On Many-Valued Partitions and Many-Valued Equivalence Relations

For a fixed integral commutative cl-monoid M = (L, ≤, *), introducing the notions of M-partitions and M-equivalence relations as the extensions of T-partitions and T- equivalence relations to the integral commutative cl-monoid M = (L, ≤, *), respectively, it is shown that the previous approaches on T-partitions and T-equivalence relations can be unified, and most of the results in these works can be easily stated based on the integral commutative cl-monoid M = (L, ≤, *). Modifying the notion of T-redundancy of finite family of fuzzy sets of a nonempty ordinary set X, it is extended to arbitrary family of L-fuzzy sets on the basis of the integral commutative cl-monoid M, and is shown that the proposed definition has more desirable properties than the previous one. Furthermore, handling the works of Hohle, Klawonn, Gebhardt and Kruse on fuzzy partitions and fuzzy equivalence relations, some of their results are improved, and several new results in this direction are pointed out.