Generalized roughness of three dimensional (∈ ,∈ q)-fuzzy ideals in terms of set-valued homomorphism

The objective of this study is to generalize the roughness of a fuzzy set in three-dimensional structure by introducing ternary multiplication. Many results and theorems of rough fuzzy ideals have been extended from semigroup and semiring, to ternary semiring by introducing the definition of a rough fuzzy subset of ternary semiring. By using the concept of set-valued homomorphism and strong set-valued homomorphism, it is proved generalized lower and upper approximations of (∈ , ∈ q) ( ∈ , ∈ ∨ q ) -fuzzy ideals (semiprime and prime ideals) of ternary semirings are (∈ ,∈ q) ( ∈ , ∈ ∨ q ) -fuzzy ideals (semiprime and prime ideals) respectively.