Transitivity of some uniformities on fuzzy sets

Daniel Jard\'on; Iv\'an S\'anchez; Manuel Sanchis

REVIEW 1 major objections 5 minor 3 cited by

Transitivity of some uniformities on fuzzy sets

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For a continuous self-map of a uniform space, weak mixing is exactly equivalent to transitivity of the Zadeh extension on fuzzy sets under any of three natural uniformities.

desk verdict A clean, modest extension of the fuzzy-hyperspace transitivity theorem from metric to uniform spaces; worth reviewing, with typos and one citation-dependent step to check. read the letter →

arxiv 2411.17037 v1 pith:VPNDJW3I submitted 2024-11-26 math.GN

Daniel Jardón , Iván Sánchez , Manuel Sanchis This is my paper · ORCID

classification math.GN MSC 54E1554B2054H20

keywords FuzzysetsLevel-wiseuniformitySkorokhodSendographZadeh'sextensionTransitivityWeaklymixingUniformspaces

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

Uniform spaces are a common generalization of metric spaces, and fuzzy sets here are normal upper semicontinuous functions from the space into [0,1] with compact support. The paper studies the induced dynamics of a continuous map f through its Zadeh extension, which moves fuzzy sets by taking suprema along fibers. The main theorem says f is weakly mixing, meaning f×f is transitive, exactly when the Zadeh extension is transitive on the fuzzy-set space under the level-wise uniformity, the Skorokhod uniformity, or the sendograph uniformity. This matters because it reduces a question about a large function space to one property of the original map and extends a previously metric-only characterization to all uniform spaces.

What carries the argument

The central object is the Zadeh extension $\widehat{f}$, defined on a fuzzy set $u$ by $\widehat{f}(u)(x)=\sup\{u(z): z \in f^{-1}(x)\}$ when the preimage is nonempty and $0$ otherwise. It carries the argument because it is level-wise: $[\widehat{f}(u)]_\alpha = f(u_\alpha)$, so image fuzzy sets are determined by images of $\alpha$-cuts, which are compact sets. The three uniformities are built from the base uniformity $\mathcal{U}$: the level-wise uniformity $\mathcal{U}_\infty$ requires every $\alpha$-cut to stay close at once, the Skorokhod uniformity $\mathcal{U}_0$ allows a small reparameterization of the level index, and the sendograph uniformity $\mathcal{U}_S$ compares the sendographs, the parts of the endographs above the support, as compact subsets of $X \times [0,1]$. The proof also leans on the hyperspace uniformity $\mathcal{K}(\mathcal{U})$ on compact subsets, whose induced topology is the Vietoris topology.

What would settle it

Take the irrational rotation $f_\theta$ on the circle with $\theta$ irrational, which is transitive but not weakly mixing. The theorem predicts its Zadeh extension is not transitive on $(\mathcal{F}(S^1),\mathcal{U}_\infty)$, $(\mathcal{F}(S^1),\mathcal{U}_0)$, or $(\mathcal{F}(S^1),\mathcal{U}_S)$. Computing whether two fuzzy sets with separated level sets can be brought into the same prescribed neighbourhood by some iterate would settle the prediction; any single transitive example here would refute Theorem 3.7.

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Extended reading notes

Core claim

The central claim is Theorem 3.7. For a uniform space $(X,\mathcal{U})$ and a continuous map $f$, weak mixing of $f$, transitivity of the induced map on the compact hyperspace with the hyperspace uniformity, and transitivity of the Zadeh extension on $(\mathcal{F}(X),\mathcal{U}_\infty)$, $(\mathcal{F}(X),\mathcal{U}_0)$, and $(\mathcal{F}(X),\mathcal{U}_S)$ are all equivalent. The proof derives the hyperspace equivalence from a cited theorem, proves the hardest implication by using weak mixing of all orders to move finitely many level sets of one fuzzy set toward the level sets of a target, then assembles the images into a fuzzy set in the prescribed neighbourhood. The reverse implication reads characteristic functions of compact sets as fuzzy sets and reconstructs hyperspace transitivity from transitivity in the sendograph uniformity.

Load-bearing premise

The load-bearing premise is the imported result that weak mixing is equivalent to transitivity on the compact hyperspace and implies weak mixing of all orders for every topological space; if that result needs extra hypotheses, the equivalence chain could fail.

