# A Study on Fuzzy Ideals ofN-Groups

Authors: B. Davvaz; O. Ratnabala Devi
Journal: Algebra
 (2013)

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## Abstract

Using the idea of the new sort of fuzzy subnear-ring of a near-ring, fuzzy subgroups, and their generalizations defined by various researchers, we try to introduce the notion of (ϵ,ϵ∨q)-fuzzy ideals of N-groups. These fuzzy ideals are characterized by their level ideals, and some other related properties are investigated.


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## Body

## 1. Introduction and Basic Definitions

The concept of a fuzzy set was introduced by Zadeh [1] in 1965, utilizing what Rosenfeld [2] defined as fuzzy subgroups. This was studied further in detail by different researchers in various algebraic systems. The concept of a fuzzy ideal of a ring was introduced by Liu [3]. The notion of fuzzy subnear-ring and fuzzy ideals was introduced by Abou-Zaid [4]. Then in many papers, fuzzy ideals of near-rings were discussed for example, see [5–11]. In [12], the idea of fuzzy point and its belongingness to and quasi coincidence with a fuzzy set were used to define (α,β)-fuzzy subgroup, where α, β take one of the values from {ϵ,q,ϵ∧q,ϵ∨q}, α≠ϵ∧q. A fuzzy subgroup in the sense of Rosenfeld is in fact an (ϵ,ϵ)-fuzzy subgroup. Thus, the concept of (ϵ,ϵ∨q)-fuzzy subgroup was introduced and discussed thoroughly in [7]. Bhakat and Das [13] introduced the concept of (ϵ,ϵ∨q)-fuzzy subrings and ideals of a ring. Davvaz [14, 15], Narayanan and Manikantan [16], and Zhan and Davvaz [17] studied a new sort of fuzzy subnear-ring (ideal and prime ideal) called (ϵ,ϵ∨q)-fuzzy subnear-ring (ideal and prime ideal) and gave characterizations in terms of the level ideals. In [18, 19], the idea of fuzzy ideals of N-groups was defined, and various properties such as fundamental theorem of fuzzy ideals and fuzzy congruence were studied, respectively. In the present paper, we extend the idea of (ϵ,ϵ∨q)-fuzzy ideals of near-rings to the case of N-groups and introduce the idea of fuzzy cosets with some results.We first recall some basic concepts for the sake of completeness.By anear-ring we mean a nonempty set N with two binary operations “+” and “·” satisfying the following axioms:(i)
(N,+) is a group,(ii)
(N,·) is a semigroup,(iii)
(x+y)·z=x·z+y·z for all x,y,z∈N.It is in fact a right near-ring because it satisfies the right distributive law. We will use the word “near-ring” to mean “right near-ring.” N is said to be zero symmetric if 0·x=x·0=0 for all x∈N. We denote x·y by xy.Note that the missing left distributive law,x·(y+z)=x·y+x·z, has to do with linearity if x is considered as a function.Example 1.
Let𝒢 be a group, and let M(𝒢) be the set of all mappings from 𝒢 into 𝒢. We define + and · on M(𝒢) by
(1)(f+g)(x):=f(x)+g(x),(f·g)(x):=f(g(x)).
Then, (M(𝒢),+,·) is a near-ring.Just in the same way asR-modules or vector spaces are used in ring theory, N-groups are used in near-ring theory.By anN-group we mean a nonempty set G together with a map Φ:N×G→G written as Φ(n,g)=ng satisfying the following conditions:(i)
(G,+) is a group (not necessarily abelian),(ii)
(n1+n2)g=n1g+n2g,(iii)
(n1n2)g=n1(n2g) for all n1,n2∈N, g∈G.Example 2.
LetN be a subnear-ring of M(𝒢). Then, 𝒢 is an N-group via function application as operation.Example 3.
