Characterizations of (U2,N)-implications generated by 2-uninorms and fuzzy negations from the point of view of material implication

Abstract

Fuzzy implications play an important role in both theoretic and applied communities of fuzzy set theory. The well-known fuzzy implications are usually constructed in appropriate ways from t-norms, t-conorms and fuzzy negations, and according to construction methods, they can be roughly classified into five classes, namely (S,N)-implications, R-implications, QL-implications including D-implications as contrapositions, Yager's implications and ordinal sum implications. So far, many further generalizations and new subclasses of fuzzy implications have been proposed from different perspectives in the literature. The present paper focuses on a specific subclass of fuzzy implications, whose members are called (U2,N)-implications and generated by disjunctive 2-uninorms U2 and fuzzy negations N from the point of view of material implication. The main results are basic properties of natural negations inherited from (U2,N)-implications and complete characterizations of (U2,N)-implications. The motivation of the work lies in the fact that such fuzzy implications can induce several kinds of natural negations, among which the 2-natural negations not only generalize the existing natural negations of general fuzzy implications just as the 2-neutral elements of 2-uninorms extend the neutral elements of uninorms, but also cooperate well with the initial generating fuzzy negations under simple conditions, and what is more important is that such 2-natural negations, viewed as counterparts of the 2-neutral elements of 2-uninorms in the framework of (U2,N)-implications, serve as key tools in deriving 2-uninorms and in characterizing (U2,N)-implications from general fuzzy implications. Finally, such fuzzy implications are extended to the setting of n-uninorms, and the corresponding properties and characterizations of the resulting (Un,N)-implications are given.