Editorial extensions

If this is right

Transitivity of the Zadeh extension under any one of the three fuzzy-set uniformities implies transitivity under all three, since weak mixing is a single property of the base map.
Weak mixing is a complete criterion for fuzzy-set transitivity, so no special structure beyond uniformity of the base space is required.
The equivalence extends the metric-space theorem to uniform spaces, carrying known examples and counterexamples from metric fuzzy dynamics into the non-metrizable setting.
The theorem identifies fuzzy-set transitivity with classical hyperspace transitivity, so methods and examples for compact hyperspaces transfer directly to the fuzzy setting.

Reading between the lines

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

A likely extension not stated in the paper: any uniformity on $\mathcal{F}(X)$ whose topology lies between the sendograph topology and the level-wise topology, and for which the Zadeh extension is continuous, should have the same equivalence; the proof uses only these topology inclusions.
The theorem predicts that a transitive but not weakly mixing map, such as an irrational rotation on the circle, has a Zadeh extension that is not transitive on any of the three uniformities; a direct verification of this prediction would provide a concrete boundary example that the paper does not carry out.
If compact support is dropped from the definition of $\mathcal{F}(X)$, the level-set approximation argument may fail because the $\alpha$-cuts would no longer be compact; constructing an example with noncompact support that breaks transitivity would clarify which hypothesis is essential.

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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 is a workmanlike extension of a known metric-space theorem to uniform spaces. The main result, Theorem 3.7, characterizes transitivity of the Zadeh extension under the level-wise, Skorokhod, and sendograph uniformities in terms of weak mixing of the base map. That characterization is new; the metric version was in the authors' earlier paper [6], and the uniform version isn't in the cited literature. The continuity theorems (2.12, 2.16, 2.19) are also new. So the paper earns its place as a genuine, if modest, contribution.

The proof follows the Banks-Peris template and adapts the authors' own prior machinery. The logic chain i)–v) is coherent, and I found no fatal error. The self-citations are transparent and not circular—they use earlier results as tools, not as a way to sneak in the conclusion. No free parameters, no invented entities.

The soft spots are mostly presentation. There are several typos: an undefined entourage W in Eq. (3.4) (from context it should be V), a mismatched K[V] in Eq. (3.5) that should be K[U], and a misprinted interval in Eq. (3.1) where beta ranges over the indices, not over the sets u_{alpha}. Lemma 2.18 is omitted as "easy to show" but it is used in the sendograph continuity proof; a two-line proof would be better. These are all fixable.

The one substantive thing I would want pinned down is the importation of Theorem 3.2, "weak mixing implies weak mixing of all orders," for arbitrary topological spaces. The proof of (ii) implies (iii) uses simultaneous return times for finitely many pairs, which is exactly what that theorem provides. If Banks's or Peris's version actually requires compactness, metrizability, or a Baire condition, then the argument as written wouldn't cover all uniform spaces. I don't see any evidence in the paper that the theorem is misstated—the citation is to the standard sources—but this is a legitimate referee question.

Bottom line: send it out. It is a careful, useful extension that with minor corrections will be publishable. The conditional verdict is right; the stress-test concern is a fair question to put to the authors, not a demonstrated flaw.

Referee Report

1 major / 5 minor

Summary. The paper studies the transitivity of the Zadeh extension f^ on the space F(X) of normal upper semicontinuous fuzzy sets with compact support over a uniform space (X,U), equipped with three uniformities: the level-wise uniformity U∞, the Skorokhod uniformity U0, and the sendograph uniformity US. The authors show that f^ is continuous with respect to each of these uniformities, and they prove Theorem 3.7: for a continuous f, weak mixing of f is equivalent to transitivity of f^ on each of the three fuzzy-set uniformities, as well as to transitivity of the induced hyperspace map on K(X). The proof proceeds via the chain (i)⇔(ii)⇒(iii)⇒(iv)⇒(v)⇒(ii), importing two external theorems: the Banks–Peris theorem on hyperspace transitivity and a theorem stating that weak mixing implies weak mixing of all orders.

