{"id":"612e4921-79f4-401b-aa0d-65d5715b95f1","arxiv_id":"1908.01983","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two algebraic entropies are defined for amenable semigroup actions on abelian groups, and an Addition Theorem and a Bridge Theorem are proved for torsion abelian groups and totally disconnected compact abelian groups.","lead":"The paper introduces two algebraic entropies for actions of cancellative right amenable semigroups on abelian groups, extending the classical entropy of a single endomorphism. It proves an Addition Theorem for torsion abelian groups and a Bridge Theorem connecting algebraic entropy to topological entropy via Pontryagin duality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Addition and Bridge Theorems are internally coherent, and reliance on the imported Ornstein-Weiss/Filling theorem is a legitimate external foundation.","rationale":"The reader's ACCEPT verdict is well supported. The paper's main results are proved from explicit constructions and from external results in [11] that are invoked in the stated regime. The weakest assumption identified by the reader, namely the availability of the Ornstein-Weiss-type limit theorem for subadditive left-subinvariant functions, is real but is a legitimate cited theorem rather than a hidden or circular assumption. I checked the main proof flow of Theorem 1.1: Proposition 6.12 supplies the lower bound via an exact-sequence argument, and Proposition 6.13 supplies the upper bound via the Filling Theorem and a careful ε-tiling estimate. The constants and subnet arguments are formally consistent, apart from easily repaired local edge cases such as the trivial finite subgroup Y = {0} in Proposition 6.13 and a factor-of-2 slack in Proposition 2.17; neither changes the mathematical content. For Theorem 1.2, the duality computations are consistent: a left action on a totally disconnected compact abelian group dualizes to a right action on a torsion discrete abelian group, and Lemma 7.6 correctly identifies |T_F(α,B)| with the index of the cotrajectory in the dual group. The open-subgroup reduction for topological entropy is justified by van Dantzig's theorem. No significant objection to the central claims was identified.","tokens_in":53403,"tokens_out":14898,"duration_ms":156779,"concrete_test":"Independently re-derive the ε-tiling step in Proposition 6.13 from Theorem 6.6 with the constants written out in full, including the case Y = {0}; if the final 5ε inequality (6.7) closes without changing the statement, the Addition Theorem proof is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central claims as: (i) Theorem 1.1, the Addition Theorem for ent on torsion abelian groups; (ii) Theorem 1.2, the Bridge Theorem for totally disconnected compact abelian groups. Both are supported by explicit proofs. The definition of entropy and all Følner limits rest on Corollary 3.2, imported from [11]; this is an external theorem, but a standard and correctly stated one, so it is not an internal gap. The proof of Theorem 1.1 uses the Filling Theorem (6.6) exactly as imported, with no visible circularity: the ε-tilings are used only to decouple the Følner growth of the trajectory sizes, and the final estimate (6.7) has the right ε-dependence. The Bridge Theorem reduces htop to h^r_alg through the Pontryagin duality computations in Lemma 7.6; the direction of the dual action and the open-subgroup parametrization are consistent. I did not find a load-bearing flaw. Minor local points, such as the constant choice in Proposition 2.17 and the unhandled Y = {0} case in Proposition 6.13, are repairable and do not affect the central claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends algebraic entropy to left actions of cancellative right amenable monoids on discrete abelian groups by endomorphisms, defining two quantities, halg and ent, which agree on torsion groups. The main results are Theorem 1.1 (Addition Theorem): for a left action of a cancellative right amenable monoid on a torsion abelian group and an invariant subgroup B, ent(alpha)=ent(alpha_B)+ent(alpha_{A/B}); and Theorem 1.2 (Bridge Theorem): for a left action of a cancellative left amenable monoid on a totally disconnected compact abelian group, the topological entropy equals the algebraic entropy of the dual right action. The paper also develops a Følner-based integration theory for subadditive functions on amenable