A Fuzzy-Equality-Based Index of T-Transitivity for Fuzzy Relations

In this paper, we introduce a novel index of transitivity for fuzzy relations based on fuzzy equality. The proposed index evaluates the supremal uniform fuzzy-equality proximity between a given relation and the family of exactly transitive relations induced by a continuous triangular norm. We establish several fundamental properties of the proposed index, including faithfulness, stability with respect to fuzzy equality, invariance under inverse relations, monotonicity under restriction, and bounds derived from transitive closures. Furthermore, we prove that the proposed index is attainable on finite universes. A detailed comparison with the traditional degree of transitivity demonstrates that the new index always provides an evaluation no smaller than the classical measure and can distinguish different aspects of approximate transitivity. Finally, we characterize the special case in which the two indices coincide for all fuzzy relations and show that this occurs precisely for the Gödel triangular norm.