On the Non-Uniqueness of Representation of $(U,N)$-Implications
This addresses a theoretical gap in fuzzy logic for researchers, but it is incremental as it refines existing characterization theorems.
The paper disproves the assumption that (U,N)-implications have a unique representation under continuous fuzzy negations, showing they can lack uniqueness even with continuity, and provides a comprehensive study of uniqueness conditions for uninorms with continuous and non-continuous functions.
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.