A revised and extended version of McShane-Whitney extensions for fuzzy Lipschitz maps

1. Problem Statement

The paper addresses a critical gap in the existing literature regarding the extension of fuzzy Lipschitz maps from a subset of a fuzzy metric space to the entire space. Specifically, it critiques a previous result published in Fuzzy Sets and Systems (2021) by the same authors.

• The Flaw: The 2021 paper established a McShane-Whitney extension theorem for maps into a Euclidean fuzzy metric space $(\mathbb{R}, M_{\phi,g}, *)$. However, the proofs in that work implicitly relied on the existence of the inverse function $\phi^{-1}$.
• The Contradiction: In the definition of a Euclidean fuzzy metric space, the function $\phi: [0, +\infty) \to [0, +\infty)$ is only required to be increasing. It is not necessarily bijective (one-to-one and onto). As noted in the paper, $\phi$ is also bounded, meaning a standard inverse $\phi^{-1}$ often does not exist.
• The Goal: The authors aim to correct these proofs and extend the theorem by replacing the non-existent inverse $\phi^{-1}$ with a mathematically rigorous alternative that works for general increasing functions $\phi$.

2. Methodology

The authors employ tools from fuzzy set theory, metric geometry, and order theory (specifically Galois connections) to resolve the issue.

• Refining the Space Definition: They introduce the concept of a left-continuous Euclidean fuzzy metric space. This requires $\phi$ to be not just increasing, but also left-continuous.
• Introduction of the Right-Adjoint ($\phi^*$):
  • Since $\phi$ may not be invertible, the authors utilize the right-adjoint (or residual) of $\phi$, denoted as $\phi^*$.
  • Defined as: $\phi^*(y) = \sup \{x \in [0, +\infty] : \phi(x) \leq y\}$.
  • Key Properties:
    1. $\text{id} \leq \phi^* \circ \phi$ (The composition is greater than or equal to the identity).
    2. If $\phi$ is increasing and left-continuous, then $\phi \circ \phi^* \leq \text{id}$.
  • This allows the authors to "invert" the inequality logic without requiring $\phi$ to be bijective.
• Handling Boundedness: The authors identify that if $\phi$ is bounded (which is true for fuzzy metrics), $\phi^*$ can take the value $+\infty$ for large inputs. To prevent this from breaking the extension, they introduce a boundedness condition on the Lipschitz constant relative to the supremum of $\phi$.

3. Key Contributions

A. Theoretical Correction

The paper corrects the definition and proofs of the McShane-Whitney extension theorem for fuzzy Lipschitz maps. It demonstrates that the assumption of $\phi$ being invertible was unnecessary and incorrect; the left-continuity of $\phi$ and the use of its right-adjoint $\phi^*$ are sufficient.

B. Revised Theorem (Theorem 5)

The authors present a corrected version of the extension theorem with a specific constraint:

• Condition: For the extension to exist, the term involving the Lipschitz constant $K(t)$, the function $g(t)$, and the fuzzy distance $(1 - M(x,y,t))$ must be strictly less than $\phi(+\infty)$.
• Significance: This ensures that the argument passed to $\phi^*$ remains within a range where it is finite, preventing the extension from becoming undefined.

C. Explicit Extension Formulas

The paper provides corrected formulas for the McShane and Whitney extensions. Instead of using $\phi^{-1}$, the formulas now utilize $\phi^*$:

• McShane Extension ($f^M$):
  $$f^M(x, t) = \sup_{s \in S} \left\{ f(s, t) - \phi^*\left( \frac{K(t)}{g(t)} (1 - M(x, s, t)) \right) \right\}$$
• Whitney Extension ($f^W$):
  $$f^W(x, t) = \inf_{s \in S} \left\{ f(s, t) + \phi^*\left( \frac{K(t)}{g(t)} (1 - M(x, s, t)) \right) \right\}$$

D. Application to Specific Examples

The authors revisit two specific examples from the original 2021 paper where $\phi(x) = \min\{x, 1\}$ (which is not bijective).

• They demonstrate that the original formulas were invalid because $\phi^{-1}$ does not exist for this function.
• They derive the corrected explicit formulas using $\phi^*$, which in this specific case acts as a piecewise function (scaling by 2 for small values and becoming infinite for large values, subject to the boundedness constraint).

4. Results

• Existence Proof: The paper proves that under the condition that $\phi$ is left-continuous and the Lipschitz condition satisfies the boundedness constraint, a fuzzy Lipschitz extension always exists.
• Preservation of Properties: The constructed extensions ($f^M$ and $f^W$) are proven to be fuzzy Lipschitz maps on the entire space $X \times (0, +\infty)$.
• Generalization: The results apply to a broader class of Euclidean fuzzy metric spaces than previously thought, removing the restrictive (and often false) assumption of bijectivity for $\phi$.

5. Significance

• Mathematical Rigor: The paper rectifies a subtle but significant logical error in the foundational theory of fuzzy Lipschitz extensions. It ensures that theorems in this field rely on necessary and sufficient conditions rather than implicit assumptions.
• Practical Applicability: By providing correct formulas for non-bijective $\phi$ functions (like $\min\{x, 1\}$), the paper makes the McShane-Whitney extension theorem applicable to a wider range of practical fuzzy metric spaces used in data analysis, image processing, and approximate reasoning.
• Theoretical Framework: The introduction of the right-adjoint $\phi^*$ as a standard tool for handling non-invertible fuzzy metric functions establishes a robust framework for future research in fuzzy metric spaces and fuzzy Lipschitz analysis.

In summary, this paper serves as a necessary correction and extension to previous work, replacing the flawed reliance on inverse functions with a rigorous order-theoretic approach using right-adjoints, thereby solidifying the theoretical foundation for extending fuzzy Lipschitz maps.