On the non-uniqueness of representation of (U, N)-implications

IF 3.1 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Fuzzy Sets and Systems Pub Date : 2026-03-01 Epub Date: 2025-11-13 DOI:10.1016/j.fss.2025.109680

Raquel Fernandez-Peralta , Andrea Mesiarová-Zemánková

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引用次数: 0

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.

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关于（U， N）表示的非唯一性——启示

模糊蕴涵函数是模糊逻辑系统中的基本运算符，是对经典条件的扩展，用于管理逻辑推理中的不确定性。在这些算子的广泛族中，经典物质蕴涵的推广已经得到了相当大的理论关注，特别是由t-适形和模糊否定构造的（S， N）蕴涵，以及它们使用析取一致子进一步推广到（U， N）蕴涵。先前的工作已经在模糊否定N连续的假设下建立了这些族的表征定理，保证了表示的唯一性。在本文中，我们对（U， N）-蕴涵证明了这最后一个事实，并表明它们不一定具有唯一表示，即使模糊否定是连续的。进一步，我们全面地研究了具有连续和非连续底层函数的一致子的唯一性条件。我们的结果为这些算子的结构特性提供了重要的理论见解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Fuzzy Sets and Systems 数学-计算机：理论方法

CiteScore

6.50

自引率

17.90%

发文量

321

审稿时长

6.1 months

期刊介绍： Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies. In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.