A Study On Transitivity And Transitivity Indicators Of φ-Additive Fuzzy Preference Structures

In this paper, in the framework of additive-fuzzy preference structures,we study the transitivity and transitivity indicators of various fuzzy preferencerelations, including the large preference relation, the strict preference relationand the indifference relation, under the conditions without incomparability andthe large preference relation strongly complete, respectively.Firstly, we review the definition of general preference structure and someconclusions on the transitivity, and give a detailed introduction to the definitionand the research status of additive-fuzzy preference structures.Secondly, we investigate the transitivity on the various kinds of preferencerelations in additive-fuzzy preference structures and obtain some conclusions.By both rotation invariant t-norm and-norm without zero divisor, we discussadditive-fuzzy preference structure without incomparability, and someresults on the relationships of the transitivity of the large preference relation，thestrict preference relation, the indifference relation, the property of T-PI and theproperty of-IP can be given. At the same time, under the condition that thelarge preference relation is strongly complete, two equivalent theorems on thetransitivity of the large preference relation can be derived by any-norm that isnot less than rotation invariant-norm.Thirdly, we carry out research on some transitivity indicators of variousfuzzy preference relations, and the conclusions of additive-fuzzy preferencestructures can be extended to the degree description. The major results are that the transitivity indicator of the large preference relation is less than or equal tothat of the strict preference relation, indicators of the property of T-PI and theproperty of-IP on rotation invariant t-norm as the incomparability relationis empty. The transitivity indicator of the large preference relation is less than orequal to that of the indifference relation for any-norm as the large preferencerelation is strongly complete.Finally, we notice that almost all conclusions related to transitivity can beextended to the fuzzy case and the indicator case. For some results which fail tobe extended, we present some counterexamples.