Research On (?,?)- Anti-fuzzy Subgroups And Subrings

Fuzzy set theory is a generalization of classical set theory.It is mainly a method to study set theory by using membership function instead of characteristic function.Fuzzy set theory maily studies the operation of fuzzy subset theory,fuzzy relation,fuzzy mapping,fuzzy recognition,fuzzy decision.Applying fuzzy set theory to group we maily studies fuzzy subgroup,fuzzy normal subgroup,homomorphism and isomorphism of fuzzy subgroup,fuzzy quotient subgroup,(?,?)-fuzzy subgroup and anti-fuzzy subgroup.Fuzzy subring of fuzzy ring,fuzzy ideal,direct products and homomorphism of ring,and various types of fuzzy group and ring.However,some areas of research are not perfect and need to futher explore.Based on the fuzzy set theory and(?,?)-fuzzy normal subgroups and(?,?)-fuzzy quotient subgroups.The main contribution of this work is summarized as follows:First,the concepts of(?,?)-normalizers and centralizers of anti-fuzzy subgroup are given,and some conclusions are obtained by combining defintions with mapping,normalities,homomorphism and isomorphism of fuzzy subgroup.Second,on the basis of the fuzzy quotient group,the anti-fuzzy quotient group is defined and the homomorphism and isomorphism of the(?,?)-anti-fuzzy subgroup are discussed under the concept.Third,definitions of(?,?)-anti-fuzzy subring,(?,?)-anti-fuzzy ideal and(?,?)-anti-fuzzy quotient ring are given and combing with the properties of fuzzy ring,Some conclusions are built.Fourth,on the basis of homomorphic mapping of fuzzy ring,the homomorphism and isomorphism of(?,?)-anti-fuzzy are built.