A lattice-theoretic framework for hesitant fuzzy convexity beyond scalar observables

This paper establishes a lattice-theoretic framework for convexity that separates domain segment structures from codomain lattice structures to demonstrate that symmetric hesitant fuzzy convexity cannot be fully reconstructed by any finite family of scalar observables, thereby revealing the limitations of scalar reductions in preserving intrinsic order-theoretic information.

The Gist

Imagine you are trying to describe the "shape" of a group of people's opinions. In the world of mathematics, this is called convexity. Usually, if you have a group of people, and you pick two people at the ends of a spectrum, everyone standing between them should have an opinion that is "in the middle" or at least as strong as the weakest of the two ends.

This paper is about how to measure that "shape" when the opinions aren't simple numbers (like 0 to 10), but are instead complex, hesitant, or fuzzy collections of numbers.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Problem: The "Scorecard" Trap

In the past, when mathematicians wanted to check if a fuzzy opinion was "convex" (well-shaped), they used a scorecard. They would take a complex opinion (like a list of possible values) and crush it down into a single number (an average or a maximum) to make it easy to compare.

• The Analogy: Imagine you have a fruit salad (a complex opinion). To decide if it's "sweet enough," you used to take a spoonful, blend it into a single liquid, and taste that.
• The Issue: The paper argues that this "blending" destroys important information. Just because the average taste is sweet doesn't mean the individual fruits are arranged correctly. You might lose the specific structure of the fruit salad in the process.

2. The Solution: The "Lattice" Framework

The authors propose a new way to look at this without crushing the data first. They use a Lattice, which is like a complex map of relationships rather than a simple ruler.

• The Analogy: Instead of blending the fruit salad, you look at the salad as a whole. You check if the "middle" fruit is compatible with the "end" fruits based on a specific set of rules (the lattice structure).
• The Goal: They separate the shape of the group (the domain) from the complexity of the opinion (the codomain). This allows them to define "intrinsic convexity"—a shape that is true to the data itself, not just to a simplified score.

3. The Big Discovery: The "Hesitant" Wall

The paper focuses heavily on Hesitant Fuzzy Sets. These are situations where a person isn't sure of their opinion and gives a list of possibilities (e.g., "I think it's between 0.4 and 0.6, but maybe 0.8 too").

The authors discovered a major roadblock when trying to use simple scores to describe these hesitant opinions:

• The Finding: You cannot perfectly reconstruct the true "shape" of a hesitant opinion using a finite number of simple scores.
• The Analogy: Imagine trying to describe a 3D sculpture using only a handful of 2D shadows. No matter how many shadows (scores) you take, if the sculpture is complex enough (specifically, if it has a certain "dimension" of complexity), you will always miss something.
• The "Three-Point" Proof: The authors proved that even with just three people in a line, you can create a scenario where a set of scores says "Yes, this is a perfect shape," but the actual complex data says "No, it's broken."
  • The Metaphor: It's like a security guard checking three IDs. The guard sees three valid-looking photos (the scores) and lets the person through. But in reality, the person is wearing a mask (the hidden structure) that the photos didn't reveal. The guard's checklist was insufficient.

4. What This Means for the Field

The paper concludes that for these specific types of complex, hesitant data:

1. Simplification is dangerous: If you rely only on averages or single numbers (scalar observables), you will inevitably miss structural flaws in the data.
2. No magic bullet: There is no finite list of "score functions" that can capture the full truth of these hesitant sets.
3. The "Symmetric" Order: The specific mathematical structure they used (called the symmetric lattice) is too complex to be flattened into a simple list of numbers without losing its essence.

Summary

Think of this paper as a warning label for data scientists working with uncertain or hesitant information. It says: "Don't just average your data and call it a day. The complex structure of 'hesitation' contains hidden dimensions that simple numbers cannot see. If you try to flatten it, you will break the shape."

The authors built a new mathematical toolkit (the lattice framework) to handle these shapes correctly, proving that for certain types of complex data, you simply cannot replace the full picture with a few summary statistics.

Technical Summary

Technical Summary: A Lattice-Theoretic Framework for Hesitant Fuzzy Convexity Beyond Scalar Observables

Problem Statement
In fuzzy set theory, the concept of convexity has traditionally been formulated through scalar reductions, utilizing score functions and aggregation operators to map complex membership degrees (such as intervals or hesitant fuzzy elements) to the unit interval $[0, 1]$. While useful, the authors argue that such reductions obscure relevant order-theoretic information inherent in the codomain, particularly in set-valued settings like Interval-Valued Fuzzy Sets (IVFSs) and Hesitant Fuzzy Sets (HFSs). Existing proposals often lack a unified structural perspective that accounts for the specific features of the fuzzy model, raising the question of whether scalar-based notions of convexity are equally appropriate for all fuzzy-set models. The core problem addressed is the potential loss of structural fidelity when replacing intrinsic lattice operations with scalar summaries.

