FUZZY-SETS, FUZZY RELATIONS, AND FUZZY GROUPS - SOME INTERRELATIONS

We define closure operator on the set of fuzzy relations on S and show that the closed hull of a fuzzy relation (S, mu) is given by GAMMA(S, mu) = (S, mu(sigmamu)). We define strong fuzzy relation and derive so me results to investigate the relationship between closed, strong, and symmetric fuzzy relations. We give a counterexample to show that the composition of two closed fuzzy relations need not be a closed fuzzy r elation. We also give a much simpler proof of Theorem 4.2 of Bhattacha rya and Mukherjee and provide counterexamples to show that their Lemma 3.4 and Theorem 3.1 of Das do not always remain true.