{"id":"521e7bb1-c858-4a54-8bc6-dd0e681ecd67","arxiv_id":"2501.05536","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For an embeddable semigroup S, the free S-group is the unique group over which every surjective continuous S-action admits a natural extension, and left reversibility characterizes when all compact extensions factor through it.","lead":"This paper builds a general theory of natural extensions for continuous actions of countable semigroups that can be embedded into groups. It shows that the free group on a semigroup is the unique receiving group that always admits such an invertible extension, and that left-reversibility is exactly the condition under which the classical smallest-extension picture works.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.6 does not produce its approximants in the natural extension: the cofinal sequence of right ideals must be chosen increasing for the subnet limit to be coherent.","rationale":"Theorem B and Theorem C rest on the intricate construction in Proposition 3.6, which appears essentially sound: the use of non-canonical right inverses for surjective shifts is delicate but the proof only needs the identity f_{vw} = f_v composed with f_w for positive v, and that identity is established by cancellation against right inverses. The genuinely load-bearing soft spot is Proposition 4.6, which supports Theorem D and the interpretation in Remark 4.9. The reader correctly identified Lemma 4.4 as the algebraic engine, but the more precise problem is that the proof applies Lemma 4.4 without the monotonicity needed to make the approximating points coherent. This is a repairable proof gap rather than a refutation of the main claims. Because the reader's conditional verdict already calls for fixing the presentation and the relevant proofs, no change of verdict is needed: the concern reinforces the conditions rather than overturning the mathematics.","tokens_in":23497,"tokens_out":32566,"duration_ms":323663,"concrete_test":"Rewrite the proof of Proposition 4.6 with an explicit monotone cofinal construction: enumerate Gamma_S = {h_1, h_2, ...}, choose g_{k+1} by Lemma 4.4 for the finite set {h_1, ..., h_{k+1}, g_k}. Then verify: (a) gamma(S)g_k is contained in gamma(S)g_{k+1} and each h_i lies in gamma(S)g_k for k >= i; (b) for every subnet limit y, the displayed equality y_{gamma(s)h} = s · y_h holds for all s and h, using that both h and gamma(s)h are eventually in the chosen ideals. If the repaired proof goes through, Proposition 4.6 stands; if some translate of an outside coordinate still forces incoherence, Theorem D needs a new argument or a counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 4.6, the proof of the factor-map property of the extension functor fixes x in X^{Gamma_S}, chooses a sequence (g_k) via Lemma 4.4 with h in gamma(S)g_k for each h eventually, and defines y_k = (y^k_h) by choosing arbitrary preimages of x_h for h outside gamma(S)g_k. It then asserts y_k belongs to Y^{Gamma_S}. This is not justified: for h outside gamma(S)g_k, the element s · y^k_h need not equal y^k_{gamma(s)h}, since gamma(s)h may lie in gamma(S)g_k while h does not. The later subnet computation only proves coherence of the limit under the extra assumption that for every h, h lies in gamma(S)g_{k(lambda)} eventually. That requires the right ideals gamma(S)g_k to be increasing and cofinal; the proof only states cofinality. The gap is repairable by choosing g_{k+1} <=_S g_k and g_k <=_S h_i for the first k enumerated elements of Gamma_S, but as printed Proposition 4.6, and hence Theorem D, is not fully proved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of natural extensions for continuous actions of countable embeddable monoids. The natural extension is defined by a universal property (Definition A) and is shown, in Proposition 2.4, to be concretely realizable as the space X^G = {x ∈ X^G : s·x_h = x_{η(s)h} for all s,h} whenever that space projects onto X. The central characterization is Theorem B: for an embeddable monoid S, a receiving S-group G has the property that every surjective continuous S-action, equivalently every surjective S-subshift, is G-extensible if and only if G is a realization of the free S-group. The paper also proves a joint universality property for the natural extension over the free S-group (Theorem C), a characterization of left reversibility through factor