The additive group(N,+) of a near-ring (N,+,·) is an N-group via the near-ring multiplication.AnidealI of N-group G is an additive normal subgroup of G such that NI⊆I and n(g+h)-ng∈I for all h∈I, g∈G, n∈N. A mapping between two N-groups G and G′ is called an N-homomorphism if f(g+h)=f(g)+f(h) and f(ng)=nf(g) for all g,h∈G, n∈N.Throughout this study, we useN to denote a zero-symmetric near-ring and G to denote an N-group.For any fuzzy subsetA of G, ImA={A(x)∣x∈G} denotes the image of A. For any subset I of G, χI denotes the characteristic function of I.Definition 4 (see [2]).
A fuzzy subsetA of a group G is called a fuzzy subgroup of G if it satisfies the following conditions:  (i)
A(x+y)≥min{A(x),A(y)},(ii)
A(-x)≥A(x),for allx,y∈G.Definition 5.
For a fuzzy subsetA of G, t∈(0,1], the subset At={x∈G∣A(x)≥t} is called a level subset of G determined by A and t.The set{x∈G∣A(x)>0} is called the support of A and is denoted by SuppA. A fuzzy subset A of G of the form
(2)A(y)={t(≠0)ify=x,0ify≠x
is said to be a fuzzy point denoted by xt. Here x is called the support point, and t is called its value. A fuzzy point xt is said to belong to (resp., quasi coincident with) a fuzzy set A written as xtϵA (resp., xtqA) if A(x)≥t (resp., A(x)+t>1). If xt∈A or xtqA, then we write xt∈∨qA. The symbols xt∈¯A, xtq¯A,xt∈∨qA¯ mean that xt∈A, xtqA,xt∈∨qA do not hold, respectively.Definition 6 (see [7, 12]).
A fuzzy subset of a groupG is said to be an (ϵ,ϵ∨q)-fuzzy subgroup of G if for all x,y∈G and t,r∈(0,1],  (i)
xt,yr∈A⇒(x+y)min{t,r}∈∨qA,(ii)
xt∈A⇒(-x)t∈∨qA.Remark 7 (see [7]).
The conditions (i) and (ii) of Definition6 are respectively equivalent to(i)
A(x+y)≥min{A(x),A(y),0.5},(ii)
A(-x)≥min{A(x),0.5},for allx,y∈G.Remark 8.
For any(ϵ,ϵ∨q)-fuzzy subgroup A of G such that A(x)≥0.5 for some x∈G, then A(0)≥0.5 and if A(0)<0.5, then A(x)<0.5 for all x∈G. So, A is just the usual fuzzy subgroup in the sense of Rosenfeld.Remark 9.
It is noted that ifA is a fuzzy subgroup then it is an (ϵ,ϵ∨q)-fuzzy subgroup of G. However the converse may not be true.Here onwards we assume thatA is an (ϵ,ϵ∨q)-fuzzy subgroup in the nontrivial sense for which case we have A(0)≥0.5.Definition 10 (see [7]).
An(ϵ,ϵ∨q)-fuzzy subgroup of a group G is said to be (ϵ,ϵ∨q)-fuzzy normal subgroup if for any x,y∈G and t∈(0,1],
(3)xt∈A⟹(y+x-y)t∈∨qA.Remark 11 (see [7]).
The condition of(ϵ,ϵ∨q)-fuzzy normal subgroup is given in the equivalent forms as(i)
A(y+x-y)≥min{A(x),0.5},(ii)
A(x+y)≥min{A(y+x),0.5},(iii)
A([x,y])≥min{A(x),0.5}, for all x,y∈G.
Here[x,y] denotes the commutator of x, y in G.In the light of this fact, the condition of Definition10 can be replaced by any one of the above conditions in Remark 8.Definition 12 (see [18]).
LetA be a fuzzy subset of an N-group G. It is called a fuzzyN-subgroup ofG if it satisfies the following conditions:  (i)
A(x+y)≥min{A(x),A(y)},(ii)
A(nx)≥A(x),for allx,y∈G, n∈N.Remark 13.
