A Study On The Relationships Of Some Indicators Of Transitivity-Related Properties Of Fuzzy Relations

In this thesis, we study the relationships between some indicators of transitivity-related properties of fuzzy relations under the condition that the involved t-norm T is continuous. Our main research and results are summarized as follows.Firstly, we review the relationships between the properties of fuzzy relations including T-transitivity, negative S-transitivity, T-S-semitransitivity, T-S-Ferrers, consistency, strong transitivity etc., which are extensiv elyemployed in fuzzy preference-based decision-making analysis.Secondly, motivated by the definitions of the indicators of reflexivity, completeness, strong completeness, we complemently define the indicators of irreflexivity, S-completeness, strong S-completeness, T-antisymmetry, antisymmetry, negative S-transitivity, T-S-semitransitivity, T-S-Ferrers, consistency and strong transitivity. Next, our mian work is to discuss the relationships between the indicators of transitivity-related properties. The research is divided into two parts. In the first part, we firstly investigate the relationships of the indicators of the properties of fuzzy relations under the operations of inverse, complement, intersection and union without requiring any links between t-norm T and t-conorm S, and then, we study the relationships between the indictors of transitivity-related properties. In the second part, we assume that (T,S,(?)) is a De Morgan triple. In this case, we discuss the relationship of some indicators of properties of fuzzy relation and the complement of a fuzzy relation. With our research, some conclusions obtained by Fodor and Wang are extended. In addition, We also provide some counterexample to illustrate the results which can not be extended.