Some Fuzzy Filters And Fuzzy Congruence Relations On Residuated Lattice

The concept of fuzzy set was proposed by professor L.A.Zadeh in 1965, it sets a solid foundation for the emergence of fuzzy logic . In 1973, professor Zadeh firstly introduced famous CRI Method of fuzzy reasoning . It has been wildly applied in the industry control and the production of electrical equipments with enormous success. However its basic theory — fuzzy reasoning , still hasn't a dependable logical base. The fuzzy logic algeraic system plays an important role in non-classical logic which is the theoretical basis of many-valued logic, fuzzy reasoning and fuzzy control. As one of the important fuzzy logic algebraic systems, Residuated lattices have broad applications. Filters and congruence relations of the residuated lattices are studied by employing muti-valued logic algebra's methods in this paper.The main results of this paper are given as follows:In the first chapter , the concept of fuzzy(P) filter is introduced. Some of its properties are investigated. The structure of fuzzy(P) filters is discussed and it is proved that the set of all fuzzy(P) filters of a residuated lattice is a distributive and complete lattice . In view of the structure of the fuzzy(P) filters, a new kind of adjoint couple (?,?) is defined in residuated lattices. A group of special fuzzy(P) filters with the adjoint couple (?) is proved to be a residuated lattice.In the second chapter, the concept of congruence relation on a residuated lattice is introduced. It is proved that the quotient algebra of a residuated lattice about the congruence relation is still a residuated lattice. Then as a generalization of the congruence relation, the concept of fuzzy congruence relation is brought in. It is proved that there is a bijection between the set of fuzzy(P) filters and the set of fuzzy congruence relations. Afterwards, the notion of normal mapping from a residuated lattice to another is defined . The Zadeh-function induced by the normal mapping is shown to be a mapping of keeping fuzzy(P) filters.In the third chapter, the concepts of several special fuzzy filters are introduced in a kind of new fuzzy logic algebra structure—regular residuated lattices, such as prime fuzzy(P) filters , fuzzy strong implicative filters and so on . The properties of...