Research About Fuzzy Normal Subgroups

Rosenfeld defined fuzzy groups in1971,Since then fuzzy algebra came into being. Normal subgroup plays an importment role in the theory of group and fuzzy normal subgroup also plays an importment role in fuzzy algebra too.we study further on fuzzy normal subgroup and give several new definition of fuzzy normal subgroup, at the same time, we obtain some algebraic properties of them.In chapter one, we introduce the developments of fuzzy normal subgroup both at home and abroad, besides, we also give the main work and innovative points of this paper.In chaptert two, we give the preliminaries which were used in this paper.In chaptert three, we first give the definition of generalized fuzzy normal subgroup in generalized fuzzy subgroup was given,at the same time we give its equivalent conditions. Besides we also give some properties of its level subset and the union of generalized fuzzy normal subgroup in generalized fuzzy subgroup. At last, some properties of homomorphic image and homomorphic preimage were given.In chapter four, we first give the definitions of R-fuzzy normal subgroup and fuzzy normal subgroup based on implication were given, at the same time we discuss the properties of intersection,union, homomorphic image and homomorphic preimage of R-fuzzy normal subgroup. The distinguish and contact of R-fuzzy normal subgroup with other types of fuzzy normal subgroup were discussed.Finally we give its equivalent form under nine specific implication operator.In chapter five, firstly,we give the definition of R-generalized fuzzy subnear-ring and give some algebraic properties of it.At the same time, the distinguish and contact of R-generalized fuzzy subnear-ring with generalized fuzzy subnear-ring were discussed.we also give its equivalent form under five specific implication operator.Finally, we give the definition of R-generalized fuzzy ideal and give some properties of its intersection,union, homomorphic image and homomorphic preimage,we also give the distinguish and contact of R-generalized fuzzy ideal with generalized fuzzy ideal and give its equivalent form under specific implication operator.In chapter six, we sum up the content of the paper and give prospect of fuzzy normal subgroup.