{"id":"36ee1e1e-49d7-4a3a-b478-6f89b6af512f","arxiv_id":"1908.04559","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Hollow S-acts are characterized as indecomposable co-uniform acts, and hollow, locally cyclic, and indecomposable act classes are shown to be strictly nested, with supplements introduced to analyze unions of hollow acts.","lead":"This paper defines hollow and co-uniform S-acts over monoids, based on the ideas of superfluous and coessential subacts. It proves structural results connecting these acts to locally cyclic, indecomposable, and local acts, and then uses a new notion of supplement to describe unions of hollow acts.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.6 is false: its proof silently uses A itself as a supplement, while Definition 5.1 requires supplements to be proper subacts; the two-element act over S={0,1} is a concrete counterexample.","rationale":"The reader correctly identified a serious error in the supplement section, specifically that Proposition 5.3's proof uses C=A even though Definition 5.1 requires properness. That is a real defect. However, the single most load-bearing failure is in Theorem 5.6, which is the advertised central characterization of unions of hollow acts. Theorem 5.6 is not merely unproved: it is false. The counterexample A=S over S={0,1} is minimal and avoids any involved monoid construction. A is cyclic and hollow, hence a union of hollow acts; it has Rad(A)={0}≤s A; the proper subact {0} has a finitely generated Rees quotient; yet {0} has no proper supplement because the only proper subact is {0} itself. This directly contradicts (i)→(ii). The proof of (i)→(ii) fails exactly because it allows the constructed union L to be A, which is not an admissible supplement. The same improper-use-of-A pattern explains Proposition 5.3's false claim. Since the central application is broken and the advertised theorem is false, the reader's REJECT verdict is correct and should stand.","tokens_in":9824,"tokens_out":6690,"duration_ms":72354,"concrete_test":"Check the two-element S-act A={0,1} over S={0,1} directly: list proper subacts (only {0}), verify Max(A)={{0}}, Rad(A)={0}≤s A, verify A/{0} is finitely generated, and verify there is no proper subact C with {0}∪C=A. This refutes Theorem 5.6 exactly. Independently, rerun the proof of (i)→(ii) on the case where the finite cover L equals A and confirm that Definition 5.1's properness condition is the only obstruction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing defect is in Theorem 5.6, not only in Proposition 5.3. Definition 5.1 requires both B and C to be proper subacts, so a supplement C of B must satisfy C≠A. In the proof of (i)→(ii), the constructed L=∪_{j=1}^n L_j is never shown to be proper; if the finite cover is A itself, then L=A. The subsequent argument only shows that a proper subact X of L cannot satisfy B∪X=A, but this does not establish L≠A. Thus the proof implicitly allows a non-proper supplement, contradicting Definition 5.1. This is not a minor gap: a finite counterexample exists. Let S={0,1} with ordinary multiplication and let A=S as a right S-act. The only proper subact is {0}. A is cyclic, hence locally cyclic and hollow; it is local with unique maximal subact {0}. Also Rad(A)={0}≤s A, so hypothesis (i) of Theorem 5.6 holds. For B={0}, the Rees quotient A/B is finitely generated, so B satisfies the hypothesis of (ii). But a supplement C of B would be a proper subact with B∪C=A; the only proper subact is {0}, and {0}∪{0}={0}≠A. Hence B has no supplement, and (ii) fails. Therefore Theorem 5.6 is false as stated. Proposition 5.3 has the same defect: its proof puts C=A, and for a hollow non-supplemented act such as the paper's chain example, no proper supplement exists.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies co-uniform and hollow right S-acts over monoids. It introduces superfluous and coessential subacts, defines hollow and co-uniform acts as acts whose proper subacts are respectively superfluous or coessential, and proves (Theorem 3.4) that hollow acts are exactly the indecomposable co-uniform acts. Section 4 studies the radical of an act and the relation between hollow, local, and cyclic acts, including a characterization of when Rad(A) is superfluous (Theorem 4.9). Section 5 introduces supplements of subacts and claims, in Theorem 5.6, that under the hypothesis Rad(A) ≤s A, being a union of hollow acts is equivalent to a supplement condition on subacts whose Rees quotients are finitely generated.","tokens_in":10076,"tokens_out":3928,"duration_ms":39680,"significance":"Sections 2 through 4 contain a number of correct-looking and potentially useful results, including the characterization of hollow acts as indecomposable co-uniform acts (Theorem 3.4), the classification of co-uniform decomposable acts (Proposition 3.5), and the radical characterization in Theorem 4.9. The chain example in Section 4 is a nice illustration that hollow acts need not be cyclic. However, the central