Methodology
The paper develops a lattice-theoretic framework for convexity on mappings $A: X \to L$, where $X$ is a point-convex segment-generated abstract convexity space and $L$ is a lattice. This approach deliberately separates the geometric structure of the domain from the order-theoretic structure of the codomain.

The methodology distinguishes between two primary perspectives:

1. Operational Perspective: Defines convexity via a binary operator $\otimes: L \times L \to L$. The paper focuses on intrinsic convexity, where $\otimes$ is the lattice meet ($\wedge$). A mapping is intrinsically convex if $A(y) \geq A(x) \wedge A(z)$ for all $y$ in the segment $[x, z]_\Gamma$. This is shown to be equivalent to the convexity of all principal cuts $[A]_a = \{x \in X : a \leq A(x)\}$.
2. Relational Perspective: Defines convexity via a preorder $\preceq$ on $L$. A mapping is $\preceq$-convex if the inverse image of every up-set is convex. This category includes observable convexity, where the preorder is induced by a scalar map (observable) $\phi: L \to [0, 1]$.

The framework analyzes the conditions under which scalar observables preserve or reconstruct intrinsic convexity. It further investigates pointwise operators $T: L^m \to L$ and characterizes those whose induced extensions preserve intrinsic convexity. Finally, the framework is specialized to Hesitant Fuzzy Sets (HFSs) endowed with the symmetric lattice $(P^*([0, 1]), \leq_0, \wedge_0, \vee_0)$, a structure introduced by Jara et al. [2023] that unifies standard fuzzy sets, interval-valued sets, and hesitant fuzzy sets.

Key Contributions and Results

1. Unified Framework: The paper establishes a common formal setting that recovers classical min-based fuzzy convexity, lattice-valued meet-based convexity, and order-based convexities as specific instances. It generalizes these concepts to non-linearly ordered domains (e.g., graph geodesics) via abstract convexity spaces.

2. Conditions for Scalar Preservation and Reconstruction:

   • Preservation: An observable $\phi$ preserves intrinsic convexity (i.e., every intrinsically convex set is $\phi$-convex) if and only if $\phi$ preserves finite meets as minima: $\phi(a \wedge b) = \min\{\phi(a), \phi(b)\}$.
   • Reconstruction: A single observable reconstructs intrinsic convexity (i.e., $A$ is intrinsically convex iff it is $\phi$-convex) if and only if $\phi$ preserves meets as minima and reflects the order (i.e., $\phi(a) \leq \phi(b) \implies a \leq b$). This implies that scalar reconstruction is impossible for non-chain lattices.
   • Families of Observables: A family of observables reconstructs intrinsic convexity if and only if the family separates the order of the codomain and each member preserves meets as minima.

3. Pointwise Operators: The paper characterizes operators $T: L^m \to L$ that preserve intrinsic convexity under pointwise extension. Such operators must satisfy a meet-compatibility condition: $T(a_1 \wedge b_1, \dots, a_m \wedge b_m) = T(a_1, \dots, a_m) \wedge T(b_1, \dots, b_m)$. The authors note that monotonicity alone is insufficient; for instance, the arithmetic mean fails to preserve min-based convexity.

4. Symmetric Convexity and Non-Representability:

   • When specialized to HFSs with the symmetric lattice, the framework recovers classical fuzzy convexity for singleton-valued profiles and interval-valued convexity for interval-valued profiles. Both lower layers admit finite scalar reconstruction (via the identity map for singletons and endpoint maps for intervals).
   • Main Structural Result: The paper proves that the symmetric order on hesitant fuzzy elements cannot be globally represented by any finite family of scalar observables. This is demonstrated by showing that the symmetric lattice contains finite subposets of arbitrarily large order dimension (specifically within the class of two-point THFEs).
   • Consequence: Symmetric hesitant convexity cannot, in general, be reconstructed by any finite family of scalar observables. Furthermore, for any finite family of monotone scalar observables, there exists a three-point hesitant profile that appears convex under all selected scalar descriptions but fails to be symmetrically convex.

Significance and Claims
The paper claims to provide a structural clarification of fuzzy convexity, distinguishing between intrinsic lattice properties and their scalar approximations. Its primary significance lies in demonstrating that scalar summaries are not globally faithful substitutes for the underlying order structure in hesitant fuzzy settings.

The authors assert that the obstruction to finite scalar representation is not an artifact of infinite cardinalities or complex profiles but appears even in simple two-point hesitant fuzzy elements. Consequently, the paper argues that finite scalar descriptions in HFS theory should be understood as partial summaries rather than complete replacements for the intrinsic hesitant structure. This challenges the reliance on score functions and aggregation operators as the sole means of defining convexity in hesitant fuzzy environments, suggesting that future research must account for the limitations of scalarization in preserving the full order-theoretic information of hesitant fuzzy sets.