Research On Some Properties Of QL-and D-Implications

Fuzzy implications play an important role in fuzzy logic and approximate reasoning. Many fuzzy rule based systems do their inference processes through these operators in fuzzy reasoning. Moreover they have proved to be useful also in other fields like fuzzy relational equations, fuzzy mathematics and so on.Because of the importance of fuzzy implication operators, many scholars research its properties from theory and practical application by typing classical logic tautology on fuzzy logic framework. Researchers define new fuzzy implication operators by different methods such as R-, S-, QL-and D-implications that generated from negations, t-norms and t-conorms. The researchers have got abundant research achievements on R-and S-implication recently. This paper will study QL-and D-implications and discuss the existence of solutions to distributive equations involving QL-and D-implications. We generalize the relevant conclusions to uninorms and get some properties of QL-and D-implications derived from uninorms. The main contents of the articles are as follows.In chapter 1, the paper describes the background and significance of the research on fuzzy implications and summarizes current researches on QL-and D-implication. Meanwhile we recall some definitions and properties of fuzzy logic connectives including fuzzy negations, t-norms, t-conorms and fuzzy implications.In chapter 2, we give a brief introduction to the definitions and properties of QL-and D-implications. Then the paper discusses the necessary conditions and sufficient conditions of QL-and D-operators to be fuzzy implications. At the same time we systematically discuss some usual properties of fuzzy implications and we get the following conclusions.First, QL-orD-implications generated from continuous negations,t-norms and t-conorms satisfies exchange principle if and only if QL-or D-implication is an S-implication.Second, QL-or D-implications generated from strong negation Nφ,t-norm T and continuous t-conorm S=(SL)φsatisfies contrapositive symmetry if and only if T is a Frank t-norm, where we prove the related conclusion by using new method.In chapter 3, we investigate the distributivity of QL-and D-implications and give the solutions to distributive equations involving QL-and D-implications, where N is strict or continuous negation.In chapter 4, the paper discuses the necessary conditions of QL-and D-operators derived from uninorms to be fuzzy implications. We systematically study some properties of them and get the following conclusions that the necessary condition of QL-or D-implications derived from conjunctive uninorm Uc, t-conorm S and continuous negation N satisfies exchange principle is Uc is a t-norm.