advertised contribution of Section 5, the supplement-based characterization of unions of hollow acts, is false as stated. Because the paper's abstract presents the supplement notion as the tool for this characterization, the defect is load-bearing and undermines the main claim of the paper. The paper is not circular and the earlier sections are largely independent, but the failure of Section 5 cannot be treated as a local typo.","major_comments":[{"comment":"The proof of Proposition 5.3 is invalid under Definition 5.1. The proof claims that for a proper subact B of an indecomposable co-uniform act A, the expression B ∪ A = A shows that A is a supplement of B. But Definition 5.1 requires a supplement C to be a proper subact of A. The paper's own example in Section 4 (S = (N, min) ∪ {ε}, A = {1,2,3,...}) is hollow and co-uniform, yet for B = {1} there is no proper subact C with B ∪ C = A, since every proper subact is a finite initial segment. Hence Proposition 5.3 is false.","section":"Section 5, Proposition 5.3"},{"comment":"The proof of (i) → (ii) in Theorem 5.6 fails to establish that the constructed subact L = L1 ∪ ... ∪ Ln is a proper subact of A, which Definition 5.1 requires for L to be a supplement. In the finite cover step, it is possible that L = A, in which case the minimality argument does not yield a supplement. This is not a technicality: take S = {0,1} with ordinary multiplication and A = S as a right S-act. A is cyclic, hence hollow and local, and Rad(A) = {0} ≤s A. For B = {0}, the Rees quotient A/B is finitely generated, so B satisfies the hypothesis of (ii). But the only proper subact of A is {0}, and {0} ∪ {0} = {0} ≠ A, so B has no supplement. Thus Theorem 5.6 is false as stated. The same defect appears in the proof of Proposition 5.3, where C = A is used despite the properness requirement.","section":"Section 5, Theorem 5.6"}],"minor_comments":[{"comment":"The final paragraph of the proof of (iii) → (i) repeats the argument that Rad(A) ≤s A after already concluding that B = A; this paragraph appears to be a leftover from a previous version and should be removed.","section":"Section 5, Theorem 5.6 proof"},{"comment":"The phrase 'union of hollow acts' in Theorem 5.6 is not formally defined; the authors should specify whether the union is over a family of subacts indexed by an arbitrary set, and how the hollow subacts are chosen.","section":"Throughout"},{"comment":"The statement of Lemma 2.2 uses 'C ∩ B ≠ ∅' as a condition, but in the category of S-acts the empty subact may require separate convention; the authors should clarify whether the empty set is considered a subact and how this affects the criterion.","section":"Section 2, Lemma 2.2"},{"comment":"The terminology 'coessential (small)' is used in the introduction but 'small' is not subsequently used; the notation would be clearer if only one term were retained.","section":"Introduction"}],"recommendation":"reject","confidential_remarks":"The flaw in Section 5 is fatal to the paper's central advertised claim. The counterexample to Theorem 5.6 is a two-element act over the trivial monoid-like semilattice, and the same issue invalidates Proposition 5.3. While Sections 2–4 appear sound and may be publishable as a separate note, the supplement-based characterization of hollow-union acts is false under the paper's own definition. A revision would require substantial changes to Definition 5.1 or to the main theorem, and even then the current arguments do not give a clear route. Therefore rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the quick take: the paper introduces genuinely new notions in S-act theory—coessential and superfluous subacts, hollow and co-uniform acts—and Sections 2–4 contain several correct and useful statements. Theorem 3.4 (hollow iff indecomposable co-uniform) and Proposition 4.6 (radical = union of superfluous subacts) are the kind of thing people in the area will want to know. That's not nothing.\n\nBut Section 5, the advertised payoff, is broken. Definition 5.1 requires a supplement to be a proper subact. Proposition 5.3 claims every co-uniform act is supplemented, and its proof literally takes C = A. That's not allowed. The authors' own chain example S = (N,min)∪{ε}, A = {1,2,3,...} is hollow and co-uniform, yet the proper subact {1} sits inside every proper subact; no proper subact C satisfies {1}∪C = A. So Proposition 5.3 is false as stated. The same defect sinks Theorem 5.6: the finitely generated cover L = L1∪...∪Ln is never shown to be proper. The stress-test note gives a concrete counterexample over S = {0,1} with A = S: A is local, hollow, Rad(A) ≤s A, but the proper subact {0} has no supplement. So the theorem's (i)→(ii) implication fails.\n\nThis is not a minor gap. The abstract says the supplement notion is introduced to characterize the union of hollow acts, and Theorems 5.3–5.6 are the whole point of Section 5. Remove or fix them and you're left with a decent structural paper; keep them and the main advertised application is false.