{"id":"4a23522e-7191-4a33-80ab-895ed6430f82","arxiv_id":"2501.03407","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For linearly orderable monoids, quasi-ACCP and almost ACCP ascend to finitary power monoids, while atomicity, near atomicity, and quasi-atomicity do not; atomic power monoids are characterized as those from atomic MCD-monoids.","lead":"This paper studies which algebraic properties, like atomicity and divisibility, survive when passing from an ordered commutative monoid to the monoid of its finite subsets under elementwise addition. It proves several preservation results and builds explicit counterexamples showing that other properties fail to survive, including a complete characterization of atomic power monoids.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3(a)⇒(c) is not proved in the paper; it is delegated as 'mutatis mutandis' to [13, Thm 3.2], so the central characterization of atomic power monoids rests on an unverified adaptation from Puiseux monoids to all linearly orderable monoids.","rationale":"The reader's conditional verdict is justified, but I locate the central pressure differently. The reader focused on Theorem 5.5 and the near-atomicity counterexample; that is a real gap but it concerns a secondary ascent result. The main theorem, Theorem 4.3, has a more fundamental issue: the proof of the forward direction is not present in the manuscript. It is replaced by a citation to [13, Theorem 3.2] with 'mutatis mutandis'. Since [13] is a Puiseux-monoid result, a skeptical reader cannot verify that the argument adapts to all linearly orderable monoids, which need not be positive, rank 1, or Archimedean. This is a proof-gap concern, not a demonstrated falsehood. The 'suitable subset' step in (b)⇒(a) is also terse, but it can likely be fixed by observing that any zero-sum pair in a linearly orderable monoid is a pair of units, so such subsets have MCD 0. I therefore keep the conditional verdict: the central characterization should be accepted only after a complete proof of (a)⇒(c) is supplied, ideally with the non-Puiseux case made explicit.","tokens_in":24210,"tokens_out":25499,"duration_ms":244084,"concrete_test":"Produce a self-contained proof of Theorem 4.3(a)⇒(b) that avoids citing [13]. As a nontrivial checkpoint, run the proof on the non-Puiseux, non-Archimedean monoid M = N0^2: show explicitly that every non-singleton finite subset of P_fin(M) is a sum of atoms of P_fin(M). If the argument used in the checkpoint relies on any step that is only valid for rank-1 positive monoids (e.g. denominator valuations or well-ordered positive cones), the generalization in Theorem 4.3 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper is Theorem 4.3: for a linearly orderable monoid M, P_fin(M) is atomic iff M is an atomic MCD-monoid. The proof of the reverse direction (b)⇒(a) is given in detail, but the forward direction (a)⇒(c) is handled by 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.' That citation is to a theorem about Puiseux monoids, i.e., rank-1 submonoids of Q≥0. No step of the Puiseux argument is stated or checked here, and the paper does not identify which parts use rank 1, nonnegativity, or Archimedeanity. Arbitrary linearly orderable monoids can be non-Archimedean and can contain negative elements, so valuation- or minimality-based decompositions from [13] need not carry over. Since (a)⇒(c) is the existence half of the characterization, this is a load-bearing proof gap. A secondary, smaller gap is the unproved reduction 'after replacing S by a suitable subset' in (b)⇒(a); this is likely repairable because a zero-sum pair in a linearly orderable monoid consists of units, but it is still an asserted step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24508,"tokens_out":10330,"duration_ms":172985,"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":[{"comment":"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":"Theorem 4.3, proof of (a)⇒(c)"},{"comment":"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.","section":"Section 5.2, Theorem 5.5 and the paragraph before Lemma 5.4"},{"comment":"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.","section":"Theorem 6.5, construction of M"},{"comment":"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.","section":"Theorem 4.3, proof of (b)⇒(a)"},{"comment":"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.","section":"Theorem 5.3, proof of the Claim"}],"minor_comments":[{"comment":"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.","section":"Theorem 6.5, proof of (2)"},{"comment":"The final sentence 'Hence we conclude that P is nearly atomic' should read 'nearly Furstenberg.'","section":"Theorem 6.2(4)"},{"comment":"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.","section":"Lemma 5.4(1)"},{"comment":"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.","section":"Theorem 4.3 and Section 2.6"},{"comment":"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.","section":"Example 