The (m, n)-fuzzy set and its application in BCK-algebras

The concept of the (m, n)-fuzzy set is introduced and compared with other types of fuzzy sets. Some operations for the (m, n)-fuzzy set are introduced, and their properties are investigated. We define (m, n)-fuzzy subalgebras in BCK-algebras and BCI-algebras and study their properties. A given (m, n)-fuzzy subalgebra is used to create a new (m, n)-fuzzy subalgebra. The intersection of two (m, n)-fuzzy subalgebras to be a (m, n)-fuzzy subalgebra is proved, and an example is given to show that the union of two (m,n)-fuzzy subalgebras may not be a (m, n)-fuzzy subalgebra. The (m,n)-cutty set is used to obtain the characterization of (m, n)-fuzzy subalgebra. The homomorphic image and preimage of (m, n)-fuzzy subalgebra is discussed. It turns out that intuitionistic fuzzy subalgebra is a subclass of (m, n)-fuzzy subalgebra.