\n\nOn the positive side: the paper is not circular, the self-citation in Lemma 3.8 is legitimate background, and the proof errors in Sections 2–4 are not evident—I'd trust Theorems 3.4, 3.6, 4.9 and Proposition 4.6. The error in Theorem 4.4 mentioned in the reader's report is real but minor; the statement looks salvageable. The citation pattern is fine.\n\nWho is this for? Specialists in S-act theory. Anyone studying covers, local acts, or radical subacts will get something from Sections 2–4. The flaw in Section 5 is instructive, but it's a liability if left as is.\n\nMy recommendation: send it to peer review, not desk-reject. A referee who knows the area can confirm the Sections 2–4 results and push the authors to either repair Section 5 or remove it. If they can't fix the supplement theorem, the paper should be accepted only for the earlier sections with the false claims retracted.","headline":"Genuinely useful sections 2–4 on hollow and co-uniform S-acts, but the advertised supplement section is broken: Proposition 5.3 and Theorem 5.6 are false under the paper's own proper-subact definition.","tokens_in":10686,"tokens_out":2814,"would_cite":false,"duration_ms":26135,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06F05","20M30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that hollow S-acts over monoids are exactly the indecomposable co-uniform acts, and introduces supplements of subacts to characterize acts that are unions of hollow acts.","keywords":["monoids","S-acts","hollow acts","co-uniform acts","superfluous subacts","coessential subacts","radical of acts","supplemented acts"],"falsifier":"Inspect the act $A = \\{1,2,3,\\dots\\}$ over $S = (\\mathbb{N},\\min)\\cup\\{\\varepsilon\\}$, where every proper subact is a finite initial segment $\\{1,\\dots,n\\}$. This act is hollow and co-uniform, yet the proper subact $\\{1\\}$ has no proper supplement, because $\\{1\\}\\cup C = A$ would force $C$ to contain all but finitely many elements, impossible for a finite initial segment. This observation settles Proposition 5.3 as stated.","tokens_in":9531,"feed_emoji":"","tokens_out":7928,"duration_ms":65352,"temperature":0.7,"pith_summary":"The paper adapts two module-theoretic notions—superfluous and coessential subobjects—to acts over monoids, and uses them to define hollow and co-uniform $S$-acts. Its central result, Theorem 3.4, says a right $S$-act is hollow exactly when it is indecomposable and co-uniform, which places hollow acts strictly between locally cyclic and indecomposable acts. The paper also studies the radical of an act (the intersection of its maximal subacts), shows that the radical equals the union of all superfluous subacts, and characterizes local acts inside the hollow class. It then introduces supplements of subacts, with the aim of describing when an act is a union of hollow acts, claiming that every co-uniform act is supplemented and that this notion drives the three-way characterization in Theorem 5.6.","feed_headline":"Hollow S-acts equal indecomposable co-uniform acts","feed_subtitle":"The theorem places hollow acts between locally cyclic and indecomposable, with supplements describing their unions.","key_machinery":"The load-bearing equivalence is Theorem 3.4, built from two definitions. A subact $B$ is superfluous ($B \\leq_s A$) when $B \\cup C \\neq A$ for every proper subact $C$, and coessential when the quotient projection $A \\to A/B$ is a coessential epimorphism, meaning no proper subact maps onto the quotient. Hollow acts are those whose proper subacts are all superfluous; co-uniform acts are those whose proper subacts are all coessential. The proof that hollow equals indecomposable co-uniform uses Lemma 2.3, which says a coessential subact of an indecomposable act is superfluous. The radical $\\mathrm{Rad}(A)$, the intersection of all maximal subacts, is then identified with the union of all superfluous subacts, and the notion of a supplement—a minimal proper subact $C$ with $B \\cup C = A$—is introduced to characterize unions of hollow acts in Theorem 5.6.","core_discovery":"The paper establishes that for right acts over a monoid, hollow and co-uniform are not automatically the same notion, as they are for modules; it proves the precise bridge: an act is hollow if and only if it is indecomposable and co-uniform. Every locally cyclic act—one in which any two elements lie in a common cyclic subact—is hollow, and every cyclic act is locally cyclic, so the hierarchy cyclic → locally cyclic → hollow → indecomposable is strict. The paper further claims that every co-uniform act is supplemented, where a supplement of a proper subact $B$ is a proper subact $C$ with $B \\cup C = A$ that is minimal with that property, and it uses this notion to prove Theorem 5.6: under the hypothesis $\\mathrm{Rad}(A) \\leq_s A$, an act is a union of hollow acts if and only if every proper subact with finitely generated Rees quotient has a supplement, if and only if every maximal subact