4.5"}],"recommendation":"major_revision","confidential_remarks":"The paper has several promising results and a clean framework, but it is not ready for publication. The proof of Theorem 4.3(a)⇒(c) is an unverified adaptation of a Puiseux-monoid argument, and Theorem 6.5 contains a concrete construction error: the monoid M actually contains −a and −b, so a and b are units and M is a group. The Section 5.2 proof of Theorem 5.5 also relies on an unproved structural claim. I would encourage the authors to supply a full proof of the missing direction in Theorem 4.3, repair or replace the Theorem 6.5 counterexample, and justify the divisibility claim in Theorem 5.5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThere 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.\n\nCitation 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":25067,"tokens_out":4745,"would_cite":true,"duration_ms":40769,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A05","13F15","13A15","13G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["finitary power monoid","atomic monoid","linearly orderable monoid","maximal common divisor","Puiseux monoid","almost atomic","Furstenberg property","TIDF monoid"],"falsifier":"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.","tokens_in":24014,"feed_emoji":"🔢","tokens_out":8882,"duration_ms":77079,"temperature":0.7,"pith_summary":"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.","feed_headline":"Power monoids are atomic under one exact condition","feed_subtitle":"For linearly orderable monoids, atomicity of P_fin(M) is equivalent to atomic M with finite MCDs.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Puiseux-monoid atomicity criterion that Theorem 4.3 generalizes, and the earlier atomic Puiseux monoid with non-atomic power monoid that Example 4.5 refines.","marker":"[13]"},{"why":"Gives the lower bound $|S+T|\\ge |S|+|T|-1$ for finitary sumsets, which drives Lemma 3.1 and the cardinality arguments throughout.","marker":"[9, Proposition 3.5]"},{"why":"Introduced the quasi-ACCP and almost ACCP conditions whose ascent to power monoids is established in Theorem 3.3.","marker":"[15]"},{"why":"Defines near atomicity, the property whose failure to ascend is proved in Theorem 5.3.","marker":"[20]"},{"why":"Introduced almost atomic and quasi-atomic monoids, the properties whose non-ascent is settled in Theorem 5.5.","marker":"[6]"},{"why":"Defines the Furstenberg property that Theorem 6.2 proves ascends to power monoids.","marker":"[7]"},{"why":"Defines the TIDF property whose ascent for positive Archimedean monoids is proved in Corollary 6.4 and whose general non-ascent is shown in Theorem 6.5.","marker":"[16]"},{"why":"Identifies finite-factorization monoids as atomic IDF-monoids, a step used in Corollary 6.4.","marker":"[19, Theorem 2]"},{"why":"Introduced the notion of k-MCD monoid that Proposition 4.1 uses to relate MCDs in M and in P_fin(M).","marker":"[24]"}],"fun_headline_variants":["Power monoid atomic iff M atomic and has finite MCDs","Atomic power monoids: exact M condition for linearly orderable","Linearly orderable M: power monoid atomic exactly then","Finitary power monoids: atomicity solved for linearly orderable","M atomic with finite MCDs gives atomic power monoid"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Power monoid atomic iff M atomic and has finite MCDs","Atomic power monoids: exact M condition for linearly orderable","Linearly orderable M: power monoid atomic exactly then","Finitary power monoids: atomicity solved for linearly orderable","M atomic with finite MCDs gives atomic power monoid"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000323,"raw_usage":{"total_tokens":1734,"prompt_tokens":785,"completion_tokens":949,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":861}},"tokens_in":401,"tokens_out":949,"duration_ms":9111,"temperature":1.0,"reasoning_tokens":861,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:55:30.147704+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the atomicity of power monoids of Puiseux monoids","cited_arxiv_id":"2401.12444","evidence_quote":"Supplies the Puiseux-monoid atomicity criterion that Theorem 4.3 generalizes, and the earlier atomic Puiseux monoid with non-atomic power monoid that Example 4.5 refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced almost atomic and quasi-atomic monoids, the properties whose non-ascent is settled in Theorem 5.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Furstenberg property that Theorem 6.2 proves ascends to power monoids."},{"cited_title":"Gotti and M","cited_arxiv_id":null,"evidence_quote":"Defines the TIDF property whose ascent for positive Archimedean monoids is proved in Corollary 6.4 and whose general non-ascent is shown in Theorem 6.5."},{"cited_title":"Roitman, Polynomial extensions of atomic domains , J","cited_arxiv_id":null,"evidence_quote":"Introduced the notion of k-MCD monoid that Proposition 4.1 uses to relate MCDs in M and in P_fin(M)."}],"review_version":1}