maps of compact extensions (Theorem D), and, in the left reversible bicancellative case, equality of amenable topological entropy and lifting of topological ergodicity, transitivity, and minimality to the natural extension (Propositions 5.3 and 5.4).","tokens_in":23676,"tokens_out":31880,"duration_ms":294779,"significance":"If the results are correct, this gives a clean and quite general framework for natural extensions beyond N^d-actions. Theorem B is a sharp, falsifiable statement: both the receiving group and the embedding matter, and in the residually finite case the characterization reduces to finite subshifts. The paper has several strengths: the universal-property definition is independent of the concrete construction, the proofs of Proposition 2.4 and Proposition 3.6 are detailed and the main algebraic steps check out, and the reliance on standard tools (Ore–Dubreil, Følner sequences, Mal'cev's embeddability theorem) is explicit and appropriate. I found no circularity in the main line of argument. The main issue identified below is a genuine but local gap in the proof of Proposition 4.6, which affects Theorem D as printed; it appears repairable without changing the paper's scope.","major_comments":[{"comment":"The proof does not justify that the subnet limit y belongs to Y_{ΓS}. The sequence (g_k) is chosen only to be cofinal in the sense that for every h ∈ ΓS there is some k with h ∈ γ(S)g_k. However, the displayed computation y^{k(λ)}_{γ(s)h} = (s s_{k(λ)})·y^{k(λ)}_{g_{k(λ)}} = s·y^{k(λ)}_h is valid only when h ∈ γ(S)g_{k(λ)}; for indices with h outside γ(S)g_{k(λ)} the coordinates y^{k(λ)}_h were chosen arbitrarily, so no coherence is forced. Cofinality alone does not imply that h ∈ γ(S)g_{k(λ)} eventually along the monotone final subnet, and therefore the limit y may fail the defining relation of Y_{ΓS}. This affects the surjectivity claim for φ_{ΓS}, the 'in particular' part of Proposition 4.6, and consequently Theorem D. The gap is local and repairable: one should choose the sequence additionally with g_{k+1} ≤_S g_k, so that the right ideals γ(S)g_k are increasing as well as cofinal; then for each h one has h ∈ γ(S)g_{k(λ)} eventually, and the displayed computation does establish y ∈ Y_{ΓS}.","section":"§4, Proposition 4.6"}],"minor_comments":[{"comment":"The equality 'ϕ′ ◦ π = τ' is a typo in the uniqueness argument; the arrow condition is π ◦ ϕ′ = τ.","section":"§3.3, Theorem C proof"},{"comment":"The direction of θ in the definition of arrows is easy to misread. For an arrow (G′, X̂′) → (G, X̂), θ is a morphism of S-groups from G to G′ and φ : Y′ → Y; writing the equivariance as φ(θ(g)·y′) = g·φ(y′) and displaying the composition rule explicitly would remove the apparent inconsistency.","section":"§3.3, category Ext_α"},{"comment":"The sentence justifying that {n : h ≤_S g_n} is finite appears to derive the conclusion from the opposite inequality. The conclusion is true, but it needs the missing argument that if h ≤_S g_n for infinitely many n, then h is ≤_S-equivalent to all sufficiently large g_n, which forces the connecting semigroup elements s_n to be units and contradicts the strict choice g_{n+1} <_S g_n.","section":"Example 4.8"},{"comment":"The reference to 'Proposition 4.4' should be to Lemma 4.4.","section":"Example 4.8"},{"comment":"The expression '∅ ≠ s s^{-1}U ∩ s^{-1}V' is confusing; the argument only needs U ∩ s^{-1}V ≠ ∅, which is exactly what topological ergodicity provides.","section":"§5.2, Proposition 5.4(i)"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Proposition 4.6 is valid and is the main reason for the major revision recommendation. The gap is local and repairable, so I do not regard it as grounds for rejection. The possible issue in Example 4.8 appears to be an omitted argument rather than a counterexample to the claimed conclusion. The paper is within the scope of the journal and, once the Proposition 4.6 proof is corrected and the presentation slips are fixed, would be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real contribution. The paper gives the first general treatment of natural extensions for countable embeddable semigroup actions, and the main new results—Theorem B (free S-group is the unique receiving pair that always admits extensions), Theorem C (terminal object), Theorem D (left reversibility characterized by factor maps)—are genuinely new, not in the cited literature. The symbolic and algebraic arguments are careful; the free S-group construction and the transfer to subshifts are sound.