Residual Quotient Fuzzy Subset in Near-rings

For any fuzzy subsets λ and μ, we introduce the notion of residual quotient fuzzy subset (λ : μ) and we have characterized residual quotient fuzzy subset in near-rings.


Introduction
In 1965, Zadeh [14] introduced the concept of fuzzy subsets and studied their properties on the parallel lines to set theory.In 1971, Rosenfeld [10] defined the fuzzy subgroup and gave some of its properties.Rosenfeld's definition of a fuzzy group is a turning point for pure mathematicians.Since then, the study of fuzzy algebraic structure has been pursued in many directions such as groups, rings, modules, vector spaces, and so on.In 1981, Das [2] explained the interrelationship between the fuzzy subgroups and their t-level subsets.Fuzzy subrings and ideals were first introduced by Wang-jin Liu [5] in 1982.Subsequently, Mukherjee and Sen [7], Swamy and Swamy [13], Dixit et al. [3], and Rajesh Kumar [4] applied some basic concepts pertaining to ideals from classical ring theory and developed a theory of fuzzy.The notions of fuzzy subnear-ring and ideal were first introduced by Abou-Zaid [1] in 1991.
Wang-jin Liu [6] introduced residual quotient fuzzy subset (λ : μ) for any two fuzzy ideals in rings.In this paper, we introduce residual quotient fuzzy subset (λ : μ) for any two fuzzy subsets, which is different from [6], and we characterize some related results in near-rings.

Preliminaries
We would like to reproduce some definitions and results proposed by the pioneers in this field earlier for the sake of completeness.Definition 2.1.A near-ring N is a system with two binary operations + and • such that (1) (N,+) is a group, not necessarily abelian; (2) (N,•) is a semigroup; (3) (x + y)z = xz + yz for all x, y,z ∈ N.
We will use the word "near-ring" to mean "right distributive near-ring."We denote xy instead of x • y.Note that 0 • x = 0 and (−x)y = −xy but in general x • 0 = 0 for some x ∈ N. Definition 2.2.Let (N,+,•) be a near-ring.A subset I of N is said to be an ideal of N if (1) (I,+) is a normal subgroup of (N,+); (2) If I satisfies ( 1) and ( 2), then it is called a right ideal of N. If I satisfies ( 1) and ( 3), then it is called a left ideal of N.
Let N be a near-ring.Given two subsets A and B of N, the product From now on, throughout this paper N will denote right distributive near-ring, unless otherwise specified.For the basic terminology and notation, we refer to Pilz [9] and Abou-Zaid [1].
A fuzzy subset μ : N → [0,1] is nonempty if μ is not the constant map which assumes the value 0. For any two fuzzy subsets λ and μ of N, λ ≤ μ means that λ(a) ≤ μ(a) for all a ∈ N. The characteristic function of N is denoted by N and, of its subset A is denoted by f A .The image of a fuzzy subset μ is denoted by Im(μ) = {μ(n) | n ∈ N}.Hereafter, we consider only nonempty fuzzy subsets of N. Definition 2.4.Let E be an N-group and let μ be a fuzzy subset of E. Then μ is called a fuzzy N-subgroup of N if for all x, y ∈ E and n ∈ N, (1) μ(x − y) ≥ min{μ(x),μ(y)}; (2) μ(nx) ≥ μ(x).
Definition 2.5.Let μ be any fuzzy subset of N. For t ∈ [0,1], the set Definition 2.6.Let f and g be any two fuzzy subsets of N. Then f ∩ g, f ∪ g, f + g, f g, and f * g are fuzzy subsets of N defined by Definition 2.11.A fuzzy ideal μ of N is called completely fuzzy prime ideal if any two fuzzy points x r , y s of N such that x r y s ∈ μ implies either x r ∈ μ or y s ∈ μ for all x, y ∈ N and for all r,s ∈ [0,1].Definition 2.12.A fuzzy ideal μ of N is called completely fuzzy semiprime ideal if any fuzzy point x r of N such that (x r ) 2 ∈ μ implies x r ∈ μ for all x ∈ N and r ∈ [0,1].Lemma 2.13 [11].Let I be a nonempty subset of N. I is an N-subgroup of N if and only if f I is a fuzzy N-subgroup of N.
Lemma 2.14 [11].Let μ be a fuzzy subset of N. μ is a fuzzy N-subgroup of N if and only if the level subset μ t , t ∈ Im(μ), is an N-subgroup of N. Lemma 2.15 [1].Let I be a subset of N. I is an (left or right) ideal of N if and only if f I is a fuzzy (left or right) ideal of N. Lemma 2.16 [1].Let μ be a fuzzy subset of N. μ is a fuzzy (left or right) ideal of N if and only if the level subset μ t , t ∈ Im(μ), is an ideal of N.
Now we introduce the notion of fuzzy bi-ideal of N. We characterize fuzzy quasi-ideal and fuzzy bi-ideal of N. Definition 2.18 [8] It is very clear that if N is a zero-symmetric near-ring, then μNμ ≤ μ for every fuzzy bi-ideal μ.
Lemma 2.20 [8].Let μ be a fuzzy subset of N. If μ is a fuzzy left ideal (right ideal, Nsubgroup, subnear-ring) of N, then μ is a fuzzy quasi-ideal of N. (2.4) We remark that if x is not expressed as Lemma 2.21.For any nonempty subsets A and B of N, (1) Proof.Proof of (1) can easily be seen in [8].
Conversely, let us assume that f Q is a fuzzy quasi-ideal of N. Let y be any element of QNQ ∩ QN * Q.Then, we have (2.5) Lemma 2.23.Any fuzzy quasi-ideal of N is a fuzzy bi-ideal of N.
Proof.Let μ be any fuzzy quasi-ideal of N.Then, we have ( Hence, μ is a fuzzy bi-ideal of N.
However, the converse of Lemma 2.23 is not true. (2.9) Similarly, we have (Nμ)(b) = 1.Thus, Lemma 2.25.Let μ be a fuzzy subset of N. If μ is a fuzzy left ideal (right ideal, N-subgroup, subnear-ring) of N, then μ is a fuzzy bi-ideal of N.
Proof.As μ is a left ideal of N and Lemma 2.20, μ is a fuzzy quasi-ideal of N. Hence by Lemma 2.23, μ is a fuzzy bi-ideal of N.
Theorem 2.26.Let μ be a fuzzy subset of N. If μ is a fuzzy quasi-ideal of N, if and only if μ t is a quasi-ideal of N, for all t ∈ Im(μ).