IfG is a unitary N-group, the above conditions are equivalent to conditions A(x-y)≥min{A(x),A(y)} and A(nx)≥A(x) for all x,y∈G, n∈N.Definition 14 (see [18, 19]).
A nonempty fuzzy subsetA of an N-group G is called a fuzzy ideal if it satisfies the following conditions:(i)
A(x-y)≥min{A(x),A(y)},(ii)
A(nx)≥A(x),(iii)
A(y+x-y)≥A(x),(iv)
A(n(x+y)-nx)≥A(y),for allx,y∈G, n∈N.Definition 15 (see [14]).
A fuzzy setA of a near-ring N is called an (ϵ,ϵ∨q)-fuzzy subnear-ring of N if for all t,r∈(0,1], and x,y∈N(i)
(a)xt,yr∈A⇒(x+y)min{t,r}∈∨qA,(b)xt∈A⇒(-x)t∈∨qA,(ii)
xt,yr∈A⇒(xy)min{t,r}∈∨qA.A is called an  (ϵ,ϵ∨q)-fuzzy ideal of N if it is (ϵ,ϵ∨q)-fuzzy subnear-ring of N and(iii)
xt∈A⇒(y+x-y)∈∨qA,(iv)
yr∈A,x∈N⇒(yx)r∈∨qA,(v)
at∈A⇒(y(x+a)-yx)t∈∨qA,for allx,y,a∈N.


## 2. Generalized Fuzzy Ideals

In this section, we give the definition of(ϵ,ϵ∨q)-fuzzy subgroup and ideal of an N-group G based on Definitions 14 and 15.Definition 16.
A fuzzy subsetA of an N-group G is said to be an (ϵ,ϵ∨q)-fuzzy subgroup of G if x,y∈G, n∈N, t,r∈(0,1],(i)
xt,yr∈A⇒(x+y)min{t,r}∈∨qA,(ii)
xt∈A⇒(-x)t∈∨qA,(iii)
xt∈A, n∈N⇒(nx)t∈∨qA.Lemma 17.
LetA be a fuzzy subset of G and t,r∈(0,1]. Then,(i)
xt,yr∈A⇒(x+y)min{t,r}ϵ∨qA⇔A(x+y)≥min{A(x),A(y),0.5},(ii)
xt∈A⇒(-x)t∈∨qA⇔A(-x)≥min{A(x),0.5}, for all x∈G,(iii)
xt∈A⇒(nx)tϵ∨qA⇔A(nx)≥min{A(x),0.5}, for all x∈G, n∈N.Proof.
(i) Letx,y∈G. Consider the case (a): min{A(x),A(y)}<0.5.
Assume thatA(x+y)<min{A(x),A(y),0.5}=min{A(x),A(y)}. Choose t such that A(x+y)<t<min{A(x),A(y)} which implies that xt∈A, yt∈A but (x+y)t∈∨qA¯ [as A(x+y)+t<1 and A(x+y)<t]. Consider the case (b): min{A(x),A(y)}≥0.5. Assume that A(x+y)<min{A(x),A(y),0.5}=0.5. Choose t such that A(x+y)<t<0.5 so that xt,yt∈A but (x+y)t∈∨qA¯.
Conversely, letxt,yr∈A⇒A(x)≥t,A(y)≥r. Then, A(x+y)≥min{A(x),A(y),0.5}≥min{t,r,0.5}. Thus A(x+y)≥min{t,r} if either t or r≤0.5 and A(x+y)≥0.5 if both t and r>0.5 which means (x+y)min{t,r}ϵ∨qA.
(ii) Letx∈G, min{A(x),0.5}≤0.5. Suppose A(-x)<min{A(x),0.5}≤0.5. Choose r such that A(-x)<r<min{A(x),0.5}≤0.5. Then, xr∈A but (-x)r∈∨qA¯ which contradicts the hypothesis. So, A(-x)≥min{A(x),0.5} for all x∈G.