semigroups, proves restriction and quotient entropy formulas for amenable group actions, and derives an Addition Theorem for topological entropy of actions on totally disconnected compact abelian groups via Pontryagin duality.","tokens_in":53635,"tokens_out":15709,"duration_ms":139875,"significance":"If correct, the results unify and generalize the classical Weiss/Peters algebraic entropy and the corresponding Bridge Theorem to a substantial class of amenable semigroup actions, going beyond the group actions treated by Virili. The proofs are explicit and are built on the published Ornstein–Weiss-type theorem of Ceccherini-Silberstein–Coornaert–Krieger and the Filling Theorem; the main arguments do not circularly reduce to the quantities being defined, and no fitted parameters are used. The paper is a serious contribution to entropy theory on groups and monoids and should be of interest to readers in algebraic entropy, topological entropy, and amenability. The strengths include the careful statement of the external tools on which the construction depends and the explicit epsilon-tracking in the proof of the Addition Theorem, modulo the presentation issues noted below.","major_comments":[],"minor_comments":[{"comment":"The proof invokes Lemma 2.7(a) to find F with |FE \\ F|/|F| ≤ 1/n, but the defining property (2.6) requires |F(E,n)s Δ F(E,n)|/|F(E,n)| ≤ 1/n for each s ∈ E. The proof should instead use |FE Δ F|/|F| ≤ 1/n (or adjust the constant by a factor of 2) so that the stated definition of a canonically indexed right Følner net is actually satisfied.","section":"2.3, Proposition 2.17"},{"comment":"The displayed inequalities in the proof of Proposition 6.13 use what appears to be multiplicative epsilons, e.g., '≤5ε ent(αB) + Halg(...)' and '≤3ε Halg(...)'. The surrounding derivation supports only additive epsilon terms of the form '≤5ε + ent(αB) + Halg(...)' and '≤Halg(...) + 3ε'. These displays should be corrected to match the argument, since the literal multiplicative reading is inconsistent with the rest of the proof.","section":"6.3, Eqs. (6.7) and (6.13)"},{"comment":"The definition of \\barε = ε/(2ℓ(Y)) is undefined when Y = {0}, since ℓ(Y) = 0. The case Y = {0} should be treated separately (its entropy contribution is zero) so that the proof covers all finite subgroups Y of A.","section":"6.3, Proposition 6.13"},{"comment":"The hypothesis is stated as 'S a right amenable monoid', but the proof applies Corollary 6.10 and Theorem 6.6, both of which require cancellativity of S. The statement should read 'cancellative right amenable monoid' in order for the quoted external theorems to apply.","section":"6.3, Proposition 6.13"},{"comment":"The statement says that γ induces a dual right action on 'the Pontryagin dual of A', but the group A has not yet been introduced at that point; the intended group is the Pontryagin dual of K, namely A = \\hat{K}. This should be corrected to avoid confusion.","section":"1, Theorem 1.2"},{"comment":"The expression 'Halg(α, ↾G,TEm(α↾H,X))' contains a misplaced comma and should be written as Halg(α↾G, T_{E_m}(α↾H,X)) for clarity.","section":"5.1, proof of Lemma 5.2"},{"comment":"The manuscript contains numerous typographical and OCR artifacts, such as 'ε2N' in Theorem 6.6, incomplete or garbled footnote material in §2.3, and inconsistent notation like 'δn,n' in Example 2.28. These should be cleaned up before publication.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical core is sound and the paper fits the journal's scope. The central claims are supported by explicit proofs relying on published external theorems (Ornstein-Weiss type lemma and Filling Theorem from [11], and Pontryagin duality). The main reason for not recommending acceptance as is is the misprinted epsilon-dependence in the key estimate of Proposition 6.13 and the unhandled Y = {0} case; both are easily repairable. No concerns about circularity or missing attribution arose: the definition of entropy depends on the imported Ornstein-Weiss theorem, which is a legitimate external foundation, and the self-citations serve only to place the results in the known framework for N-actions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two things you should know: this is a genuinely substantial generalization of algebraic entropy from N-actions and amenable group actions to cancellative right amenable semigroup actions, and the core arguments are coherent. The Addition Theorem (ent for torsion abelian groups) and the Bridge Theorem (topological entropy equals algebraic entropy of the dual) are the real payoffs, and they appear to be proved correctly. The stress-test note is right: no load-bearing flaw surfaced on a careful pass, and the reliance on the imported Ornstein–Weiss/Filling machinery from [11] is legitimate external support, not a hidden assumption.