A Study On T-transitive-consistency And Its Indicator Of Fuzzy Relations

This paper systematically investigates T-transitive-consistency and its indicator of the fuzzy relations.Our main work and results are summarized as follows.Firstly,we briefly review the concepts and properties of crisp relations,fuzzy logic connective operations and fuzzy relations.Secondly,we systematically summarize the research results related to transitive-consistency of crisp relations and discuss their properties.As a consequence,the research of transitive-consistency of crisp relations is completed.Thirdly,we generalize the concept of transitive-consistency of crisp relation to the fuzzy condition by means of t-norms,and discuss the relationships of T-transitive-consistency,T-transitivity,T-quasi-transitivity and T-acyclicity of fuzzy relations.Meanwhile,we make a detailed investigation on the properties of T-transitive-consistency of fuzzy relations.Finally,we give a definition of T-transitive-consistency indicator of fuzzy relation.At the same time,we make a research on the relationships of T-transitive-consistency indicator,T-transitivity indicator,T-quasitransitivity indicator and T-acyclicity indicator of fuzzy relation.Furthermore,some properties of T-transitive-consistency indicator of fuzzy relations are given.It is worth noting that almost all of conclusions concerning transitive-consistency of crisp relation can be extended to fuzzy case and indicator case under our assumptions.However,there are still some conclusions are no longer valid.Thus,we give some examples to illustrate these conditions.In summary,we have a further understanding of T-transitiveconsistency and its indicator of fuzzy relation by our research.They are of great significance to the development of fuzzy decision theory.