Fuzzy bi-quasi interior ideals of Γ-semirings

Abstract

Generalization of ideals and fuzzy ideals of algebraic structures has shown to be both exciting and valuable to mathematics. In this paper, we introduce the notion of a fuzzy bi-quasi interior ideal as a generalization of fuzzy ideals, fuzzy bi-quasi ideals, fuzzy quasi-interior ideals, and fuzzy bi-interior ideals of a $\Gamma$-semiring. We prove that every fuzzy quasi-interior ideal is a fuzzy bi-quasi interior ideal. We also show that union and intersection of fuzzy bi-quasi interior ideals are fuzzy bi-quasi interior ideals. We characterize the regular $\Gamma$-semiring in terms of fuzzy bi-quasi interior ideals and study some of the properties. We prove that if $M$ is a regular $\Gamma$-semiring and $\mu$ is a $\Gamma$-subsemiring, then fuzzy ideal, fuzzy quasi-interior ideal, fuzzy bi-quasi interior ideal, and fuzzy tri-ideal are equivalent.