揭示$(U,N)$-蕴含函数在连续否定下仍不唯一，挑战了既有理论。
On the Non-Uniqueness of Representation of $(U,N)$-Implications
- 通过反例证明即使否定连续，$(U,N)$-蕴含也不一定有唯一表示
- 系统分析了连续与非连续情形下表示唯一的充要条件
- 为模糊逻辑算子结构研究提供关键理论支撑，适合模糊数学研究者
模糊蕴含函数是模糊逻辑系统中的基础运算符，用于处理逻辑推理中的不确定性。其中，基于t-可和与模糊否定构造的$(S,N)$-蕴含及其推广到使用析取性双射的$(U,N)$-蕴含受到广泛关注。以往研究在假设模糊否定$N$连续的前提下建立了表征定理，认为其具有唯一表示。本文推翻了这一结论，证明$(U,N)$-蕴含即使在否定连续的情况下也未必具有唯一表示。我们进一步全面研究了在底层函数连续与非连续情形下的唯一性条件。研究成果为这些算子的结构特性提供了重要理论洞见。
原文摘要 · Abstract (English)
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.
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