Rough Prime Bi-Γ-hyperideals and Fuzzy Prime Bi-Γ-hyperideals of Γ-semihypergroups

Abstract

In this paper, we introduce the concept of prime bi-Γ-hyperideals, rough prime bi-Γ-hyperideals and fuzzy prime bi-Γ-hyperideals of Γ-semihypergroups. We prove that the lower approximation of a prime bi-Γ-hyperideal is a prime bi-Γ-hyperideal and the upper approximation of a prime bi-Γ-hyperideal is a prime bi-Γ-hyperideal. Also the rough set theory is applied to prime bi-Γ-hyperideals in the quotient Γ-semihypergroups. In the end, the notion of fuzzy prime bi-Γ-hyperideals of Γ-semihypergroups has been introduced, and we proved that a bi-Γ-hyperideal B of a Γ-semihypergroup H is prime (resp. strongly prime) if and only if the characteristic function χ_{B} of B is a fuzzy prime (resp. fuzzy strongly prime) bi-Γ-hyperideal of H.