Weak T -transitivity and weak closures of interval-valued fuzzy relations

In this paper a new weak transitive property for interval-valued fuzzy relations (IVFRs) is introduced. Weak T -transitivity is equivalent to T -transitivity when the IVFR is a fuzzy relation and T = (T, T) for a triangular norm T. Otherwise it is a weaker property than T -transitive one relaxing the need that all intervals must be comparable and must satisfy all the transitive inequalities by just the need of not having comparable non T -transitive cycles. This paper also defines a weak concept of closure, and it is proved that it exists is just one T -transitive and weak T -transitive closure, it does not exists a T -transitive weak closure, but there are many weak T -transitive weak closures of an IVFRs.