Conversely, letxt∈A. Then, A(x)≥t. But we have A(-x)≥min{A(x),0.5}≥min{t,0.5}⇒A(-x)≥t or A(-x)≥0.5 according as t≤0.5 or t>0.5⇒(-x)t∈∨qA.
(iii) Letx∈G and min{A(x),0.5}≤0.5. Suppose A(nx)<min{A(x),0.5}≤0.5. Choose r such that A(nx)<r<min{A(x),0.5}≤0.5. Then, A(x)>r that is, xr∈A, but (nx)r∈∨qA¯ as A(nx)<r and A(nx)+r≤1.
Conversely letxt∈A, n∈N; then A(x)≥t. But A(nx)≥min{A(x),0.5}≥min{t,0.5}⇒A(nx)≥t or A(nx)≥0.5 according as t≤0.5 or r>0.5⇒A(nx)≥t or A(nx)+t>1⇒(nx)t∈∨qA.Theorem 18.
LetA be a fuzzy subset of G. Then, A is an (ϵ,ϵ∨q)-fuzzy subgroup of G if and only if the following conditions are satisfied:  (i)
A(x+y)≥min{A(x),A(y),0.5},(ii)
A(-x)≥min{A(x),0.5},(iii)
A(nx)≥min{A(x),0.5},for allx,y∈G, n∈N.Proof.
It follows from the previous lemma.Definition 19.
A fuzzy subsetA of an N-group G is said to be (ϵ,ϵ∨q)-fuzzy ideal of G if it is an (ϵ,ϵ∨q)-fuzzy subgroup and satisfies the following conditions:(i)
xt∈A⇒(y+x-y)t∈vqA,(ii)
at∈A⇒(n(x+a)-nx)t∈vqA,for anyn∈N, x,a∈G.Lemma 20.
LetA be a fuzzy subset of G and t,r∈(0,1]. Then,(i)
xt∈A⇒(y+x-y)t∈vqA⇔A(y+x-y)≥min{A(x),0.5},(ii)
at∈A⇒(n(x+a)-nx)t∈vqA⇔A(n(x+a)-nx)≥min{A(x),0.5}.Proof.
(i) Assume thatA(y+x-y)<min{A(x),0.5}. Choose t such that A(y+x-y)<t<min{A(x),0.5}. But min{A(x),0.5}≤A(x) or 0.5 according as A(x)<0.5 or A(x)≥0.5. So, A(x)>t or A(x)≥0.5⇒xt∈A or x0.5∈A. But (y+x-y)t∈vqA¯ or (y+x-y)0.5∈vqA¯, respectively, which contradicts the hypothesis.
Conversely, assume thatxt∈A, then A(x)≥t. For any y∈G, we have A(y+x-y)≥min{A(x),0.5}≥min{t,0.5}⇒A(y+x-y)≥t or 0.5 according as t<0.5 or t≥0.5⇒(y+x-y)t∈A or A(y+x-y)+t>1. So, ⇒(y+x-y)t∈vqA.
(ii) Assume thatA(n(x+a)-nx)<min{A(a),0.5}=A(a) or 0.5 for some n∈N, x,a∈G. According as A(a)<0.5 or A(a)≥0.5. Choose t∈(0,1] such that A(n(x+a)-nx)<t<min{A(a),0.5}≤0.5. In either case, A(n(x+a)-nx)<t and A(n(x+a)-nx)+t<1. So, (n(x+a)-nx)t∈vqA¯, a contradiction.
Conversely, assume thatA(n(x+a)-nx)≥min{A(a),0.5} for all a,x∈G, n∈N. Let at∈A. Then, A(a)≥t. So, A(n(x+a)-nx)≥min{t,0.5}≤0.5=t or 0.5 according as t≤0.5 or t>0.5. So, A(n(x+a)-nx)t∈vqA.Theorem 21.
