Introducing strong T-transitivity in approximate fuzzy preorders and equivalences

Any fuzzy preorder or equivalence is or is not a fuzzy preorder or equivalence. Through these pages we present two ways of regarding any arbitrary fuzzy relation as a fuzzy preorder or equivalence, at least to some extent. The two ways are the axiomatic approach, which deals with relaxed versions of reflexivity, symmetry and T-transitivity, and the similarity based approach, which looks into the proximity between a given arbitrary relation and a prototype - a fuzzy preorder or equivalence in the standard fuzzy sense. The relationship between the two views on the problem is studied. As a result, strong-T-transitivity is introduced and shown to be a more suitable choice than standard T-transitivity.