{"id":"25029591-8d9d-4f5e-9130-671d762408df","arxiv_id":"2607.26548","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quasi-central sets in commutative adequate partial semigroups are exactly the return-time sets of jointly intermittently uniformly recurrent pairs in a dynamical system.","lead":"This mathematics paper studies pattern-rich central sets in adequate partial semigroups, where combining two elements is sometimes allowed. It proves new dynamical descriptions for these sets, extending a long line of results in Ramsey theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.4's forward direction builds a non-total shift action on {0,1}^{S∪{e}}; this gap is inherited by Theorem 5.10, though a repair via Lemma 4.3 is available.","rationale":"I read Theorem 5.10 as the central claim. It depends on Theorem 4.4 and Lemma 5.9. The reader correctly flags Lemma 5.9 as underproved; however, its assertion is true: K is a filter and \\bar K=cl K(δS) follows from Definition 5.4(1) once the algebraic criterion for piecewise syndeticity is read as \\bar A∩K(δS)≠∅. Thus I do not expect that concern to be fatal. The more serious defect is the model system in Theorem 4.4: composing with right multiplication by s is only a partial operation on {0,1}^{S∪{e}}, so the construction given does not produce a dynamical system in the sense of Definition 2.15. Since both directions of Theorem 5.10 invoke Theorem 4.4, the current text does not fully support the quasi-central characterization. I nevertheless believe the theorem is true: the shift action of Lemma 4.3 on the adjoined-identity partial semigroup supplies a total action and yields exactly the required set equality, so this is a substantial but repairable proof gap rather than a counterexample. A focused rewrite of Theorem 4.4(⇒) along those lines would settle the issue. No false conclusion is apparent, so conditional acceptance remains the appropriate disposition.","tokens_in":14277,"tokens_out":21967,"duration_ms":180258,"concrete_test":"Rewrite the proof of Theorem 4.4(⇒) using Lemma 4.3 with index partial semigroup R=S∪{e}: define T_s(f)(r)=f(r·s) if s∈ϕ(r), and 0 otherwise. Verify (i) T_s is continuous and T_s∘T_t=T_{st} whenever t∈ϕ(s); (ii) for x=1_A, y=T_r(x), and U={w∈X:w(e)=1}, one has y∈U and A={s∈S:T_s(x)∈U}; (iii) T_r(x)=y=T_r(y), so by Lemma 4.2 the pair is jointly K-recurrent. If all three checks pass, the proof gap in Theorem 4.4 is closed; any failure would leave Theorem 5.10 without a valid forward direction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.10 is proved solely by applying Theorem 4.4 with K={B⊆S:S\\B is not piecewise syndetic}. The load-bearing step in Theorem 4.4 is the forward direction: for A∈r with r an idempotent in \\bar K, one must produce a dynamical system (X,⟨T_s⟩), points x,y, and a neighbourhood U with (x,y) jointly K-recurrent and A={s∈S:T_s(x)∈U}. The submitted proof defines X={0,1}^R with R=S∪{e} and T_s(f)=f∘ρ_s, where ρ_s is right multiplication by s. Since S is only a partial semigroup, ρ_s is not a total function R→R when some r·s is undefined, so T_s is not a well-defined self-map of X and the cited [18, Theorem 19.14] does not apply. This is not a cosmetic issue: without a total action the expressions T_r(x) and A={s∈S:T_s(x)∈U} have no meaning. Consequently, as written, Theorem 4.4—and hence Theorem 5.10—lacks a valid proof. Lemma 5.9's one-line proof is a separate under-specification; its claim can, however, be verified directly from Definition 5.4(1).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies adequate partial semigroups and aims to give algebraic and dynamical characterizations of quasi-central sets in the commutative case. After collecting background on the Stone-Čech compactification and topological dynamics for partial semigroups, Section 3 develops minimal dynamical systems and characterizes uniformly recurrent points via minimal left ideals of δS. Section 4 states a dynamical characterization of members of idempotent ultrafilters for an adequate partial semigroup, and Section 5 defines quasi-central sets as members of idempotents in cl K(δS), asserts that such sets satisfy the Central Sets Theorem, and concludes with the main theorem (Theorem 5.10): A is quasi-central iff A is the return set of a jointly intermittently uniformly recurrent pair in some dynamical system. The proof of Theorem 5.10 is a direct application of Theorem 4.4 to the filter K of sets whose complement is not