A Study On A Family Of Fuzzy Implications

In this paper, we study the properties of a family of fuzzy implications I T ,N and its extension I U ,N in detail. The main research work is summarized as follows.Firstly, we present an overview of the current study on fuzzy implications and uninorms which mainly includes the detailed investigation into the research on the properties of three families of usual implications (S-implications, R-implications and QL-implications), fuzzy implications and operators generated from uninorms, and their characterizations. Then, we briefly introduce some concepts of fuzzy logic connectives(negations, t-norms, t-conorms, fuzzy implications).Secondly, we study the properties of a family of fuzzy implications I T ,N in detail. We give the conditions under which I T, N is a fuzzy implication and the conditions under which I T ,N is an S-implication, R-implication or QL-implication(main part). Meanwhile, we also investigate the conditions under which a function I is I T ,N and the conditions under which I T ,N satisfies the properties I ( x,N(x))= N(x) and I ( x , y )= I ( x , I ( x ,y )) . In addition, we discuss the necessary and sufficient condition under which TI ,N is a t-norm.Finally, we study the properties of an extension I U ,N of I T ,N, In view that I U ,N is not a fuzzy implication generally, we give some conditions under which I U ,N is fuzzy implication, and the conditions under which I U ,N satisfies fuzzy implications and prove that I U ,N generated from representable uninorm U is an RU - implication. At last, we discuss the necessary and sufficient condition under which U I ,N is a uninorm.