Significance. If correct, Theorem 3.7 is a substantial generalization of the metric-space result in [6] to arbitrary uniform spaces, and it provides a unified characterization for three distinct uniformities on fuzzy sets. The continuity theorems (2.12, 2.16, 2.19) are useful in their own right. A particular strength of the paper is that the proof of (ii)⇒(iii) constructs explicit fuzzy sets w and z from the hyperspace return points, rather than invoking abstract machinery. The paper contains no fitted parameters and no circular reasoning; the main result is a falsifiable equivalence that can be checked in concrete examples. The principal caveat is the reliance on Theorem 3.2, which is not proved in the text and may require hypotheses beyond an arbitrary topological space.

major comments (1)

[Section 3, proof of Theorem 3.7, implication (ii)⇒(iii)] The proof applies Theorem 3.2 to the hyperspace (K(X), τ_V) to obtain simultaneous return times K_i and L_i. Theorem 3.2 is stated for an arbitrary topological space X, but the citation [1, Theorem 1] (Banks) is, in many accounts, proved for compact metric spaces, and the authors do not give a proof of the general statement. Since (X,U) is an arbitrary uniform space, K(X) need not be compact or a Baire space. If Theorem 3.2 requires such hypotheses, the implication (ii)⇒(iii) is not established as written and the characterization in Theorem 3.7 would only hold under unstated additional assumptions. The authors should either supply a proof of Theorem 3.2 in the stated generality or restrict the main theorem to a class of spaces where the cited theorem is known to hold. The same verification should be provided for Theorem 3.1, although reference [9] is generally regarded as covering arbitrary topological spaces for that result.

minor comments (5)

[Eq. (3.4)] The entourage W in (3.4) is undefined; it should be V to make the subsequent claim (f^m(w_{α_i}), z_{α_i}) ∈ K[V] true.
[Eq. (3.5)] The relation (w_{α_i}, u_{α_i}) ∈ K[V] should read K[U], matching the earlier line from Proposition 3.4 and enabling the later K[U]^2 ⊆ K[U^2] ⊆ K[U_1] step.
[Eq. (3.1)] In the display preceding (3.1), the interval should be β ∈ (α_k, α_{k+1}], not (u_{α_k}, u_{α_{k+1}}], since β is a level index in [0,1].
[Proof of Theorem 3.7, (ii)⇒(iii)] The entourages U and V should be chosen symmetric, or the authors should justify why K[U] is symmetric for the applications of Proposition 3.4 in the construction of w and z.
[Section 3, after Proposition 2.20] The implication iii) ⇒ iv) ⇒ v) relies on the fact that transitivity on a finer topology implies transitivity on a coarser topology; this is correct but should be stated explicitly for completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniform-space transitivity theorem is proved from external dynamical results and prior level-set lemmas, not from its own conclusion.

full rationale

Theorem 3.7 is a new uniform-space generalization. The equivalence (i)⇔(ii) is imported from the external Banks–Peris theorem (Theorem 3.1), not re-derived from the paper's own equations. The new implications (ii)⇒(iii) and (v)⇒(ii) are constructive: the former builds fuzzy sets w and z from compact level sets using the entourage relations K[U] and K[V], and the latter reduces transitivity of the Zadeh extension under U_S to transitivity of the induced map on K(X) via characteristic functions χ_K and χ_L. The cited prior work by the same authors ([4], [5], [6]) supplies definitions and general level-set lemmas (Propositions 3.3–3.6, 2.20) whose statements do not include the target characterization; they are parameter-free and are applied as independent facts, so per the review rules they do not raise the circularity score. The correctness concerns flagged by the skeptical pass—the imported arbitrary-space hypotheses of Theorems 3.1 and 3.2, the undefined entourage W in (3.4), the mismatched K[V] in (3.5), and the omitted proof of Lemma 2.18—are genuine rigor or typo issues, but they are not instances of a claim reducing to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters. The theorem rests on standard topology plus several cited results from the authors' earlier work and from Banks and Peris; none of these is an ad hoc assumption invented for this paper.

assumptions (5)

domain assumption Banks-Peris theorem: f is weakly mixing iff the induced map on K(X) is transitive, for arbitrary topological spaces (Theorem 3.1, cited [1,9]).
Used at the start of the proof of Theorem 3.7 to get (i) iff (ii); the paper does not reprove it or state precise hypotheses.
domain assumption Weak mixing implies weak mixing of all orders (Theorem 3.2, cited [1]).
Used to find m and n compact sets simultaneously in the proof of (ii) implies (iii).
domain assumption Alpha-cut reconstruction: any decreasing family of compact sets satisfying left-continuity conditions determines a fuzzy set (Proposition 3.3, cited [5]).
Used to build w and z in the proof of Theorem 3.7; imported from the authors' earlier paper.
domain assumption The families {F[U]}, {G[U,epsilon]}, {S[U,epsilon]} are bases for the uniformities U_infinity, U_0, U_S with tau(U_S) subset tau(U_0) subset tau(U_infinity) (Propositions 2.9, 2.13, 2.20, cited [4]).
These are the objects the theorem is about; they are taken from the authors' previous paper.
domain assumption Zadeh extension respects alpha-cuts and level reparametrizations: [f^(u)]_alpha = f(u_alpha) and f^(tu) = t f^(u) (Proposition 2.1, Theorem 2.15, cited [5]).
Used throughout the continuity proofs; imported from the authors' earlier paper.