Residual quotient fuzzy subset in N
Lemma 3.1.If μ is a fuzzy left ideal of N, then μ(n 0 x) ≥ μ(x) for all x ∈ N and n 0 ∈ N 0 .
Lemma 3.2.N is zero-symmetric near-ring if and only if each fuzzy left ideal of N is a fuzzy N-subgroup of N.
Proof.Assume that N = N 0 .Let μ be a fuzzy left ideal of N. As μ is a fuzzy left ideal of N, by Lemma 3.1, μ(n 0 x) ≥ μ(x) for all x ∈ N and n 0 ∈ N 0 = N.Thus, μ is a fuzzy Nsubgroup of N.
Conversely, let us assume that each fuzzy left ideal of N is a fuzzy N-subgroup of N. Let L be a left ideal of N.Then, f L is a fuzzy N-subgroup of N.This implies f L (nx) ≥ f L (x) for all n,x ∈ N. In particular, x ∈ L and n ∈ N, then NL ⊆ L. Taking L as {0}, we have N{0} ⊆ {0}.This implies N • 0 = {0} and hence N = N 0 .Now we introduce the notion of residual quotient fuzzy subset (λ : μ) for any two fuzzy subsets λ and μ and annihilator, ann(μ), of fuzzy subset μ of N. Definition 3.3.Let λ and μ be any two fuzzy subsets of N. The residual quotient fuzzy subset (λ : μ) of N is defined as where (λ t : Definition 3.4.Let O be a fuzzy subset defined as O(0) = 1 and O(x) = 0 for all x = 0 ∈ N. Then (O : μ) is the annihilator of μ and it is denoted by ann(μ).
Remark 3.7.Let λ and μ be any two fuzzy subsets of N. If λ is a fuzzy left ideal of N, then (λ : μ) is not necessarily fuzzy ideal of N as the following example shows.Now we find the conditions under which (λ : μ) is a fuzzy ideal of N. Theorem 3.9.Let λ and μ be any two fuzzy subsets of N. If λ is a fuzzy left ideal and μ is a fuzzy N-subgroup of N, then (λ : μ) is a fuzzy ideal of N.
Proof.Let λ be a fuzzy completely semiprime ideal of N. Let t = 0 ∈ Im(λ).Let x ∈ N such that x 2 ∈ λ t .This implies (x t ) 2 ∈ λ.As λ is a fuzzy completely semiprime ideal of N, x t ∈ λ.This implies x ∈ λ t .
Conversely, let us assume that λ t is a completely semiprime ideal of N, t ∈ Im(λ).Suppose y s 2 ∈ λ.Then, y 2 ∈ λ s and y s ∈ λ.Thus, λ is a fuzzy completely semiprime ideal of N. Theorem 3.14.Let λ be a fuzzy ideal of N. If a r b s ∈ λ implies b s a r ∈ λ for any fuzzy points a r ,b s of N, then for any fuzzy subset μ of N, (λ : μ) is a fuzzy ideal of N.

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First
Round of ReviewsMarch 1, 2009 Definition 2.7.For any x ∈ N and t ∈ (0,1], define a fuzzy point x t as If x t is a fuzzy point and μ is any fuzzy subset of N and x t ≤ μ, then we write x t ∈ μ.Note that x t ∈ μ if and only if x ∈ μ t where μ t is a level subset of μ.If x r and y s are fuzzy points, then x r y s = (xy) min{r,s} ., then it is called a fuzzy left ideal of N. If μ is both fuzzy right and fuzzy left ideal of N, then μ is called a fuzzy ideal of N.An ideal P of N is called completely prime if any two elements a, b of N such that ab ∈ P implies either a ∈ P or b ∈ P.An ideal P of N is called completely semiprime if any element a of N such that a 2 ∈ P implies a ∈ P for all x ∈ N.