\n\nThe strongest part is the free S-group characterization. The proof of Proposition 3.6 is detailed and checks out. I also like the concrete examples (BS(1,2), BS(2,3) non-Hopfian) showing that group choice and embedding both matter.\n\nSoft spots:\n\n1. Proposition 4.6 has a real gap as printed. The proof picks a cofinal sequence (g_k) and defines y^k by choosing arbitrary preimages outside γ(S)g_k. But the coherence computation for the subnet limit assumes that for each h, both h and γ(s)h lie in γ(S)g_{k(λ)}—not just that each h is covered eventually. The proof never ensures the right ideals are increasing, so the equality y^k_{γ(s)h} = (s s_k)·y^k_{g_k} is unjustified for those λ where γ(s)h falls outside γ(S)g_k. The fix is straightforward: choose g_{k+1} ≤_S g_k with g_k ≤_S h_i for the first k enumerated elements, which Lemma 4.4's downward directedness allows. Then for each h, h ∈ γ(S)g_k eventually, and the subnet computation goes through. But as printed, Theorem D's first direction is not fully proved. This is repairable, not fatal.\n\n2. Minor: the definition of arrows in Extα is confusing but actually consistent—for an arrow (G′,X̂′)→(G,X̂), θ maps G→G′, contrary to what one might first guess. The proof of Theorem C matches this. So I'd call it a presentation issue, not an error.\n\n3. Example 4.8 cites \"Proposition 4.4\" for the cofinal sequence; should be Lemma 4.4. Also the expression \"ss−1U\" in Proposition 5.4(i) is garbled, though the intended inequality is clear given surjectivity.\n\nThe cited Lemma 4.4 is the engine for several later results. It is from Hilgert–Neeb and standard; I don't see a problem with relying on it. The paper cites the sparse prior work honestly, including the folklore status of some results.\n\nWho this is for: people working on semigroup actions, symbolic dynamics over monoids, or topological dynamics of non-invertible maps. It deserves a serious referee. Send it to review; the referee will ask for the Proposition 4.6 fix and some cleaning up, but the core is solid.","headline":"A genuinely new and mostly sound theory of natural extensions for semigroup actions, with one repairable gap in Proposition 4.6 that should be fixed before publication.","tokens_in":24223,"tokens_out":6820,"would_cite":true,"duration_ms":56735,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M30","37B02","20M05","37B10","20F05","20M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the free group on an embeddable semigroup is the unique receiving group that always converts a surjective semigroup action into an invertible group action.","keywords":["countable semigroup","semigroup action","natural extension","free S-group","left reversible semigroup","topological entropy","topological transitivity","subshift"],"falsifier":"Take the subshift $X = \\{ x \\in \\Gamma^S : x(s) = \\gamma(s)x(1_S) \\}$ constructed in Proposition 3.8 for a proper quotient G of the free S-group Γ, and compute whether its G-extension space $X_G$ is empty. The theorem predicts it is empty for every nontrivial element of the kernel of the quotient map Γ → G; one nonempty $X_G$ would disprove Theorem B.","tokens_in":120,"feed_emoji":"🔄","tokens_out":11110,"duration_ms":212048,"temperature":0.7,"pith_summary":"This paper develops a theory of natural extensions for continuous actions of countable embeddable semigroups: given a semigroup S embedded in a group G, an S-action extends to G by recording, at each group element, the future of the orbit. The central result, Theorem B, says that a receiving group G extends every surjective continuous S-action—equivalently, every surjective S-subshift—if and only if G is a realization of the free S-group, the group generated by S with no relations beyond those already in S. This shows that both the choice of the group and the choice of the embedding are essential, and that a surjective action can fail to have any invertible extension to a particular generating group precisely when the semigroup is not left reversible. In the left-reversible case the paper proves that the natural extension is the smallest compact extension, and in the amenable case topological entropy, ergodicity, transitivity, and minimality lift from S to the group action.","feed_headline":"Only