piecewise syndetic.","tokens_in":14518,"tokens_out":7685,"duration_ms":67544,"significance":"If Theorem 5.10 is correct, it gives a complete topological-dynamical characterization of quasi-central sets in commutative adequate partial semigroups, extending the Burns–Hindman characterization for semigroups and complementing Ghosh's combinatorial treatment of C-sets in partial semigroups. Section 3 also extends the Hindman–Strauss–Zamboni minimal-systems results to this setting. The overall strategy is natural and the main theorem is plausible. The paper does not rely on fitted parameters, and the main proof reduces to a known template once the partial-semigroup action issues are fixed. However, as written, several load-bearing proofs contain genuine gaps, so the manuscript is not yet acceptable.","major_comments":[{"comment":"The forward direction defines X={0,1}^{S∪{e}} and T_s(f)=f∘ρ_s, where ρ_s is right multiplication by s. When S is only a partial semigroup, ρ_s is a partial function: for r∈S with r·s undefined, ρ_s(r) is undefined, so T_s is not a well-defined self-map of X. The citation to [18, Theorem 19.14] therefore does not apply, and the expressions T_r(x), T_p(x), and A={s∈S:T_s(x)∈U} have no meaning. Since Theorem 5.10 is proved by applying Theorem 4.4, this gap is load-bearing for the main characterization. A repair is available: work on R=S∪{e}, use the total action of Lemma 4.3 on {0,1}^R, and take U={w:w(e)=1}; the authors should rewrite the proof accordingly and verify the required ultrafilter identities in δR.","section":"§4, Theorem 4.4(⇒)"},{"comment":"The proof is a single garbled line, 'By the construction of K and Definition 5.4(2), we have K= TK(δS)', followed by unexplained citations. The filter property is not demonstrated, and the citation should be to Definition 5.4(1), where piecewise syndetic is defined, not Definition 5.4(2). Because Lemma 5.9 is the bridge that allows Theorem 4.4 to be applied with K={A:S\\A is not piecewise syndetic}, a complete proof is required. The statement is verifiable directly: A∈K iff S\\A∩K(δS)=∅ gives closure under finite intersections and upward closure, and \\bar K={p∈βS: every member of p is piecewise syndetic}=cl K(δS).","section":"§5, Lemma 5.9"},{"comment":"In the proof of (4)⇒(3), the ultrafilter p appearing in 'δSrp' and 'T_qp(x)' is never introduced; the argument appears to need p to be an idempotent in the minimal left ideal L, such as the q from the hypothesis or the idempotent supplied by Lemma 3.6. The assertion δSrp=L also needs a justification that δSrp is a left ideal contained in L. In (6)⇒(3), the group identity e of L∩δSr is invoked without specifying the minimal left ideal L containing q, and the step T_e(T_q(y))=T_eq(y) relies on the ultrafilter composition rule for δS, which should be stated explicitly. These gaps affect the minimal-systems results of Section 3, though they do not directly enter the proof of Theorem 5.10.","section":"§3, Lemma 3.7"}],"minor_comments":[{"comment":"There are numerous typos and disfluencies: 'Hausdroff' for Hausdorff, 'define define', 'quasi-central neare' in Theorem 5.10, 'setm' in its proof, 'and and' in Corollary 3.13, and 'x∈δS' versus 'x∈X' in Theorem 3.11(2).","section":"Throughout"},{"comment":"The proof contains empty citations 'so by ,' and 'By ,'; the intended references to Lemma 4.2 should be inserted.","section":"Theorem 4.4"},{"comment":"The proof cites 'Definition 5.4(2)' for piecewise syndeticity, but piecewise syndetic is Definition 5.4(1); Definition 5.4(2) defines J-sets.","section":"§5, Lemma 5.9"},{"comment":"The reference list is serviceable but includes an arXiv preprint [6] that should be updated if a published version exists, and some citation formats for conference/arXiv items could be completed.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is plausible and the main defects are local and repairable: Theorem 4.4(⇒) can be fixed with the total action of Lemma 4.3, and Lemma 5.9 has a direct proof. I am recommending major revision rather than rejection because the repair path is evident and the paper's scope and main claim are appropriate for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does something new: it defines quasi-central sets for adequate partial semigroups and gives them a dynamical characterization, alongside minimal-system results and a characterization of idempotent ultrafilters. The results are plausible and extend the known semigroup theory in a natural way. I believe the core theorems are likely true, but the manuscript is not ready to appear as is.