LetA be an (ϵ,ϵ∨q) fuzzy subgroup of G. Then, A is an (ϵ,ϵ∨q)-fuzzy ideal of G if and only if  (i)
A(y+x-y)≥min{A(x),0.5}, for all x,y∈G,(ii)
A(n(x+a)-nx)≥min{A(a),0.5}, for all n∈N, x,a∈G.Proof.
It is immediate from Lemma20.By definition, a fuzzy ideal ofG is an (ϵ,ϵ∨q)-fuzzy ideal of G. But the converse is not true in general as shown by the following example.Example 22.
ConsiderG=𝕊3={i,ρ1,ρ2,τ1,τ2,τ3} (written additively) to be a ℤ-group. Define a fuzzy subset A of G as A(i)=1, A(ρ1)=A(ρ2)=A(τ2)=A(τ3)=0.6, A(τ1)=0.8 which is not fuzzy ideal as A[2(τ1+τ2)-2τ2]=A(ρ1)=A(ρ2)=0.6<A(τ1); it contradicts the condition (iv) of Definition 14. As A(x-y), A(nx), A(y+x-y) and A(n(x+a)-nx)=0.6 or 0.8≥min{0.5,0.6 or 0.8}=0.5, thus, the notion of (ϵ,ϵ∨q)-fuzzy ideal is a successful generalization of fuzzy ideals of G as introduced in [18].Theorem 23.
Let{Ai,i∈J} be any family of (ϵ,ϵ∨q)-fuzzy ideals of G. Then, A=⋂‍Ai is an (ϵ,ϵ∨q)-fuzzy ideal of G.Proof.
It is straightforward.Theorem 24.
A nonempty subsetI of G is an ideal of G if and only if χI is an (∈,∈∨q)-fuzzy ideal of G.Proof.
IfI is an ideal of G, it is clear from [18, Proposition 2.11] that χI is fuzzy ideal of G. Since every fuzzy ideal is (ϵ,ϵ∨q)-fuzzy ideal, χI is (ϵ,ϵ∨q)-fuzzy ideal of G.
Conversely, letχI be an (ϵ,ϵ∨q)-fuzzy ideal of G. Let x,y∈I, χI(x-y)≥min{χI(x),χI(y),0.5}=0.5. So, χI(x-y)=1⇒x-y∈I. Let n∈N, x∈I, χI(nx)≥min{χI(x),0.5}=0.5⇒χI(nx)=1⇒nx∈Iy∈G, x∈I, χI(y+x-y)≥min{χI(x),0.5}=0.5⇒χI(y+x-y)=1⇒y+x-y∈In∈N, x∈I, y∈G, χI(n(y+x)-ny)≥min{χI(x),0.5}=0.5⇒χI(n(y+x)-ny)=1⇒n(y+x)-ny∈I. Then, I is an ideal of G.Theorem 25.
A fuzzy subsetA of G is an (ϵ,ϵ∨q)-fuzzy (subgroup) ideal of G if and only if the level subset At is a (subgroup) ideal for 0<t≤0.5.Proof.