the free S-group always extends semigroup actions","feed_subtitle":"Every surjective action of an embeddable semigroup extends to its free group; other receiving groups fail on some subshift.","key_machinery":"The central object is the free S-group (Γ,γ): the group presented by the generators and relations of S, existing for every semigroup and characterized as the initial object among S-groups. The concrete construction is the space $X_G = \\{ (x_h)_{h\\in G} \\in X^G : s\\cdot x_h = x_{\\eta(s)h} \\text{ for all } s\\in S,\\, h\\in G \\}$, equipped with the shift action of G and the projection π(x)=x_{1_G}; Proposition 2.4 identifies X_G with the natural G-extension exactly when π is surjective, and Proposition 2.10 rewrites it as a G-subshift by reading the same forbidden patterns over G instead of S. The proof of Theorem B separates into a feasibility direction, in which every surjective S-action is shown to extend to the free S-group by lifting through the free semigroup on a presentation, and an infeasibility direction, in which any proper quotient G of Γ is shown to admit a surjective S-subshift with X_G empty. For Theorem D the engine is the cited Lemma 4.4, which equates the property that G is the group of right fractions of S with downward directedness of the pre-order $g \\leq_S h \\iff hg^{-1} \\in \\eta(S)$ and with thickness of η(S) in G; this provides the translates of finite sets into η(S) that carry the compact-extension factor maps and the dynamical lifting arguments.","core_discovery":"The load-bearing claim is Theorem B: for an embeddable monoid S and a realization (Γ,γ) of the free S-group, a receiving S-group G has the property that every surjective continuous S-action (equivalently, every surjective S-subshift) is G-extensible if and only if G is a realization of the free S-group. When Γ is residually finite, the characterization reduces to finite surjective S-subshifts: any proper quotient of Γ is betrayed by a finite symbolic subshift whose G-extension space is empty. The paper also establishes Theorem C, that the natural Γ-extension is the terminal object in the category of all invertible extensions, so it encodes every other extension; and Theorem D, that the classical demand that every compact G-extension factor onto the natural extension holds exactly when S is left reversible and G is its group of right fractions. For left reversible bicancellative monoids, the natural extension to the group of right fractions preserves amenable topological entropy and lifts topological ergodicity, transitivity, and minimality.","pith_inferences":["The paper leaves implicit that Theorem B supplies a dynamical certificate of non-universality: to show a proposed receiving pair (G,η) is not the free S-group, it suffices to exhibit any surjective S-subshift whose G-extension space is empty, a check that may be easier than comparing algebraic presentations.","In the non-left-reversible case, natural extensions are only unique relative to a fixed receiving group; different embeddings of the same semigroup into the same non-Hopfian group can therefore produce genuinely different invertible envelopes of the same dynamics, and the paper's examples suggest this is a real phenomenon rather than a pathology.","The residual-finiteness criterion suggests a computational route: for semigroups whose free S-group is residually finite, verifying universality of a receiving group can be reduced to testing finite surjective subshifts, which is a finite-state search in practice.","The entropy equality raises the question, not addressed here, of whether finer invariants—measure-theoretic entropy, periodic orbit data, or the maximal equicontinuous factor—also lift through the natural extension in the amenable left-reversible case."],"forward_implications":["Every surjective continuous action of an embeddable monoid extends to the free S-group, while no other receiving S-group can play this role for all actions simultaneously.","Failure of extensibility to a given generating group is an algebraic obstruction: the group must be a proper quotient of the free S-group, and when the free S-group is residually finite a finite symbolic subshift certifies the failure.","For a left reversible bicancellative semigroup the receiving group is unique up to isomorphism—the group of right fractions—and the natural extension is the smallest compact