\n\nWhat the paper does well is more than just state two big theorems. The Fubini-type theorem for the integral of subadditive functions (Theorem 3.10) is a genuinely useful tool, and the proof of the Addition Theorem deliberately avoids the heavy structure theory of abelian groups used in earlier N-action proofs. The authors are also honest about what they cannot do: the Addition Theorem is proved for torsion groups only, and the Bridge Theorem for totally disconnected compact abelian groups; both limitations are stated explicitly, and the conjectural extensions are flagged as such. The citation pattern is fine—self-citations go to prior work that establishes the classical cases, and they are not used to force the new theorems.\n\nSoft spots are local and repairable. The paper is dense, and the Følner machinery makes full verification painful; I could not check every epsilon. The minor issues the stress-test flagged—the constant choice in Proposition 2.17 and the unhandled Y = {0} case in Proposition 6.13—are exactly that: minor, and they do not threaten the main results. Proposition 6.13's proof would benefit from a sentence treating the trivial quotient, but that is a detail. A referee should also push on whether the integral framework in Section 3 can be simplified, since the present exposition is heavier than the underlying ideas require.\n\nThis paper is for researchers in algebraic entropy, amenable group actions, and topological entropy on abelian groups. It will likely become standard machinery in that niche. I would cite it and would bring it to a reading group. The right call is to send it to peer review: it deserves a serious referee, not a desk reject, even though the referee will need to do some work to verify the technical core.","headline":"A robust generalization of algebraic entropy to amenable semigroup actions; the Addition and Bridge Theorems hold up under scrutiny and the paper deserves serious refereeing.","tokens_in":54145,"tokens_out":1265,"would_cite":true,"duration_ms":16087,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20K30","20M20","37A35","37B40","43A07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines algebraic entropy for actions of cancellative right amenable semigroups on abelian groups and proves that, for torsion groups, entropy is additive over invariant subgroups and equals the topological entropy of the dual…","keywords":["algebraic entropy","amenable semigroup actions","Følner nets","Addition Theorem","Bridge Theorem","Pontryagin duality","topological entropy","torsion abelian groups"],"falsifier":"Compute the defining trajectory limits for the $\\mathbb N$-action on $A=\\bigoplus_{i\\in\\mathbb Z}(\\mathbb Z/2\\mathbb Z)$ with shifts $e_i\\mapsto e_{i+s}$, taking $B=\\mathrm{span}\\{e_i:i\\ge 0\\}$. The Addition Theorem predicts $\\operatorname{ent}(\\alpha)=\\log 2$, $\\operatorname{ent}(\\alpha_B)=\\log 2$, and $\\operatorname{ent}(\\alpha_{A/B})=0$; a direct count gives $|T_n(\\{e_0\\})|=2^n$ and $|T_n(\\{e_{-1}+B\\})|=2$. Any mismatch in the identity, or any dependence of the limits on the chosen right Følner net, would refute Theorem 1.1 or the imported mean theorem.","tokens_in":53228,"feed_emoji":"🔗","tokens_out":31540,"duration_ms":324935,"temperature":0.7,"pith_summary":"This paper defines algebraic entropy—a measure of how fast finite sets grow under a group action—for left actions $S\\curvearrowright A$ of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms, generalizing the classical case $S=\\mathbb N$. Its central claim is an Addition Theorem: whenever $A$ is torsion and $B$ is an $\\alpha$-invariant subgroup, $\\operatorname{ent}(\\alpha)=\\operatorname{ent}(\\alpha_B)+\\operatorname{ent}(\\alpha_{A/B})$. Its second central claim is a Bridge Theorem: for actions of cancellative left amenable monoids on totally disconnected compact abelian groups, the topological entropy of the action equals the algebraic entropy of the Pontryagin-dual action. These two theorems together yield an