On the role of transitivity in fuzzy relational calculus

The formulation of the transitivity property of a fuzzy relation usually depends on the choice of a particular triangular norm T. Applications of this notion of T-transitivity in various fields such as clustering, control and preference modelling are identified. A fuzzy counterpart of the classical one-to-one correspondence between equivalence relations and partitions is discussed, shedding a Eight on the relationships between fuzzy rule-based systems and fuzzy controllers. The non-triviality of the mathematical study of additive fuzzy preference structures is illustrated by means of some transitivity-related properties of fuzzy preference relations.