A study on the transitivity of probabilistic and fuzzy relations

Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for important particular cases. We also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables. We focus on uniform continuous random variables.

►We consider fuzzy relations and the associated probabilistic relations.► We study the connection between the transitivity of fuzzy and probabilistic relations.► Special attention is paid to the most important operators defining transitivity.►The transitivity verified if the alternatives are uniformly distributed r.v. is studied.