Addition Theorem for the topological entropy. The reader should care because the paper shows that entropy behaves like a length function on a natural algebraic category, and because the bridge makes algebraic and topological entropy computable from each other.","feed_headline":"For torsion abelian groups, entropy splits across invariant subgroups","feed_subtitle":"Via Pontryagin duality, the identities also bridge to topological entropy of compact group actions.","key_machinery":"The central object is the functional $H_S(f)=\\lim_i f(F_i)/|F_i|$ on increasing, subadditive, left-subinvariant functions on the finite subsets of a cancellative right amenable monoid $S$, where $(F_i)$ is a right Følner net; its net-independence is imported from the semigroup mean theorem (Theorem 3.1). Applied to $f_X(F)=\\log|\\sum_{s\\in F}\\alpha(s)(X)|$, it produces $H_{\\mathrm{alg}}(\\alpha,X)$. The proof of the Addition Theorem is carried by the Filling Theorem (Theorem 6.6), which for any $\\varepsilon$ finds finitely many translates $P_jF_j$ that almost cover a Følner set $D$ and are almost disjoint, so entropy sums split over the pieces; a Fubini-type theorem (Theorem 3.10) then controls the comparison of entropies over different right Følner nets. The Bridge Theorem is carried by Pontryagin duality: the annihilator correspondence $B\\mapsto B^\\perp$ bijects finite subgroups of the torsion discrete group with open subgroups of its compact dual, converting trajectory size $\\log|T_F(B)|$ into the index $\\log[\\widehat A : C_F(\\widehat\\alpha,B^\\perp)]$ that defines topological entropy.","core_discovery":"The paper introduces two variants of algebraic entropy, $\\operatorname{ent}$ and $h_{\\mathrm{alg}}$, both defined by taking the supremum, over finite subgroups (for $\\operatorname{ent}$) or finite subsets (for $h_{\\mathrm{alg}}$), of the Følner average of $\\log|T_F(\\alpha,X)|$ along right Følner nets; the two coincide on torsion abelian groups. Its principal claim is that on torsion abelian groups this entropy is additive over invariant subgroups (Theorem 1.1), the proof of which uses a Filling Theorem to break an arbitrary Følner set into finitely many translates of fixed shapes, avoiding the classification of torsion abelian groups. Its second principal claim is the Bridge Theorem (Theorem 1.2): if $S$ is a cancellative left amenable monoid acting by continuous endomorphisms on a totally disconnected compact abelian group $K$, then the topological entropy of the action equals the algebraic entropy of the induced right action on the discrete Pontryagin dual $\\widehat K$; since the duals of totally disconnected compact groups are exactly the torsion abelian groups, this connects the two entropies on the full class where both are defined.","pith_inferences":["The Filling-Theorem proof does not use the structure theory of torsion abelian groups, so the same template may prove the Addition Theorem for any additive invariant that is continuous under direct limits and behaves like a length function on module categories—beyond the logarithmic one used here.","The conjectures in Section 8 suggest the results should survive when $A$ is not torsion (Addition Theorem) and when $K$ is not totally disconnected (Bridge Theorem); a positive answer would make algebraic and topological entropy of amenable-group actions on locally compact abelian groups completely parallel.","The frequent vanishing for $\\mathbb N^d$ and matrix group actions indicates that for genuinely multidimensional amenable semigroup actions, nonzero algebraic entropy is concentrated in shift-type actions and in phenomena invisible to the Følner average, a setting where alternative invariants such as receptive entropies would be tested."],"forward_implications":["For every left action of a cancellative right amenable monoid on a torsion abelian group, entropy is additive over invariant subgroups, making $\\operatorname{ent}$ a length function on the category of torsion $\\mathbb Z[S]$-modules that is also continuous under direct limits.","For cancellative left amenable monoids acting on totally disconnected compact abelian groups, topological entropy satisfies the same addition identity; this is a direct corollary of the Addition and Bridge Theorems.","Finite-index subgroups obey a Logarithmic Law: if $[G:H]=k$, then $h_{\\mathrm{alg}}(\\alpha\\!