We prove the result for(ϵ,ϵ∨q)-fuzzy ideal S. Let A be an (ϵ,ϵ∨q)-fuzzy ideal of G. Let t≤0.5, x,y,i∈At, n∈N.  (i)
A(x-y)≥min{A(x),A(y),0.5}≥min{t,0.5}=t⇒x-y∈At,(ii)
A(nx)≥min{A(a),0.5}≥min{t,0.5}=t⇒nx∈At,(iii)
A(y+x-y)≥min{A(x),0.5}≥min{t,0.5}=t⇒y+x-y∈At, y∈G(iv)
A(n(y+x)-ny)≥min{A(x),0.5}≥min{t,0.5}=t⇒n(y+x)-ny∈At, y∈G, n∈N.Hence, At is an ideal of G. Again, let At be an ideal of G for all t≤0.5. If possible, let there exist x,y∈G such that A(x-y)<t<min{A(x),A(y),0.5}. Let t be such that A(x-y)<t<min{A(x),A(y),0.5}⇒x, y∈At and x-y∉At, a contradiction. So, A(x-y)≥min{A(x),A(y),0.5}, for all x,y∈G. For n∈N, x∈G let A(nx)<min{A(x),0.5}. If possible let t be such that A(nx)<t<min{A(x),0.5}. This implies x∈At, but nx∉At, a contradiction. Similarly, we can prove that A(y+x-y)≥min{A(x),0.5}, A[n(y+x)-ny]≥min{A(x),0.5}, x,y∈G, n∈N.Remark 26.
Fort∈(0.5,1), A may be an (ϵ,ϵ∨q)-fuzzy ideal of G, but At may not be an ideal of G. Let t=0.8 in Example 22. Then, At={i,τ1}. At is not an ideal of 𝕊3 as it is not a normal subgroup of 𝕊3.We are looking for a corresponding result whenAt is an ideal of G for all t∈(0.5,1].Theorem 27.
LetA be a fuzzy subset of an N-group G. Then, At≠ϕ is an ideal of G for all t∈(0.5,1] if and only if A satisfies the following conditions:  (i)
max{A(x-y),0.5}≥min{A(x),A(y)},(ii)
max{A(nx),0.5}≥A(x),(iii)
max{A(y+x-y),0.5}≥A(x),(iv)
max{A(n(y+x)-ny),0.5}≥A(x),for allx,y∈G, n∈N.Proof.
Suppose thatAt≠ϕ is an ideal of G for all t∈(0.5,1]. In order to prove (i), suppose that for some x,y∈G, max{A(x-y),0.5}<min{A(x),A(y)}. Let t=min{A(x),A(y)}. So, x,y∈At and t∈(0.5,1]. Since At is an ideal, x-y∈At. So, A(x-y)≥t>max{A(x-y),0.5}, a contradiction. In order to prove (ii), suppose that x∈G, n∈N and max{A(nx),0.5}<A(x)=t (say). Then, x∈At⇒nx∈At⇒A(nx)≥t>max{A(nx),0.5}, a contradiction. Similarly, we can prove (iii) and (iv).
Conversely, suppose that conditions (i) to (iv) hold. We show thatAt is an ideal of G for all t∈(0.5,1]. Let x,y∈At. Then, 0.5<t≤min{A(x),A(y)}≤max{A(x-y),0.5}=A(x-y). So, x-y∈At. Let n∈N, x∈At. Then, 0.5<t≤A(x)≤max{A(x),0.5}=A(x) so nx∈At. For x∈At, y∈G, 0.5<t≤A(x)≤max{A(y+x-y),0.5}=A(y+x-y)⇒y+x-y∈At. Also, if n∈N, x∈At, y∈G, 0.5<t≤A(x)≤max{A(n(y+x)-ny),0.5}=A(n(y+x)-ny). Hence, n(y+x)-ny∈At. Then, At is an ideal of G.A definition for the previous kind of fuzzy subset was given for the case of near-rings in [17]. Now, we give the definition for N-groups.Definition 28.
A fuzzy subset ofG is called an (ϵ¯,ϵ¯∨q¯)-fuzzy subgroup of G if for all t,r∈(0,1] and for all x,y∈G, n∈N,(i)
(a)(x+y)min(t,r)∈¯A implies xt∈¯vq¯A or yr∈¯vq¯A,(b)(-x)t∈¯A implies xt∈¯vq¯A,(ii)
(nx)t∈¯A implies xt∈¯vq¯A.Moreover,A is called an  (ϵ¯,ϵ¯∨q¯)-fuzzy ideal of G if A is (ϵ¯,ϵ¯∨q¯)-fuzzy subgroup of G and(iii)
(y+x-y)t∈¯A implies xt∈¯vq¯A,(iv)
(n(x+y)-ny)t∈¯A implies xt∈¯vq¯A.Theorem 29.
