T-transitive closures, openings and approximations of similarity relations

Given a t-norm T belonging to the class of copulas, we propose three algorithms operating on a similarity relation R on a finite universe: an algorithm for computing the T-transitive closure of R, an algorithm that nearly always generates a T-transitive opening of R, and an algorithm for approximating R by a T-transitive similarity relation or T-equivalence that is ´close´ to it. The latter method consists of a cascade of T´-closure and T´-opening operations, where the t-norm T´ gradually progresses to T