(U~2,N)-Implication And Its Description

Fuzzy implications play an important role in both theoretic and applied as-pects of fuzzy set theory.Implications are usually constructed in appropriate ways from t-norms,t-conorms and fuzzy negations,and according to construction meth-ods,they can be roughly classified into five classes,namely(S,N)-implications,R-implications,QL-implications,Yager’s implications and ordinal sum implication-s.So far,many further generalizations and new subclasses of fuzzy implications have been proposed from different perspectives in the literature.The present the-sis focuses on a specific subclass of fuzzy implications,whose members are called(U2,N)-implications and generated by disjunctive 2-uninorms U2 and fuzzy nega-tions 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-uminorms,and the corre-sponding properties and characterizations of the resulting(Un,N)-implications are given.The concrete content is as follows:Chapter One:Preliminaries.In this chapter,some basic concepts and relevant properties of fuzzy negation.2-uninorms and n-uninorm,fuzzy implication are given.Chapter Two:(U2,N)-implication and its characterization.Firstly,the notion of(U2,N)-implication is proposed and some examples of(U2,N)-implications are given.Secondly,we discusses main properties and proves,where the natural negations,particularly the 2-natural negations inherited from(U2,N)-implications are emphasized,and lots of verifying and illustrating examples are constructed.Finally,some characterizations of(U2,N)-implications are given.Chapter Three:(Un,N)-Impliation and its characterization.We extend the results of Section 3 to the setting of(Un,N)-implications generated by n-uninorms and fuzzy negations from the point of view of material implication.