Fuzzy derivations $(m,n)$-fold $BCC$-ideals on $BCC$-algebras

Fuzzy derivation concepts have been given on some algebra types and their basic properties have been examined. In this paper, we observe that, the notions of derivation $(m,n)$-fold $BCC$-ideals and fuzzy derivation $(m,n)$-fold $BCC$-ideals, for each positive integers $m,~ n$, are indeed the natural generalization of $BCC$-ideals and fuzzy $BCC$-ideals, respectively. A characterization of derivation $(m,n)$-fold $BCC$-ideals and fuzzy derivation $(m,n)$-fold $BCC$-ideals is given, and conditions for which an ideal (respectively fuzzy ideal) is an derivation $(m,n)$-fold $BCC$-ideal (respectively fuzzy derivation $(m,n)$-fold $BCC$-ideal) are studied. We also establish extension properties for derivation $(m,n)$-fold $BCC$ -ideals and fuzzy derivation $(m,n)$-fold $BCC$-ideals. Furthermore, this paper focuses on application on fuzzy left (right) derivation $(m, n)$-fold ideals in $BCC$- algebras, left-right derivation $(m, n)$-fold $BCC$-ideals of a $BCC$-algebra, the homomorphic image and the pre-image of left-right derivation $(m n)$-fold $BCC$-ideals of a $BCC$-algebra, the Cartesian product left-right derivation $(m, n)$-fold $BCC$-ideals of a $BCC$-algebra. There are applications of this work in the field of medicine, engineering, industry, statistics, etc.