Multi – Fuzzy Ideals of Г - Near Ring

Multi – fuzzy set theory is an extension of fuzzy set theory. In this paper, we define the multi fuzzy ideals of Г - near ring. Also, the notion of anti multi fuzzy ideals of a Г - near ring is introduced and investigated some related properties. This concept of multi fuzzy ideals of a Г - near ring is a generalization of the concept of fuzzy ideals in Г - near rings. Also, we define the multi-level subsets and multi anti level subsets of a multi fuzzy sub Г - near ring of a Г-near ring. In this paper we define the multi level subset and multi anti level subset of AUB. The purpose of this study establishes the algebra of multi fuzzy Г-near ring.


Introduction
In 1965, Zadeh [1] proposed the notion of fuzzy set.Later A. Rosenfeld [5] developed fuzzy groups in 1971.The concept of ring, a generalization of a ring in algebra was introduced and studied first by Nobusawa [14] in 1964 and generalization by Barnes [15] in1966.A generalization of both the concepts near ring and the ring namely near ring was introduced by Bh.Satyanarayana [10,11,12], in 1999.They developed theoretically some important concepts in near ring.Later the authors S. Ragamai, Y. Bhargavi, T. Eswarlal [17] developed theory of fuzzy and L fuzzy ideals of near rings.Many authors developed concepts of fuzzy theory and applications in various fields.Many extensions and generalizations of Zadeh's fuzzy set theory are developed so far.But fuzzy set is not enough to study some reality problems.Characterization problems like complete colour characterization of colour images, taste recognition of food items, decision making problems with multi aspects etc. cannot completely be characterized by a single membership function of Zadeh's fuzzy sets.Some of these problems can completely be characterized by multi-membership functions of suitable multi-fuzzy sets.To consider such situations Yager defined a fuzzy bag to be a crisp bag of X x [0,1] in 1986.Miyamoto [16] later redefined it as fuzzy multi sets in 2000.Further studied concept of multi fuzzy sets by Sabu Sabestain [2,3,4] and re defined multi fuzzy sets is a generalisation of theories of fuzzy sets, fuzzy sets and intuitionistic fuzzy sets.K.Hemabala and B. Srinivas kumar[18,19,20] established algebraic properties of neutrosophic multi fuzzy sets.In this paper, we define the multi fuzzy sets of Г -near ring and verified union and intersection of multi fuzzy ideals in Г -near ring.Also, the notion of multi fuzzy ideals of Cartesian product and verified multi anti fuzzy ideals of Гnear ring.We introduced multi anti level fuzziness.

Preliminaries
In this section we site the fundamental definitions that will be used in the sequel.
Definition: Let be a non empty set and be a fuzzy set over is defined by = { } where [0,1].
Definition: Let X be a non empty set.A multifuzzy set in X is defined as a set of ordered sequence = {(,,………….): } where : X→ [0,1] for all Where ………….one can append any number of zeros at the right end of a finite sequence of the membership values of x.

Remarks:
1.If the sequences of the membership functions have only k-terms (finite number of terms), i is called the dimension of A.
2. A = B if and only if An = Bn for all i = 1,2,…….i.Definition: A non -empty set N with two binary operations '+'(addition) and '.'(multiplication) is called a near ring if it satisfies the following axioms ) is a semi group 3 (x + y) .z = x .z + y. z for all x, y, z ϵ N Precisely speaking it is a right nearring, because it satisfies the right distributive law.We will use the word "near-ring" to mean "right near ring".We denote x y instead of x. y.Moreover, a near ring N is said to be a zerosymmetric if r.0=0 for all rϵ N, where 0 is the additive identity in N Definition: Let (R, +) be a group and Г be a non -empty set then R is said to be a Г -near ring if there exists a mapping Rx Г x R R (the image of (x, α, y) is denoted by (x α y) satisfying the following Conditions 1. (x + y) α z= x α z + y α z

Proof:
Let  and  are two multi fuzzy Γ-near rings of ℛ.

Proof:
Let  and  are two multi fuzzy Γ-near ring of ℛ Let ,  ϵ ℛ and  ϵ Г Similarly iif  ⊆  we get  ∪  is a multi fuzzy Γ-near ring of ℛ.
Let A be a multi fuzzy Г -near ring defined by A(x) = (0.9,0.