A fuzzy subsetA of G is an (ϵ¯,ϵ¯∨q¯)-fuzzy ideal of G if and only if (1)
(a)max{A(x+y),0.5}≥min{A(x),A(y)},(b)max{A(-x),0.5}≥A(x),(2)
max{A(y+x-y),0.5}≥A(x),(3)
max{A(nx),0.5}≥A(x),(4)
max{A(n(y+x)-ny),0.5}≥A(x).Proof.
(
i
)
a
⇔
(
1
)
a. Let x,y∈G be such that max{A(x+y),0.5}<min{A(x),A(y)}. Let t=min{A(x),A(y)}; then 0.5<t≤1, (x+y)t∈¯A. So we must have xt∈¯vq¯A or yt∈¯vq¯A. But xt∈A and yt∈A. Here xtq¯A or ytq¯A then t≤A(x) and A(x)+t≤1 or t≤A(y) and A(y)+t≤1 then t≤0.5, a contradiction.
Conversely, let(x+y)min(t,r)∈¯A. Then, A(x+y)<min(t,r). If A(x+y)≥min{A(x),A(y)}, then min{A(x),A(y)}<min(t,r). Hence, either A(x)<t or A(y)<r which implies xt∈¯A or yr∈¯A. Thus, xt∈¯vq¯A or yr∈¯vq¯A.
Again ifA(x+y)<min{A(x),A(y)}, then by (1)a(4)0.5≥min{A(x),A(y)}>A(x+y).
Suppose that xt∈A or yr∈A then t≤A(x)≤0.5 or r≤A(y)≤0.5. It follows that either xtq¯A or ytq¯A, and thus xt∈¯vq¯A or yr∈¯vq¯A.
(
i
)
b
⇔
(
1
)
b: Suppose that there exists x∈G such that max{A(-x),0.5}<A(x). If A(x)=t then 0.5<t≤1 and A(-x)<t so that (-x)t∈¯A. But then we must have either (x)t∈¯A or xtq¯A. Also we have (x)t∈A. So, A(x)+1≤1 which means that t≤0.5, a contradiction.
Conversely, suppose that(x)t∈¯A then A(-x)<t. If A(-x)≥A(x), then A(x)<t which gives xt∈¯vq¯A. Again if A(-x)<A(x) by (1)b we have 0.5≥A(x). Putting (x)t∈A, then t≤A(x)≤0.5 so that xtq¯A which means that xt∈¯vq¯A. Similarly, we can prove the remaining parts.Theorem 30.
A fuzzy subsetA of G is an (ϵ¯,ϵ¯vq¯)-fuzzy ideal if and only if At(≠ϕ) is an ideal of G for all t∈(0.5,1].


## 3. Fuzzy Cosets and Isomorphism Theorem

In this section, we first study the properties of(ϵ,ϵ∨q)-fuzzy ideals under a homomorphism. Then, we introduce the fuzzy cosets and prove the fundamental isomorphism theorem on N-groups with respect to the structure induced by these fuzzy cosets.Theorem 31.
LetG and G′ be two N-groups, and let f:G→G′ be an N-homomorphism. If f is surjective and A is an (ϵ,ϵ∨q)-fuzzy ideal of G, then so is f(A). If B is a (ϵ,ϵ∨q)-fuzzy ideal of G′, then f-1(B) is a fuzzy ideal of G.Proof.
We assume thatA is an (ϵ,ϵ∨q)-fuzzy ideal of G. For any x,y∈G′; it follows that
(5)f(A)(x+y)=supx+y=f(z){A(z)}≥supf(u)=x,f(v)=y{A(u+v)}≥supf(u)=x,f(v)=y{min{A(u),A(v),0.5}}=min{supf(u)=x{A(u)},supf(v)=y{A(v)},0.5}=min{f(A)(x),f(A)(y),0.5}.
