Generalized Anti Fuzzy Ideals in Near-Rings

In this paper, we generalize anti-fuzzy ideals of near-rings, introduce the concept of (<,< ) ∨Υ -fuzzy ideals, prime (<,< ) ∨Υ fuzzy ideals, semiprime (<,< ) ∨Υ -fuzzy ideals of near-rings and discuss some properties of such ideals.


Introduction
The notion of near-ring was first introduced by Dickson and Leonard in 1905 [1]. We note that the ideals of near-rings play a central role in the structure theory, however, they do not in general coincide with the usual ring ideals of a ring.
The concept of a fuzzy set was first introduced by Zadeh [2]. Fuzzy set theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. The fuzzy algebraic structures play a major role in mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, real analysis, measure theory etc. The notions of fuzzy subnear-ring and ideal were first introduced by Abou-Zaid in [3]. In Hong et al. [4] and Kim et al. [5] introduced the concept of fuzzy ideals of near-ring and obtained some useful results in near-rings.
In Biswas [6] introduced the concept of anti-fuzzy subgroups of groups, Kim and Jun studied the notion of anti-fuzzy R-subgroups of near-ring in [7], and Kim et al. studied the notion of anti-fuzzy ideals in near-rings in [8].
Indian Journal of Science and Technology | Print ISSN: 0974-6846 | Online ISSN: 0974-5645 www.indjst.org | Vol 6 (8) | August 2013 We will use the word "near-ring" to mean "right distributive near-ring" and xy instead of x. y. Note that 0x = 0 and (−x)y = xy but in general 0x ≠ 0 for some x ∈ N. The sets N 0 = {x ∈ N | x0 ∈ 0} and N c = {x ∈ N | x0 ∈ x} = {x ∈ N | xy = x, for all y ∈ N} are called zero symmetric and constant part of N, respectively. N 0 and N c are near-rings. .
If A satisfies (1) and (2), then it is called a right ideal of N.
If A satisfies (1) and (3), then it is called a left ideal of N.
In what follows, N will denote right distributive near-ring, unless otherwise specified. If S and T are subsets of N, we denote the set { : , ) st s S t T ∈ ∈ by ST. An ideal P of N is called a prime ideal if for any ideal I and J of N, IJ P ⊆ implies either I P ⊆ or J P ⊆ . An ideal I of N is called a semi-prime ideal if for any ideal J of N, J I 2 ⊆ implies that J I ⊆ . A fuzzy subset f of a universe set X is a function from X into the unit closed interval f g ≤ means that, for all x X ∈ , f x g x ( ) ( ) ≤ . Through out this paper max, min, sup, and inf are used for maximum, minimum, sup remum and inf imum respectively.
x ∈ X. f c is called the complement of f, f ∩ g is called the intersection of f and g, and f ∪ g is called the union of f and g. Let A be a subset of X the characteristic function of A is the function C A of X into {0, 1} defined by A fuzzy set f in X is said to be non unit if there exists x ∈ X such that f(x) < 1. Definition 4. [16] An anti-fuzzy point x s is said to beside to (resp. be non-quasi coincident with) a fuzzy set f, denoted by . We say that < (resp. ϒ) is a beside to (resp. non-quasi coincident with) relation between anti-fuzzy points and fuzzy sets.
does not hold.

Definition 5. [8]
A fuzzy subset f of a near-ring N is called an anti-fuzzy subnear-ring of N if for all x, y ∈ N:  (1), (2) and (3), then it is called an anti-fuzzy right ideal of N. If f satisfies (1), (2) and (4), then it is called an anti-fuzzy left ideal of N.

Definition 7.
For any two anti-fuzzy ideals f and g of N. The product f g  is defined by N y x ∈ and s 1 , s 2 ∈ 0,1).

