(H,l)-implication And Its Functional Equation

Fuzzy implication is an important operator in fuzzy logic theory,which general-izes the classical Boolean implication.How to construct different fuzzy implications is one of the most concerned problems in fuzzy implication theory.In 2004,R.R.Yager introduced two families of fuzzy implications generated by unary functions,namely f-implication and g-implication.In 2006,B.Jayaram introduced another new family of fuzzy implications,namely h-implication.This thesis will extend the family of h-implications into a new large family of fuzzy implications and then study this new family of fuzzy implications.The main contents of thesis are arranged as follows:Chapter One:Preliminaries.In this chapter,some basic concepts and relevant conclusions in Triangular Norm Theory and Fuzzy Implication Theory are given.Chapter Two:(h,l)-implications and their properties.Firstly,the concept of(h,l)-implication is introduced and some examples are given.Secondly,the proper-ties of natural negation of(h,l)-implications are discussed.Finally,the basic prop-erties of(h,l)-implications are studied,it is also proved that two(h,l)-implications generated from the same continuous,strictly increasing function l are equal if and only if their h.-generators are also equal.Chapter Three:The relationships between the family of(h,l)-implications with other families of fuzzy implications.Firstly,the relationships between three families of generated implications f-implications,g-implications,h-implications with their respective families of generalized generated implications(f,g)-implications,(g,g)-implications,(h,l)-implications are discussed.Secondly,the relationships be-tween(f,g)-implications,(g,f)-implications,(h,l)-implications are studied.Finally,the relationships between(h,l)-implications and(S,N)-implications,R-implications are discussed.Chapter Four:Functional equations of(h,l)-implications.The law of impor-tation and four distributive equations of(h,l)-implications are mainly studied,and the corresponding characterization theorems are also given in this chapter.