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

Abstract

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.