(T,φ)-Fuzzy Quasi-Order And Weak Order Structures

In this paper, we deal with the (T,φ)-fuzzy quasi-order and weak order structures in the framework of additive j-fuzzy preference structures, and carry out the research on their indicators.In studying (T,φ)-fuzzy quasi-order structures, we firstly present the definition of a (T,φ)-fuzzy structures. Then some conditions are presented to guarantee the validity of two inclusions, JPPPSTUo í and JIIISTUo í. Finally, we define the indicator of (T,φ)-fuzzy quasi-order structures and extend the above inclusions to the indicator version.In dealing with (T,φ)-fuzzy weak order structures, we mainly investigate the relationships among the following statements:(1)(P,I,J)is a(T,j)-fuzzy weak order structure;(2) J =? and R is T-transitive;(3)R is strongly S-complete and T-transitive;(4) J =?and P is T-asymmetric,negatively S-transitive. Meanwhile, someconditions for these statements to be equivalent are presented. Finally,the indicator to measure (T,φ)-fuzzy weak order structures is defined and some results on the indicator are obtained mainly under the condition J =?.