extension, with factor maps lifting to it.","Amenable topological entropy is preserved by the natural extension: $h(X,S)$ equals $h(X_{\\Gamma_S}, \\Gamma_S)$, so entropy can be computed in the invertible group setting.","Topological ergodicity, topological transitivity, and minimality all lift from the semigroup action to the group action on its natural extension."],"supporting_citations":[{"why":"Supplies the free S-group (Γ,γ) whose universal property is the algebraic core of the extensibility characterization.","marker":"[7, Construction 12.3]"},{"why":"Gives the criterion that an embeddable semigroup is exactly one whose free S-group embedding γ is injective, framing the whole problem.","marker":"[7, Theorem 12.4]"},{"why":"Provides Lemma 4.4, the equivalence among being the group of right fractions, downward directedness of the pre-order, and thickness of η(S); this lemma drives Theorem D and the entropy and dynamical lifting results.","marker":"[13, §3.6]"},{"why":"Proves that left reversibility plus bicancellativity implies embeddability, the classical entrance to the left-reversible case.","marker":"[18, Theorem 1]"},{"why":"Establishes the converse: left reversibility is necessary and sufficient for the existence of the group of right fractions, which Theorem D uses to isolate the case where the natural extension is smallest.","marker":"[10]"},{"why":"Rephrased as Lemma 4.2, it shows the group of right fractions of a subsemigroup behaves well and yields uniqueness of the receiving group for left reversible semigroups.","marker":"[8, Lemma 4]"},{"why":"Supplies the definition and existence of amenable topological entropy for Følner sequences, which Proposition 5.3 compares between S and ΓS.","marker":"[5]"},{"why":"Characterizes left amenability of a cancellative semigroup by the existence of a left Følner sequence, the framework used in Proposition 5.1 and the entropy section.","marker":"[17, Corollary 3.6]"}],"fun_headline_variants":["Only free S-group guarantees extension of every semigroup action","Left reversibility is key to natural extensions of semigroup actions","Free group is the unique universal extension for embeddable semigroups","Natural extension fails outside left reversible semigroups","Terminal natural extension exists only for left reversible actions"],"cache_read_input_tokens":26368,"weakest_assumption_plain":"The left-reversible half of the paper rests on a cited algebraic equivalence: for a left reversible semigroup, a receiving group being the group of right fractions is the same as being able to move any finite set of group elements into the semigroup by one right translation; if that equivalence fails, the compact-extension characterization and the entropy and dynamical lifting theorems no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Only free S-group guarantees extension of every semigroup action","Left reversibility is key to natural extensions of semigroup actions","Free group is the unique universal extension for embeddable semigroups","Natural extension fails outside left reversible semigroups","Terminal natural extension exists only for left reversible actions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000287,"raw_usage":{"total_tokens":1679,"prompt_tokens":930,"completion_tokens":749,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":667}},"tokens_in":546,"tokens_out":749,"duration_ms":7407,"temperature":1.0,"reasoning_tokens":667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:16:11.317269+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the subshift $X = \\{ x \\in \\Gamma^S : x(s) = \\gamma(s)x(1_S) \\}$ constructed in Proposition 3.8 for a proper quotient G of the free S-group Γ, and compute whether its G-extension space $X_G$ is empty. The theorem predicts it is empty for every nontrivial element of the kernel of the quotient map Γ → G; one nonempty $X_G$ would disprove Theorem B.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the converse: left reversibility is necessary and sufficient for the existence of the group of right fractions, which Theorem D uses to isolate the case where the natural extension is smallest."},{"cited_title":"Ceccherini-Silberstein, M","cited_arxiv_id":null,"evidence_quote":"Supplies the definition and existence of amenable topological entropy for Følner sequences, which Proposition 5.3 compares between S and ΓS."}],"review_version":1}