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A revised and extended version of McShane-Whitney extensions for fuzzy Lipschitz maps
Pith reviewed 2026-05-09 21:03 UTC · model grok-4.3
The pith
McShane-Whitney extensions for fuzzy Lipschitz maps hold when φ is increasing and left-continuous.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The revised McShane-Whitney extension theorem applies to fuzzy Lipschitz maps with values in (R, M_φ,g, *) whenever φ is increasing and left-continuous, removing the need for invertibility assumed in earlier results.
What carries the argument
The McShane-Whitney extension construction for maps into the Euclidean fuzzy metric space, relying on order properties preserved by left-continuous increasing φ.
If this is right
- The extension theorem now covers a larger family of φ functions used to define fuzzy metrics.
- Preservation of the Lipschitz constant holds under the weaker left-continuity condition.
- Applications of fuzzy metric extensions become available without requiring bijective φ.
Where Pith is reading between the lines
- This change may allow non-strictly increasing φ in models of fuzzy distances.
- Similar continuity relaxations could apply to other extension results in fuzzy analysis.
- Explicit checks with left-continuous but non-invertible φ would show the added generality.
Load-bearing premise
The function φ defining the fuzzy metric on the reals is increasing and left-continuous.
What would settle it
A counterexample showing a fuzzy Lipschitz map that cannot be extended when φ is increasing but not left-continuous would disprove the revised theorem.
read the original abstract
In the paper [E. Jim\'enez-Fern\'andez, J. Rodr\'{\i}guez-L\'opez, E. A. S\'anchez-P\'erez, Fuzzy Sets and Systems 406 (2021),66-81], a McShane-Whitney extension theorem is presented for real-valued fuzzy Lipschitz maps between fuzzy metric spaces. Specifically, the codomain space is considered as a so-called Euclidean fuzzy metric space $(\mathbb{R},M_{\phi,g},\ast).$ However, while the function $\phi$ is only required to be increasing, some results of the paper implicitly assume that $\phi$ is invertible, even though this is not explicitly stated. We propose here an alternative possibility that only requires $\phi$ to be also left-continuous.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript identifies an implicit assumption of invertibility for the increasing function φ in the authors' 2021 paper on McShane-Whitney extensions of fuzzy Lipschitz maps into the Euclidean fuzzy metric space (ℝ, M_{φ,g}, ∗). It proposes that the extension theorem continues to hold if φ is required only to be increasing and left-continuous, without needing invertibility.
Significance. If the revised conditions are shown to suffice, the result would broaden the class of admissible codomain fuzzy metrics for which the McShane-Whitney theorem applies, including non-strictly increasing φ with flat intervals. This corrects a gap in the prior work and strengthens the foundation for extensions in fuzzy metric spaces.
major comments (2)
- [§3] §3 (main extension theorem and proof): The argument that left-continuity of φ replaces the use of φ^{-1} from the 2021 paper must be made explicit. In particular, the verification that the extended map f̂ satisfies the fuzzy Lipschitz condition with respect to M_{φ,g} needs to be checked step-by-step when φ is constant on an interval; the original steps relying on order recovery via the inverse may not carry over directly.
- [§2] Definition of M_{φ,g} and the triangle inequality (likely §2): It is unclear whether left-continuity alone guarantees that the pseudometric induced by M_{φ,g} satisfies the required inequalities in the extension construction when φ has flat segments. A concrete check or counter-example ruling out failure of the fuzzy triangle inequality under the new hypotheses is needed.
minor comments (2)
- [Abstract and §1] The abstract and introduction should explicitly recall the precise statement of the 2021 theorem (including the implicit invertibility) so that the correction is self-contained.
- [§2] Notation for the fuzzy metric M_{φ,g} and the t-norm ∗ should be restated in §2 for readers who have not consulted the 2021 paper.
Simulated Author's Rebuttal
We thank the referee for the careful reading and valuable comments, which help strengthen the clarity of our revision to the McShane-Whitney theorem. We address each major comment below and will incorporate the suggested clarifications in the revised manuscript.
read point-by-point responses
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Referee: [§3] The argument that left-continuity of φ replaces the use of φ^{-1} from the 2021 paper must be made explicit. In particular, the verification that the extended map f̂ satisfies the fuzzy Lipschitz condition with respect to M_{φ,g} needs to be checked step-by-step when φ is constant on an interval; the original steps relying on order recovery via the inverse may not carry over directly.
Authors: We agree that the proof requires greater explicitness on this point. Left-continuity of φ allows us to work directly with left-limits in the definition of the extension, avoiding any need for inversion. In the revised version we will expand the proof of the main theorem in §3 with a dedicated subsection that verifies the fuzzy Lipschitz condition case by case. When φ is constant on an interval [a,b], the left-continuity ensures that for any value attained in the flat segment the supremum/infimum operations in the McShane-Whitney formula remain compatible with the fuzzy metric inequalities, because the metric values cannot jump discontinuously from the left. We will insert the missing intermediate inequalities that replace the order-recovery step of the 2021 paper. revision: yes
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Referee: [§2] It is unclear whether left-continuity alone guarantees that the pseudometric induced by M_{φ,g} satisfies the required inequalities in the extension construction when φ has flat segments. A concrete check or counter-example ruling out failure of the fuzzy triangle inequality under the new hypotheses is needed.
Authors: We will add the requested verification in §2. The fuzzy triangle inequality for M_{φ,g} continues to hold under the sole assumptions that φ is increasing and left-continuous. The key is that left-continuity precludes right-discontinuities that could otherwise violate the inequality when the argument of φ lands inside a flat segment. In the revision we will include a short lemma (or remark) that explicitly checks the triangle inequality for three points whose pairwise distances fall inside a flat interval of φ; the monotonicity of φ together with left-continuity guarantees that the t-norm composition remains valid. No counter-example arises, and we will exhibit the algebraic steps that confirm this. revision: yes
Circularity Check
No significant circularity; revision supplies independent proof under weaker assumption
full rationale
The manuscript identifies an implicit invertibility assumption in the authors' 2021 paper and replaces it with the explicit requirement that φ be increasing and left-continuous. The new McShane-Whitney extension construction is then verified directly from the definition of the fuzzy metric M_φ,g and the left-continuity property, without any step that reduces the claimed extension map or Lipschitz constant to a fitted quantity, a self-referential definition, or an unverified self-citation. The 2021 reference is used only for historical context and to state the correction; the load-bearing arguments of the present work are self-contained against the fuzzy-metric axioms and the stated regularity on φ.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Standard axioms of fuzzy metric spaces (M1-M5 or equivalent)
- domain assumption φ is increasing and left-continuous
Reference graph
Works this paper leans on
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[1]
Eklund, J
P. Eklund, J. Guti´ errez-Garc´ ıa, U. H¨ ohle, and J. Kortelainen,Semigroups in complete lattices. Quantales, modules and related topics, Springer, 2018
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[2]
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V. Gregori, S. Morillas, and A. Sapena,On a class of completable fuzzy metric spaces,Fuzzy Sets and Systems 161 (2010), 2193–2205
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[3]
Jim´ enez-Fern´ andez, J
E. Jim´ enez-Fern´ andez, J. Rodr´ ıguez-L´ opez, and E. A. S´ anchez-P´ erez, McShane-Whitney extensions for fuzzy Lipschitz maps, Fuzzy Sets and Sys- tems 406 (2021), 66-81. (E. Jim´ enez-Fern´ andez)Departamento de Teor´ıa e Historia Econ´omica, Campus Universitario de Cartuja, Universidad de Granada, 18071 Granada, Spain Email address:edjimfer@ugr.es ...
2021
discussion (0)
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