Fuzzy Subgroups And Fuzzy Subrings With Thresholds (?,?) Of Operators

This paper generalizes the theories of the fuzzy group and the fuzzy ring.And proposing the concepts of the fuzzy subgroup,intuitionistic fuzzy subgroup,fuzzy subring and intuitionistic fuzzy subring with thresholds(?,?)of operators.And discussing some of their elementary characterizations.The results of this article are listed as follows:1.Proposing the definitions of the fuzzy subgroup,fuzzy normal subgroup and fuzzy quotient subgroup with thresholds(?,?)of operators,and discussing some of their elementary characterizations.Besides,in the sense of M-group homomorphism,studying the homomorphic images and preimages of the fuzzy subgroup(fuzzy normal subgroup)with thresholds(?,?)of operators.2.Proposing the definitions of the intuitionistic fuzzy subgroup and intuitionistic fuzzy normal subgroup with thresholds(?,?)of operators,and discussing some of their elementary characterizations.Besides,in the sense of M-group homomorphism,studying the homomorphic images and preimages of the intuitionistic fuzzy subgroup(intuitionistic fuzzy normal subgroup)with thresholds(?,?)of operators.3.Proposing the definitions of the fuzzy subring,fuzzy ideal and fuzzy quotient subring with thresholds(?,?)of operators,and discussing some of their elementary characterizations.Besides,in the sense of M-ring homomorphism,studying the homomorphic images and preimages of the fuzzy ring(fuzzy ideal)with thresholds(?,?)of operators.4.Proposing the definitions of the intuitionistic fuzzy subring and intuitionistic fuzzy ideal with thresholds(?,?)of operators,and discussing some of their elementary characterizations.Besides,in the sense of M-ring homomorphism,studying the homomorphic images and preimages of the intuitionistic fuzzy subring(intuitionistic fuzzy ideal)with thresholds(?,?)of operators.