Definition 9.
A fuzzy set f of N is called a (<, <)-fuzzy right(resp, left) ideal of N if for all x y N , , ∈ and s 1 , s 2 ∈ 0,1).
A fuzzy set which is a (<, <)-fuzzy right and left ideal of N is called a (<, <)-fuzzy ideal of N. < and hence f is a (<,<)-fuzzy subnearring of N.
Then, clearly (N, +, .) is a near-ring. Now, let define f as: For this fuzzy subset f, let 0 This completes the proof.
Then, a f f a x y a xy N + − ∈ . This completes the proof.

Theorem 7: Let M be a subnear-ring of N and f a fuzzy set of N such that:
Theorem 8: Let I be a right ideal of N and f a fuzzy set of N such that: (b) Now, let x y N , ∈ and s ∈0,1) be such that x f s < . Then, f(x) ≤ s. Thus, x ∈ I and y ∈ N, and so y + (c) Now, let x y N , ∈ and s ∈0,1) be such that x s < f. Then, f(x) ≤ s. Thus, x ∈ I and y ∈ N, and so xy ∈ I, since I is right ideal of N. Consequently f(xy) ≤ 0.5. If s ≥ 0.5, then f(xy) ≤ 0.5 ≤ s and so (xy) s < f. If s < 0.5, then f(xy) + s < 0.5 + 0.5 = 1 and so (xy) 5 ∨ϒf . Therefore, ( ) xy f s < ∨ϒ . Hence, f is a (<, <)-fuzzy right ideal of N. This completes the proof. (c) Now, let x y a N , , ∈ and s ∈0,1) be such that a A s ∈ . Then, f a s ( )≤ . Thus, a ∈ I and x y N , ∈ , and so x y a xy I ( ) (a) x s 1 and y f

Theorem 9: Let I be a left ideal of N and f a fuzzy set of N such that:
,0.5} , for all x y N , ∈ and s s 1 2 ,   ∈ and x f y x y f s s . This completes the proof.
Proof: Let f be a (<, <)-fuzzy left ideal of N such that (c) and, f xy f xy f xy Hence, f is a (<, <)-fuzzy right ideal of N. This completes the proof.
The remaining proof follows from Theorem 16.
For any fuzzy set f in N and . Then, Thus .    Conversely, suppose that f is a fuzzy set in N such that Proof: Let f is a <,< ∨ ( )      In this section, we describe semiprime and prime ( , ) < < ∨ -fuzzy ideals of near-rings and investigate some properties of these ideals.
Theorem 24. Let f be a prime <,< ∨ ( ) ϒ -fuzzy ideal of N. Then, the set : is a prime ideal of N.
Proof: The proof follows from Theorem 5.
is a semiprime ideal of N. implies that xy P ( )∈ . Since, P is a prime, so P x ∈ or P y ∈ , which implies that ( ) 0.5

Conclusion
In the study of the structure of an anti-fuzzy algebraic system, we notice that anti-fuzzy ideals with special properties always play an important role. In this paper, we define ) < (<, ϒ ∨ -fuzzy subnear-ring(left, right two-sided) ideals of near-rings and investigated the relationship among these generalized anti-fuzzy subnear-ring (left, right) ideals of near-rings. Finally, we define prime(semiprime) ) < (<, ϒ ∨ -fuzzy ideals of near-rings and investigated some important results. Some characterization theorems of prime(semiprime) ) < (<, ϒ ∨ -fuzzy ideals of near-rings are obtained. We hope that the research along this direction can be continued, and in fact, this work would serve as a foundation for further study of the theory of near-ring, it will be necessary to carry out more theoretical research to establish a general framework for the practical application. In our future study of anti-fuzzy structure of near-rings, may be the following topics should be considered: (1): To focus on other types of near-ring with similar nature and study their mutual relationships; (2): To establish an antifuzzy spectrum of near-ring; (3): To consider the structure of quotient near-ring by using ) < (<, ϒ ∨ -fuzzy subnearring (left, right) ideals; (4): To describe fuzzy soft near-ring and its applications; (5): To describe anti-fuzzy soft nearrings and its applications in informations sciences and general systems.