Vague Bi-Quasi-Interior Ideals of Γ -Semirings

. In this paper, we introduce and study the concept of vague bi-quasi-interior ideals of Γ - semirings as a generalization of vague bi-ideals, vague quasi-ideals, vague interior ideals, vague bi-quasi-interior ideals, and vague bi-quasi-interior ideals of Γ -semirings.


Introduction
In 1965, Zadeh [13] introduced the study of fuzzy sets.Mathematically, a fuzzy set in a non-empty set X is a mapping µ from X into the interval [0, 1]; for x in X, µ(x) is called the membership of x belonging to X.This membership function gives only an approximation for belonging, but it does not give any information about not belonging.To counter this problem and obtain a better estimation and analysis of data decision-making, Gau and Buehrer [7] have initiated the study of vague sets with the hope that they form a better tool to understand, interpret and solve real-life problems.Further, in 1995, Rao [10] introduced the concept of Γ-semirings which is a generalization of Γ-rings, ternary semirings, and semirings, and after that, he introduced and studied the ideals of Γ-semirings.
Ideals play an important role in advanced studies and using algebraic structures.Generalization of ideals in algebraic structures is necessary for further study of algebraic structures.Many mathematicians proved important results and characterization of algebraic structures by using the concept and the properties of a generalization of ideals in algebraic structures.Rao et al. [9,11,12] introduced the concepts of left (resp., right) bi-quasi-ideals and bi-interior ideals of Γ-semirings and studied the properties of left bi-quasi ideals.However, Bhargavi et al. [1][2][3][4][5][6]8] developed the theory of vague sets on Γ-semirings.
This paper is a sequel to our study.We introduce and study the concept of vague bi-quasi-interior ideals of Γ-semirings as a generalization of vague bi-ideals, vague quasi-ideals, and vague interior ideals and characterize the vague bi-quasi-interior ideals of Γ-semirings to the crisp bi-quasi-interior ideals of Γ-semirings.

Preliminaries
In this section, we recall some of the fundamental concepts and definitions which are necessary for this paper.
Throughout this paper, GSR stands for a Γ-semiring, VGSR stands for a vague Γ-semiring, VBQII stands for a vague bi-quasi-interior ideal, and VI stands for a vague ideal.

Vague Bi-Quasi-Interior Ideals of Γ-Semirings
In this section, we introduce and study VBQII as a generalization of vague bi-ideals, vague quasiideals, and vague interior ideals of GSRs and characterize the vague bi-quasi-interior ideals of GSRs to crisp bi-quasi-interior ideals of GSRs.
Throughout this section, Ṙ stands for a GSR with unity, and δ stands for the vague characteristic set of Ṙ unless otherwise mentioned.
Every VBQII of Ṙ need not be a vague bi-ideal, a vague quasi-ideal, a vague interior ideal, a bi-interior ideal, and a bi-quasi-ideals of Ṙ in general.
Example 3.1.Let Ṙ be the set of all negative integers and Γ be the set of all negative even integers.
Then Ṙ and Γ are additive commutative semigroups.Define the mapping and Then Φ is a VBQII of Ṙ.
Theorem 3.2.Every vague interior ideal of Ṙ is a VBQII of Ṙ.
Theorem 3.4.Every right VI of Ṙ is a VBQII of Ṙ.
Proof.The proof is similar to the above theorem.
Corollary 3.1.Every VI of Ṙ is a vague bi-interior ideal of Ṙ.
Theorem 3.5.The intersection of a VBQII and a right VI of Ṙ is always a VBQII of Ṙ.
Proof.Let Φ be a VBQII and ξ be a right VI of Ṙ.
Theorem 3.6.The intersection of a vague bi-ideal and a vague interior ideal of Ṙ is always VBQII of Ṙ.
Proof.Let Φ be a vague bi-ideal and ξ be a vague interior ideal of Ṙ.
Proof.Let Φ and ξ be vague bi-quasi-interior ideals of Ṙ.

Definition 2 . 5 .
The functions t Φ and f Φ are called true membership function and false membership function, respectively.For a vague set Φ