quote":"Introduces hereditary atomicity in integral domains and supplies the conjecture that Theorem 4.2 answers."},{"cited_title":"Gotti and J","cited_arxiv_id":null,"evidence_quote":"Settles the torsion-free case of the ACCP conjecture and poses the general monoid conjecture answered here."},{"cited_title":"Geroldinger and F","cited_arxiv_id":null,"evidence_quote":"Provides the standard monoid and factorization background, including the fact that ACCP implies atomicity used in the corollaries."},{"cited_title":"Coykendall and W","cited_arxiv_id":null,"evidence_quote":"Introduces length-factoriality (as other-half-factoriality), the property classified in Theorem 6.3."}],"review_version":1}