\\upharpoonright\\!H)=k\\,h_{\\mathrm{alg}}(\\alpha)$, and the same holds for $\\operatorname{ent}$.","Many natural higher-rank actions have zero algebraic entropy: actions of $\\mathbb N^d$ ($d>1$) on torsion-free finite-rank abelian groups, and the multiplicative action of upper-triangular subgroups of $\\mathrm{GL}_n(K)$ on $K^n$ for infinite $K$.","Whenever an action factors through a quotient by an infinite normal subgroup lying in the kernel, its algebraic entropy is zero."],"supporting_citations":[{"why":"Supplies the semigroup mean theorem (their Theorem 3.1) and the Filling Theorem (their Theorem 3.8) that make the entropy well-defined and drive the Addition Theorem.","marker":"[11]"},{"why":"Proves the Addition Theorem for endomorphisms of torsion abelian groups (S=N), the classical case extended here to monoids.","marker":"[30]"},{"why":"Provides the definition and properties of the algebraic entropy halg for single endomorphisms on which the two new invariants are modelled.","marker":"[28]"},{"why":"Introduces the Bridge Theorem for endomorphisms, connecting algebraic and topological entropy; generalized here as Theorem 1.2.","marker":"[77]"},{"why":"Establishes the bridge for discrete abelian groups via Pontryagin duality, the route followed in Section 7.","marker":"[64]"},{"why":"The classical work on entropy for amenable group actions whose semigroup analogue the paper imports.","marker":"[62]"}],"fun_headline_variants":["Entropy splits for torsion groups in semigroup actions","Algebraic entropy bridges to topological via duality","Semigroup entropy addition theorem proven for torsion groups","For torsion abelian groups, entropy is additive over subgroups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported theorem that subadditive left-subinvariant functions on cancellative right amenable semigroups have a Følner-net-independent mean; if that mean depended on the net, neither entropy would be well-defined.","fun_headline_variants_meta":{"raw":{"variants":["Entropy splits for torsion groups in semigroup actions","Algebraic entropy bridges to topological via duality","Semigroup entropy addition theorem proven for torsion groups","For torsion abelian groups, entropy is additive over subgroups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000329,"raw_usage":{"total_tokens":1826,"prompt_tokens":926,"completion_tokens":900,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":839}},"tokens_in":542,"tokens_out":900,"duration_ms":8763,"temperature":1.0,"reasoning_tokens":839,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:57:04.566175+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the defining trajectory limits for the $\\mathbb N$-action on $A=\\bigoplus_{i\\in\\mathbb Z}(\\mathbb Z/2\\mathbb Z)$ with shifts $e_i\\mapsto e_{i+s}$, taking $B=\\mathrm{span}\\{e_i:i\\ge 0\\}$. The Addition Theorem predicts $\\operatorname{ent}(\\alpha)=\\log 2$, $\\operatorname{ent}(\\alpha_B)=\\log 2$, and $\\operatorname{ent}(\\alpha_{A/B})=0$; a direct count gives $|T_n(\\{e_0\\})|=2^n$ and $|T_n(\\{e_{-1}+B\\})|=2$. Any mismatch in the identity, or any dependence of the limits on the chosen right Følner net, would refute Theorem 1.1 or the imported mean theorem.","supporting_citations":[{"cited_title":"Ceccherini-Silberstein, M","cited_arxiv_id":null,"evidence_quote":"Supplies the semigroup mean theorem (their Theorem 3.1) and the Filling Theorem (their Theorem 3.8) that make the entropy well-defined and drive the Addition Theorem."},{"cited_title":"Dikranjan, B","cited_arxiv_id":null,"evidence_quote":"Proves the Addition Theorem for endomorphisms of torsion abelian groups (S=N), the classical case extended here to monoids."},{"cited_title":"Dikranjan, A","cited_arxiv_id":null,"evidence_quote":"Provides the definition and properties of the algebraic entropy halg for single endomorphisms on which the two new invariants are modelled."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Bridge Theorem for endomorphisms, connecting algebraic and topological entropy; generalized here as Theorem 1.2."},{"cited_title":"Peters, Entropy on discrete abelian groups , Adv","cited_arxiv_id":null,"evidence_quote":"Establishes the bridge for discrete abelian groups via Pontryagin duality, the route followed in Section 7."},{"cited_title":"Ornstein, B","cited_arxiv_id":null,"evidence_quote":"The classical work on entropy for amenable group actions whose semigroup analogue the paper imports."}],"review_version":1}