The Research On The Transitivity-Related Properties Of Fuzzy Relations

Fuzzy relation theory is one of the most important branches of fuzzy mathematics and is extensively used in many fields especially in the area of decision-making. For example, fuzzy clustering analysis, choice problem under fuzzy preferences, ordering of fuzzy quantities and fuzzy preference structures are all based on fuzzy relations. In the research of fuzzy relations, the investigation of transitivity-related properties plays a central role. In fact, the above mentioned applications are all concerned with some certain transitivity properties.In this research, using the theory of t-norm, t-conorm and fuzzy relations, we systematically discuss transitivity-related properties of operations of fuzzy relations, relationships between T -transitivity, negative S -transitivity, T-S -semitransitivity and T-S-Ferrers properties, relationships of transitivity-related properties between the large preferencerelation and strict preference, indifference relations, and transitivity of the limit of a sequence of fuzzy relations with certain properties. The main conclusions are as follows.Firstly, among operations of fuzzy relations, the inverse does not change any transitivity at all and the complement changes some transitivity properties whereas the operations union and intersection fail to preserve most transitivity properties.Secondly, we carry out a detailed investigation into the relationships of T -transitivity, negative S -transitivity, r-S-semitransitivity and T-S -Ferrers properties under some conditions, such as completeness, strong De Morgan triple, positive /-norm etc., and have some expected conclusions.Thirdly, based on our definition of fuzzy preference structure without incomparability, we find out relationships between transitivity properties of large preference relation and strict preference and indifference relation with ^-transformation of Lukasiewicz t -norm. For example, we point out that the W -transitivity of large preference relation can derive the same transitivity of strict preference and indifference relation.Finally, the transitivity of the limit of a sequence of fuzzy relations is discussed from the theoretic view. We find that thetransitivity of limit of a sequence of fuzzy relations is preserved if the involved t-norm and t-conorm are continuous. Also some ideal results are obtained when the sequence is monotonic and the involved t-norm and t-conorm are left continuous t-norm and right continuous t-conorm respectively. In addition, some results are still available without continuity.The research is of important significance for the deep insight into relationships between the transitivity-related properties. At the moment, how to define some particular preference structures reasonably is unresolved. The key problem is how to choose the transitivity properties. The research can serve as a guidance to define fuzzy preference structures.