\n\nThe main problem is in Theorem 4.4, forward direction. The proof builds an action on {0,1}^{S∪{e}} using right multiplication maps ρ_s, which are not total functions when S is a partial semigroup. So the T_s are not well-defined self-maps, and the cited theorem from [18] does not apply. This is a load-bearing issue because Theorem 5.10 leans directly on Theorem 4.4. The good news is that Lemma 4.3 already gives a correct partial-semigroup action, so the proof is very likely repairable. This is a substantive gap but not a sign the theorems are false.\n\nThere are other soft spots that are less severe. Lemma 3.7 uses an undefined symbol r and a group identity step that needs justification. Lemma 5.9 has a one-line proof that is essentially garbled; the claim itself looks correct and checkable directly from Definition 5.4, but the written proof is not informative. There are also typos and a couple of references that do not line up with the actual argument.\n\nI want to give credit where it is due. The paper follows the established Hindman-Strauss-Zamboni and Johnson templates carefully, and the new definitions make sense. The authors are not hand-waving about the nature of the contribution; they are extending known machinery to a class where it hasn't been done before. The reliance on earlier work, including their own, is normal for this area and not circular.\n\nWho is this for? Specialists in Ramsey theory and ultrafilter dynamics on partial semigroups. If you work in that area, you will want to know these results and will probably use them once the proof issues are fixed.\n\nMy recommendation: engage with it, but only after a major revision. I would not cite the current arXiv version in my own work; I would wait for a corrected version. For peer review, though, this paper deserves a serious referee. The flaws are fixable and the results are new and worth the community's attention.","headline":"A genuinely new but currently unpolished extension of quasi-central set theory to adequate partial semigroups, with a fixable gap in the main proof.","tokens_in":15095,"tokens_out":3973,"would_cite":false,"duration_ms":35492,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","22A15","54D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For commutative adequate partial semigroups, the paper proves that a set is quasi-central if and only if it is the visit set of a jointly intermittently uniformly recurrent pair in a dynamical system.","keywords":["Algebra in the Stone-Čech compactification","Adequate partial semigroups","Dynamical system","Quasi-central set","Uniform recurrence","Proximality","Piecewise syndetic"],"falsifier":"Find two subsets $A$ and $B$ of a commutative adequate partial semigroup such that neither $S\\setminus A$ nor $S\\setminus B$ is piecewise syndetic but $(S\\setminus A)\\cup(S\\setminus B)$ is piecewise syndetic; that would violate the finite-intersection property of the filter in Lemma 5.9 and remove the foundation of Theorem 5.10. A complementary check is to supply a real derivation of the asserted identity $\\overline{K}=\\operatorname{cl}K(\\delta S)$, since the printed proof of Lemma 5.9 is only that single equality.","tokens_in":13997,"feed_emoji":"🔄","tokens_out":18760,"duration_ms":142912,"temperature":0.7,"pith_summary":"The paper studies quasi-central sets in commutative adequate partial semigroups, where a partial product is defined only for compatible pairs and adequacy means every finite collection of elements has a common element that can multiply all of them. Quasi-central sets are defined algebraically by membership in an idempotent ultrafilter in the closure of the smallest ideal of the Stone-Čech compactification, and they satisfy the Central Sets Theorem even when they are not central. The central result is a dynamical characterization: a set is quasi-central if and only if it is the set of times at which a point of a compact dynamical system enters a fixed neighbourhood of another point while the pair is jointly intermittently uniformly recurrent. The paper also gives a general characterization of members of idempotent ultrafilters by joint recurrence and develops the minimal-system theory for adequate partial semigroups. If right, this carries the classical Ramsey-theoretic connection between combinatorial size and topological recurrence from ordinary semigroups to the