The Operations And Approximate Sets Of Anti-fuzzy Subgroups

Both the rough set theory and the fuzzy set theory generalize the classical set theory in dealing with uncertainty issues. The two theories are different in the starting point and focus, and thus are strongly complementary rather than exclusive. It is of important theoretical and practical significance to combine two theories. The research is conducted based on rough sets and fuzzy sets, and the details are as follows:Firstly, based on the current research on fuzzy subgroups, we obtain some necessary and sufficient conditions for anti-fuzzy subgroups and anti-fuzzy normal subgroups. Then, with the definitions of complement,inverse, union, product and intersection of anti-fuzzy subgroups, we discuss the properties of these operations. The research enriches the investigation of anti-fuzzy subgroups.Afterwards, we give some further results on the relationships of upper and lower approximation sets of fuzzy rough sets between the different definitions of fuzzy rough sets coined respectively by Z. Pawlak,M. Banerjee, G. L. Liu and P. Kong. Meanwhile, a condition for a fuzzy set to be a definable set is proposed. Then, with the Liu’s definition, weprove the decomposition theorems of upper and lower approximations and give some properties of upper and lower approximations of anti-fuzzy subsemigroups, anti-fuzzy subgroups and anti-fuzzy normal subgroups. At last, we give the definition of (λ,μ)-fuzzy equivalence relation,),(aml-- cut and strong),(aml-- cut, generalize the definition of the upper and lower approximation sets of a fuzzy rough set and investigate the properties of upper approximate sets of (λ,μ)- fuzzy subgroups and (λ,μ)-fuzzy normal subgroups.