Recurrence in a dynamical system over adequate partial semigroups

Md Moid Shaikh; Shivam Kumar; Sourav Kanti Patra

REVIEW 3 major objections 4 minor 27 references

Recurrence in a dynamical system over adequate partial semigroups

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A genuinely new but currently unpolished extension of quasi-central set theory to adequate partial semigroups, with a fixable gap in the main proof. read the letter →

arxiv 2607.26548 v1 pith:P3LEIYY2 submitted 2026-07-29 math.CO

Md Moid Shaikh , Shivam Kumar , Sourav Kanti Patra This is my paper · ORCID

classification math.CO MSC 05D1022A1554D35

keywords AlgebraintheStone-ČechcompactificationAdequatepartialsemigroupsDynamicalsystemQuasi-centralsetUniformrecurrenceProximalityPiecewisesyndetic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $s$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This 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.

The 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.

There 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.

I 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.

Who 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.

My 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[§4, Theorem 4.4(⇒)] 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.
[§5, Lemma 5.9] 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).
[§3, Lemma 3.7] 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.

minor comments (4)

[Throughout] 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).
[Theorem 4.4] The proof contains empty citations 'so by ,' and 'By ,'; the intended references to Lemma 4.2 should be inserted.
[§5, Lemma 5.9] 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.
[References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 5.10 is a direct application of internal Theorem 4.4 to the filter dual to piecewise syndetic sets; self-citations are not load-bearing and no prediction reduces to an input by construction.

full rationale

The derivation chain for the main theorem is: Definition 5.5 defines quasi-central sets as members of idempotents in cl K(δS); Lemma 5.9 identifies the filter K = {B : the complement of B is not piecewise syndetic} with cl K(δS), using the definition of piecewise syndetic and standard closure properties via external reference [18]; Theorem 4.4 gives a general dynamical characterization of members of idempotents in Kbar for any filter K whose closure is a compact subsemigroup; and Theorem 5.10 instantiates Theorem 4.4 with this K, observing that joint K-recurrence with L(K) equal to the piecewise syndetic sets is exactly joint intermittent uniform recurrence. The forward direction of Theorem 4.4 constructs a shift system whose return set is the given set A, but this is a witness construction from a set A already known to belong to an idempotent; it is not fitting a parameter, nor is quasi-centrality defined in terms of the dynamical condition. The proof does contain a genuine correctness gap: T_s(f) = f composed with right multiplication by s may not be well-defined because the partial semigroup operation is not total, and Lemma 5.9's one-line proof is compressed. These are correctness and completeness defects, not circular reductions. The self-citations to [6] and [26] occur mainly in Section 3 for preliminary definitions and lemmas; Theorem 5.10 does not depend on those self-citations, relying instead on internal Theorem 4.4 and external results [16] and [18]. No step exhibits an equation equal to its input by definition, and no fitted value is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The paper introduces no new exotic entities; it re-purposes known ultrafilter and dynamical-system machinery. The main unproved inputs are standard theorems and the filter Lemma 5.9, whose proof is not coherently written.

assumptions (4)

standard math Ellis's theorem: every compact Hausdorff right topological semigroup has a smallest ideal and contains idempotents.
Used to guarantee idempotents in δS and in subsemigroups (Section 2, Theorem 2.14 consequence, and Lemma 4.2).
domain assumption δS is a compact Hausdorff right topological semigroup for every adequate partial semigroup S (Theorem 2.14).
Invoked throughout Sections 3-5 as the base semigroup for ideals and idempotents; cited to Hindman-McCutcheon [15, Theorem 2.10].
domain assumption Prior results cited as black boxes: [16, Corollary 3.4] (J(S) compact ideal), [16, Theorem 3.6] (Central Sets Theorem for adequate partial semigroups), [6, Lemma 2.18] (uniform recurrence equivalence), and [20, Theorem 3.3] (joint recurrence characterization).
These external theorems carry much of the derivation weight in Sections 3-5; they are taken as given.
ad hoc to paper The family K of sets whose complement is not piecewise syndetic is a filter, and its closure in δS is cl K(δS), a compact subsemigroup (Lemma 5.9).
This is the bridge for Theorem 5.10. The proof in the paper is a corrupted one-liner, so this premise is not actually established in the text.

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Pith. "Pith review of Recurrence in a dynamical system over adequate partial semigroups." pith.science (2026). https://pith.science/paper/P3LEIYY2

@misc{pith2026260726548,
  author       = {Pith},
  title        = {Pith review of: Recurrence in a dynamical system over adequate partial semigroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3LEIYY2}},
  note         = {Machine review of arXiv:2607.26548}
}

abstract

Using tools from topological dynamics, H.~Furstenberg introduced the notion of \emph{central sets} and established the celebrated Central Sets Theorem. The sets which satisfy the conclusion of the Central Sets Theorem are called $C$-sets. Hindman, Maleki, and Strauss first brought the concept of an important type of $C$-sets called the quasi-central sets which are not central sets. In 2017, A. Ghosh gave the combinatorial treatment of $C$-sets in commutative adequate partial semigroups, where $C$-sets are the sets which satisfy the conclusion of Central Sets Theorem for commutative adequate partial semigroups. In this work, we discuss the Quasi-central sets algebraically and dynamically for commutative adequate partial semigroups. We give dynamical characterization of members of idempotent ultrafilters for commutative adequate partial semigroups, also we study the minimal dynamical systems for an adequate partial semigroup.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Hindman and D

N. Hindman and D. Strauss,Algebra in the Stone- ˇCech compactification - theory and application, W. de Gruyter and Co., Berlin, 2012

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Bergelson, A

V. Bergelson, A. Blass, N. Hindman,Partition theorems for spaces of variable words, Proc. London Math. Soc. 68 (1994) 449–476

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Bergelson and N

V. Bergelson and N. Hindman,Nonmetrizable topological dynamics and Ramsey the- ory, Trans. Amer. Math. Soc. 320 (1990), 293-320

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S. Burns, N. Hindman,Quasi-central sets and their dynamical characterization, Topology Proc. 31 (2007), 445-455

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D. De, N. Hindman, and D. Strauss,A new and stronger Central Sets Theorem, Fundamenta Mathematicae 199 (2008), 155–175

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Dynamical characterization of central sets in adequate partial semigroups

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Ellis,Lectures on Topological Dynamics, Benjamin, New York, 1969

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