has a supplement.","pith_inferences":["Read literally, Definition 5.1 makes Proposition 5.3 break: the proof uses the whole act as the supplement of a proper subact, although the definition requires supplements to be proper. In the paper's own chain example $S = (\\mathbb{N},\\min)\\cup\\{\\varepsilon\\}$, $A=\\{1,2,3,\\dots\\}$, the proper subact $\\{1\\}$ has no proper supplement, since every proper subact is a finite initial segment.","A plausible repair is to allow the supplement to equal $A$, or to restrict supplementation to proper subacts with finitely generated Rees quotient; Theorem 5.6 may survive under that weaker reading.","The equality between the radical and the union of superfluous subacts transfers a standard module-theoretic fact to $S$-acts and suggests that local acts are the natural finite analogue of hollow acts, just as local modules sit inside hollow modules."],"forward_implications":["Every locally cyclic and every cyclic $S$-act is hollow, and every finitely generated hollow act is cyclic.","A monoid satisfies condition (A)—every right $S$-act contains a minimal generating set—if and only if every hollow $S$-act is cyclic.","Every cover of a hollow act is indecomposable; consequently projective covers of hollow acts are cyclic and strongly flat covers are locally cyclic.","When a maximal subact exists, an act is hollow exactly when it is cyclic and local.","If $\\mathrm{Rad}(A) \\leq_s A$, then an act is a union of hollow acts exactly when every maximal subact has a supplement."],"supporting_citations":[{"why":"Supplies the classical notion of hollow modules whose dualization motivates hollow S-acts.","marker":"[2]"},{"why":"Provides the preliminaries and basic results on monoids and S-acts used throughout the paper.","marker":"[6]"},{"why":"Introduced uniform acts over semigroups, the dual notion that leads to co-uniform acts here.","marker":"[9]"},{"why":"Established that a cover of a locally cyclic act is indecomposable and gave the condition (A) equivalence that the paper extends to hollow acts.","marker":"[5]"},{"why":"Supplies the module-theoretic background on superfluous submodules and radicals that the S-act analogues are modeled on.","marker":"[10]"}],"fun_headline_variants":["Hollow S-acts = indecomposable & co-uniform","Hollow acts are the co-uniform indecomposables","Cyclic → locally cyclic → hollow → indecomposable","Supplements characterize hollow-act unions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise of the supplemented-acts half of the paper is that the whole act may serve as the supplement of any proper subact, but Definition 5.1 explicitly requires supplements to be proper; if properness is enforced, the paper's own chain example gives a co-uniform hollow act whose proper subact $\\{1\\}$ has no proper supplement.","fun_headline_variants_meta":{"raw":{"variants":["Hollow S-acts = indecomposable & co-uniform","Hollow acts are the co-uniform indecomposables","Cyclic → locally cyclic → hollow → indecomposable","Supplements characterize hollow-act unions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002759,"raw_usage":{"total_tokens":10463,"prompt_tokens":843,"completion_tokens":9620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":9557}},"tokens_in":459,"tokens_out":9620,"duration_ms":70273,"temperature":1.0,"reasoning_tokens":9557,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:40:05.627059+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Inspect the act $A = \\{1,2,3,\\dots\\}$ over $S = (\\mathbb{N},\\min)\\cup\\{\\varepsilon\\}$, where every proper subact is a finite initial segment $\\{1,\\dots,n\\}$. This act is hollow and co-uniform, yet the proper subact $\\{1\\}$ has no proper supplement, because $\\{1\\}\\cup C = A$ would force $C$ to contain all but finitely many elements, impossible for a finite initial segment. This observation settles Proposition 5.3 as stated.","supporting_citations":[{"cited_title":"Pacific J","cited_arxiv_id":null,"evidence_quote":"Supplies the classical notion of hollow modules whose dualization motivates hollow S-acts."},{"cited_title":", Knauer, U","cited_arxiv_id":null,"evidence_quote":"Provides the preliminaries and basic results on monoids and S-acts used throughout the paper."},{"cited_title":"and Sedaghatjoo, M.: On uniform acts over semig roups","cited_arxiv_id":null,"evidence_quote":"Introduced uniform acts over semigroups, the dual notion that leads to co-uniform acts here."},{"cited_title":"and Sedaghatjoo, M.: Strongly flat and condi- tion (P) covers of acts over monoids","cited_arxiv_id":null,"evidence_quote":"Established that a cover of a locally cyclic act is indecomposable and gave the condition (A) equivalence that the paper extends to hollow acts."},{"cited_title":"Gordon an d Breach Science Publishers, Reading (1991) 14","cited_arxiv_id":null,"evidence_quote":"Supplies the module-theoretic background on superfluous submodules and radicals that the S-act analogues are modeled on."}],"review_version":1}