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Pith. "Pith review of Transitivity of some uniformities on fuzzy sets." pith.science (2026). https://pith.science/paper/VPNDJW3I

@misc{pith2026241117037,
  author       = {Pith},
  title        = {Pith review of: Transitivity of some uniformities on fuzzy sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPNDJW3I}},
  note         = {Machine review of arXiv:2411.17037}
}

abstract

Given a uniform space $(X, \mathcal{U})$, we denote by $\mathcal{F}(X)$ to the family of all normal upper semicontinuous fuzzy sets $u \colon X \to [0,1]$ with compact support. In this paper, we study transitivity on some uniformities on $\mathcal{F}(X)$: the level-wise uniformity $\mathcal{U}_{\infty}$, the Skorokhod uniformity $\mathcal{U}_{0}$, and the sendograph uniformity $\mathcal{U}_S$. If $f \colon (X, \mathcal{U}) \to (X, \mathcal{U})$ is a continuous function, we mainly characterize when the induced dynamical systems $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{\infty}) \to (\mathcal{F}(X), \mathcal{U}_{\infty})$, $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{0}) \to (\mathcal{F}(X), \mathcal{U}_{0})$ and $\widehat{f} \colon (\mathcal{F}(X), \mathcal{U}_{S}) \to (\mathcal{F}(X), \mathcal{U}_{S})$ are transitive, where $\widehat{f}$ is the Zadeh's extension of $f$.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Li-Yorke chaos on fuzzy dynamical systems
math.DS 2025-10 conditional novelty 6.0 of 10

Li-Yorke and distributional chaos transfer from a dynamical system to its hyperspace and fuzzy-set extensions, while a new Cantor-dense variant transfers back under completeness and linearity assumptions.
Topological dynamics for the endograph metric II: Extremely radical properties
math.DS 2025-10 accept novelty 6.0 of 10

For the endograph-metric Zadeh extension, contractivity iff f is constant, expansivity-type iff X is a singleton, and chain recurrence/transitivity/mixing iff f has dense range.
Topological dynamics for the endograph metric I: Equivalences with other metrics
math.DS 2025-10 conditional novelty 6.0 of 10

Topological A-transitivity, (ℓ,A)-recurrence, Devaney chaos, and the specification property are shown to be equivalent across the hyperspace and the supremum, Skorokhod, sendograph, and endograph fuzzy extensions.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages · cited by 3 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [6]

Jard´ on, I

D. Jard´ on, I. S´ anchez and M. Sanchis, Transitivity in fuzzy hy per- spaces, Mathematics 8 (11): 1862 (2020). https://doi.org/10.3390/math8111862

work page doi:10.3390/math8111862 2020

[2] [9]

Peris, Set-valued discrete chaos, Chaos, Solitons and Fract als 26 (2005) 19–23

A. Peris, Set-valued discrete chaos, Chaos, Solitons and Fract als 26 (2005) 19–23. https://doi.org/10.1016/j.chaos.2004.12.039

work page doi:10.1016/j.chaos.2004.12.039 2005

[3] [1]

Banks, Chaos for induced hyperspace maps, Chaos Solitons a nd Frac- tals 25 (2005) 1581–1583

J. Banks, Chaos for induced hyperspace maps, Chaos Solitons a nd Frac- tals 25 (2005) 1581–1583. https://doi.org/10.1016/j.chaos.2004 .11.089

work page doi:10.1016/j.chaos.2004 2005

[4] [2]

Bourbaki, General Topology, Springer Berlin, Heidelberg, 199 5

N. Bourbaki, General Topology, Springer Berlin, Heidelberg, 199 5. https://doi.org/10.1007/978-3-642-61701-0. 14

doi:10.1007/978-3-642-61701-0

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Jard´ on, I

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work page arXiv 2023

[7] [5]

Jard´ on, I

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work page 2020

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Kupka, On fuzziﬁcations of discrete dynamical systems, Inf ormation Sciences 181 (2011) 2858–2872

J. Kupka, On fuzziﬁcations of discrete dynamical systems, Inf ormation Sciences 181 (2011) 2858–2872. https://doi.org/10.1016/j.ins.2011.02.024

work page doi:10.1016/j.ins.2011.02.024 2011