REVIEW 2 major objections 4 minor 10 references
Co-uniform and hollow S-acts over monoids
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 5, Proposition 5.3] 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 5, Theorem 5.6] 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.
minor comments (4)
- [Section 5, Theorem 5.6 proof] 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.
- [Throughout] 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 2, Lemma 2.2] 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.
- [Introduction] 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.
Circularity Check
No significant circularity: the claimed characterizations are derived from definitions and prior independent results, with only peripheral self-citations.
full rationale
The main derivation chain is self-contained. Theorem 3.4 (hollow iff indecomposable co-uniform) is proved directly from Definitions 2.1 and 3.1 together with Lemma 2.3, which is established within the paper; it does not presuppose the theorem. Proposition 3.3 (locally cyclic implies hollow) is a direct argument from the definition of superfluous subacts. The radical results in Section 4 follow from Lemma 4.5 and Proposition 4.6 proved in the text. Section 5's supplemented-act claims are attempted proofs from these earlier results, not restatements of their conclusions. The only self-citations are to the authors' earlier paper [5]: the condition (A) characterization in Lemma 3.8 and the cover fact in Lemma 3.9. Both are used as background equivalences about locally cyclic acts and covers; they are parameter-free, do not state hollow acts as their conclusion, and are not the target claims being established, so under the rule that independent support does not create circularity they do not raise the score. No fitted parameters or predictions are involved. (Correctness caveat, not circularity: Proposition 5.3's proof invokes C = A as a supplement although Definition 5.1 requires supplements to be proper subacts, and the same issue affects Theorem 5.6; this is a mathematical flaw in the proof, not a circular derivation.)
Assumptions & free parameters
assumptions (3)
- standard math Zorn's lemma is used to establish the existence of maximal subacts in cyclic and finitely generated acts.
- domain assumption Standard background on S-acts, Rees quotient acts, covers, and essential subacts is taken from Kilp, Knauer, and Mikhalev [6].
- domain assumption The equivalence that a monoid satisfies condition (A) iff every right S-act contains a minimal generating set is taken from [5].
Cite this review
Pith. "Pith review of Co-uniform and hollow S-acts over monoids." pith.science (2026). https://pith.science/paper/Q2DMF476
@misc{pith2026190804559,
author = {Pith},
title = {Pith review of: Co-uniform and hollow S-acts over monoids},
year = {2026},
howpublished = {\url{https://pith.science/paper/Q2DMF476}},
note = {Machine review of arXiv:1908.04559}
}
read the original abstract
In this paper, we first introduce the notions of superfluous and coessential subacts. Then hollow and co-uniform S-acts are defined as the acts that all proper subacts are superfluous and coessential, respectively. Also it is indicated that the class of hollow S-acts is properly between two classes of indecomposable and locally cyclic S-acts. Moreover, using the notion of radical of an S-act as the intersection of all maximal subact, the relations between hollow and local S-acts are investigated. Ultimately, the notion of a supplement of a subact is defined to characterize the union of hollow S-act.
Reference graph
Works this paper leans on
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Reviewed August 14, 2026 · model on record in the stance chip above.
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