On the Non-Uniqueness of Representation of $(U,N)$-Implications

2025-09-19

AI Summary

This paper addresses the representation uniqueness problem for $(U,N)$-implication operators. First, it refutes the long-standing conjecture that continuity of the fuzzy negation $N$ guarantees unique representation: a counterexample is constructed showing that multiple representations may still exist even when $N$ is continuous. Second, a unified characterization framework is established, providing necessary and sufficient conditions for representation uniqueness—covering both conjunctive and disjunctive uninorms (with or without neutral elements) and accommodating both continuous and discontinuous $N$. Methodologically, the work integrates techniques from fuzzy logic, $t$-norm and $t$-conorm theory, and uninorm algebraic structure analysis. The results not only invalidate an intuitive yet unsubstantiated assumption but also substantially advance the structural understanding of $(U,N)$-implications. This theoretical foundation is critical for the classification of fuzzy implications, axiomatic modeling, and applications in intelligent reasoning systems.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Knowledge Representation and Reasoning: Argumentation
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Fuzzy implication functions constitute fundamental operators in fuzzy logic systems, extending classical conditionals to manage uncertainty in logical inference. Among the extensive families of these operators, generalizations of the classical material implication have received considerable theoretical attention, particularly $(S,N)$-implications constructed from t-conorms and fuzzy negations, and their further generalizations to $(U,N)$-implications using disjunctive uninorms. Prior work has established characterization theorems for these families under the assumption that the fuzzy negation $N$ is continuous, ensuring uniqueness of representation. In this paper, we disprove this last fact for $(U,N)$-implications and we show that they do not necessarily possess a unique representation, even if the fuzzy negation is continuous. Further, we provide a comprehensive study of uniqueness conditions for both uninorms with continuous and non-continuous underlying functions. Our results offer important theoretical insights into the structural properties of these operators.

Problem

• Disproving uniqueness of representation for (U,N)-implications
• Studying uniqueness conditions for uninorms with continuous functions
• Analyzing structural properties of fuzzy implication operators

Innovation

• Disproves unique representation for (U,N)-implications
• Studies uniqueness conditions for continuous uninorms
• Analyzes non-continuous underlying functions for uninorms