partial-semigroup setting.","feed_headline":"Dynamical recurrence characterizes quasi-central sets","feed_subtitle":"In commutative adequate partial semigroups, quasi-central sets are exactly the sets visited by a jointly recurrent pair.","key_machinery":"The load-bearing mechanism is the compact semigroup $\\delta S$ formed inside the Stone-Čech compactification of the adequate partial semigroup $S$: it is the intersection of the closures of all domains $\\phi(x)$, and it is where idempotent ultrafilters live. The quasi-central characterization is carried by the filter $K=\\{A\\subseteq S:S\\setminus A\\text{ is not piecewise syndetic}\\}$, whose closure $\\overline{K}=\\operatorname{cl}K(\\delta S)$ is claimed in Lemma 5.9 to be a compact subsemigroup of $\\delta S$, together with the general joint-recurrence theorem (Theorem 4.4) that converts membership in an idempotent of such a compact subsemigroup into a neighbourhood-visit set in a dynamical system. The forward direction uses the explicit shift system $X=\\{0,1\\}^R$ with $T_s(f)=f\\circ\\rho_s$, where $\\rho_s$ is right multiplication by $s$.","core_discovery":"Inside the Stone-Čech compactification $\\beta S$ of a discrete adequate partial semigroup $S$, the set $\\delta S=\\bigcap_{x\\in S}\\operatorname{cl}(\\phi(x))$ is a compact right topological semigroup; its smallest ideal $K(\\delta S)$ houses central sets, and the closure $\\operatorname{cl}K(\\delta S)$ houses quasi-central sets. The paper's main claim is Theorem 5.10: $A\\subseteq S$ is quasi-central exactly when there exists a dynamical system $(X,\\langle T_s\\rangle_{s\\in S})$, points $x,y\\in X$, and a neighbourhood $U$ of $y$ such that $(x,y)$ is jointly intermittently uniformly recurrent and $A=\\{s\\in S:T_s(x)\\in U\\}$. Intermittent uniform recurrence means that for every neighbourhood $U$ of $y$, the set of times $s$ with both $T_s(x)\\in U$ and $T_s(y)\\in U$ is piecewise syndetic. The proof goes through Theorem 4.4, which states the same equivalence for any filter $K$ whose closure $\\overline{K}$ is a compact subsemigroup of $\\delta S$: membership of $A$ in an idempotent of $\\overline{K}$ is equivalent to $A$ being the visit set of a jointly $K$-recurrent pair. The paper also shows that quasi-central sets satisfy the Central Sets Theorem for adequate partial semigroups and that in the canonical shift system the points of $K(\\delta S)$ are exactly the uniformly recurrent points, with left cancellability making the description of minimal left ideals complete.","pith_inferences":["Because Theorem 4.4 is stated for an arbitrary filter with compact-subsemigroup closure, the same proof scheme should give dynamical characterizations for other algebraically defined Ramsey classes in adequate partial semigroups, such as $J$-sets or $C$-sets, by substituting the appropriate dual filter.","The explicit shift system of Lemma 4.3 makes quasi-centrality testable in principle on finitely generated adequate partial semigroups: one could search for a pair whose joint visit times are piecewise syndetic, a check that the ultrafilter definition alone does not suggest.","If Lemma 5.9 turns out to require an extra hypothesis, the algebraic and dynamical definitions of quasi-centrality would split, and the natural repair would be to replace $\\operatorname{cl}K(\\delta S)$ by the closure of a provably ideal-closing filter, preserving Theorem 4.4 while changing which sets count as quasi-central."],"forward_implications":["Quasi-central sets in commutative adequate partial semigroups are partition regular, since one piece of any finite partition of a member of an idempotent ultrafilter must itself lie in that ultrafilter.","Every quasi-central set satisfies the conclusion of the Central Sets Theorem for adequate partial semigroups, so the dynamical description does not lose the combinatorial content.","Central sets are quasi-central but not vice versa; the dynamical characterization separates the two by requiring joint intermittent uniform recurrence rather than proximality to a uniformly recurrent point.","In the canonical system $(\\beta S,\\langle \\lambda_s\\rangle)$, uniform recurrence of a point is equivalent to lying in $K(\\delta S)$, and minimal subsystems are exactly the orbits of minimal left ideals, so algebraic minimality has a direct dynamical reading.","Any filter whose closure is a compact subsemigroup of $\\delta S$ produces a family of sets with the same visit-set dynamical description, so Theorem 4.4 applies to other algebraic notions of largeness."],"supporting_citations":[{"why":"supplies the Stone-Čech compactification algebra, including the filter–compact-subspace correspondence and the shift-system construction used in Theorem 4.4","marker":"[18]"},{"why":"gives the ordinary-semigroup quasi-central dynamical characterization that Theorem 5.10 transfers to adequate partial semigroups","marker":"[4]"},{"why":"defines dynamical systems over adequate partial semigroups and provides the minimal-left-ideal, proximality, and central-set lemmas reused in Sections 3–5","marker":"[6]"},{"why":"introduces quasi-central sets algebraically and supplies the syndetic and filter background behind Definitions 5.5 and 5.8","marker":"[14]"},{"why":"provides the J-set ideal $J(S)$ and the Central Sets Theorem for adequate partial semigroups used in Theorem 5.7","marker":"[16]"},{"why":"proves the jointly $K$-recurrent characterization of members of idempotent ultrafilters that Theorem 4.4 adapts","marker":"[20]"},{"why":"is the minimal-dynamical-systems paper whose $U(x)$-ideal results Section 3 generalizes","marker":"[19]"},{"why":"defines syndetic and piecewise syndetic size notions in partial semigroups, which Definition 5.8 and Lemma 5.9 use","marker":"[22]"}],"fun_headline_variants":["Joint recurrence pins down quasi-central sets","Adequate partial semigroups: quasi-central = jointly recurrent","Quasi-central sets are exactly joint recurrence sets","Dynamical proof: idempotents give quasi-central sets","Recurrence in adequate partial semigroups defines quasi-central"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim depends on the unproved filter property in Lemma 5.9 — that the sets whose complements are not piecewise syndetic form a filter whose closure is a compact subsemigroup; if that property fails, the dynamical characterization collapses.","fun_headline_variants_meta":{"raw":{"variants":["Joint recurrence pins down quasi-central sets","Adequate partial semigroups: quasi-central = jointly recurrent","Quasi-central sets are exactly joint recurrence sets","Dynamical proof: idempotents give quasi-central sets","Recurrence in adequate partial semigroups defines quasi-central"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00054,"raw_usage":{"total_tokens":2637,"prompt_tokens":1043,"completion_tokens":1594,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":1514}},"tokens_in":659,"tokens_out":1594,"duration_ms":9328,"temperature":1.0,"reasoning_tokens":1514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:25:32.110652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find two subsets $A$ and $B$ of a commutative adequate partial semigroup such that neither $S\\setminus A$ nor $S\\setminus B$ is piecewise syndetic but $(S\\setminus A)\\cup(S\\setminus B)$ is piecewise syndetic; that would violate the finite-intersection property of the filter in Lemma 5.9 and remove the foundation of Theorem 5.10. A complementary check is to supply a real derivation of the asserted identity $\\overline{K}=\\operatorname{cl}K(\\delta S)$, since the printed proof of Lemma 5.9 is only that single equality.","supporting_citations":[{"cited_title":"Burns, N","cited_arxiv_id":null,"evidence_quote":"gives the ordinary-semigroup quasi-central dynamical characterization that Theorem 5.10 transfers to adequate partial semigroups"},{"cited_title":"Dynamical characterization of central sets in adequate partial semigroups","cited_arxiv_id":"2406.16918","evidence_quote":"defines dynamical systems over adequate partial semigroups and provides the minimal-left-ideal, proximality, and central-set lemmas reused in Sections 3–5"},{"cited_title":"Hindman, A","cited_arxiv_id":null,"evidence_quote":"introduces quasi-central sets algebraically and supplies the syndetic and filter background behind Definitions 5.5 and 5.8"},{"cited_title":"Hindman and K","cited_arxiv_id":null,"evidence_quote":"provides the J-set ideal $J(S)$ and the Central Sets Theorem for adequate partial semigroups used in Theorem 5.7"},{"cited_title":"Hindman, D","cited_arxiv_id":null,"evidence_quote":"is the minimal-dynamical-systems paper whose $U(x)$-ideal results Section 3 generalizes"},{"cited_title":"McLeod,Some notions of size in partial semigroups, Topology Proc","cited_arxiv_id":null,"evidence_quote":"defines syndetic and piecewise syndetic size notions in partial semigroups, which Definition 5.8 and Lemma 5.9 use"}],"review_version":2}