Also,
(6)f(A)(-x)=supf(z)=-x{A(z)}=supf(-z)=x{A(z)}=supf(z)=x{A(-z)}≥supf(z)=x{min{A(u),0.5}}=min{supf(z)=x{A(z)},0.5}=min{f(A)(x),0.5}.
Again,
(7)f(A)(nx)=supf(z)=nx{A(z)}≥supf(u)=x{A(nu)}≥supf(u)=x{min{A(u),0.5}}=min{supf(u)=x{A(u)},0.5}=min{f(A)(x),0.5},f(A)(y+x-y)=supf(z)=y+x-y{A(z)}≥supf(v)=x,f(u)=y{A(u+v-u)}≥supf(v)=x{min{A(v),0.5}}=min{supf(v)=x{A(v)},0.5}=min{f(A)(x),0.5},f(A)(n(y+x)-ny)=supf(z)=n(y+x)-ny{A(z)}≥supf(v)=x,f(u)=y{A(n(u+v)-nu)}≥supf(v)=x,f(u)=y{min{A(u),0.5}}=min{f(A)(x),0.5}.
Therefore, f(A) is an (ϵ,ϵ∨q)-fuzzy ideal of G. Similarly, we can show that f-1(A) is an (ϵ,ϵ∨q)-fuzzy ideal of G.Definition 32.
LetA be (∈,∈∨q)-fuzzy subgroup of G. For any x∈G, let Ax be defined by Ax(g)=min{A(g-x),0.5} for all g∈G. This fuzzy subset Ax is called the (∈,∈∨q)-fuzzy left coset of G determined by A and x.Remark 33.
LetA be an (ϵ,ϵ∨q)-fuzzy subgroup of G. Then, A is an (ϵ,ϵ∨q)-fuzzy normal if and only if min{A(g-x),0.5}=min{A(-x+g),0.5} for all x,g∈G. If A is an (ϵ,ϵ∨q)-fuzzy ideal, we simply denote fuzzy coset by Ax.Lemma 34.
LetA be an (ϵ,ϵ∨q)-fuzzy ideal of G. Then, Ax=Ay if an only if A(x-y)≥0.5.Proof.
Assume thatA(x-y)≥0.5 and g∈G. Ax(g)=min{A(g-x),0.5} ≥min{A(g-y),A(y-x),0.5}≥min{A(g-y),A(x-y),0.5}=min{A(g-y),0.5}=Ay(g) which implies that Ax≥Ay. Similarly, we can verify that Ax≤Ay. Conversely we assume that Ax=Ay. Then, Ay(x)=Ax(x)⇒min{A(x-y),0.5}=0.5⇒A(x-y)≥0.5.Proposition 35.
Every fuzzy cosetAx is constant on every coset of G0={x∈G∣A(x)=A(0)}.Proof.
Lety+y0∈y+G0. Now, we have Ax(y+y0)=min{A(y+y0-x),0.5} ≥min{A(y-x),A(x+y0-x),0.5}≥min{A(y-x),A(y0),0.5}=min{A(y-x),0.5}=Ax(y). Also Ax(y)=min{A(y-x),0.5}≥min{A(-x+y+y0-y0),0.5}≥min{A(-x+y+y0),A(-y0),0.5} =min{A(-x+y+y0),0.5}≥min{A(y+y0-x),0.5}=Ax(y+y0). Thus Ax(y+y0)=Ax(y) for all y0∈G0.Theorem 36.
For any(ϵ,ϵ∨q)-fuzzy ideal A of G, GA the set of all fuzzy cosets of A in G is an N-group under the addition and scalar multiplication defined by Ax+Ay=Ax+y, n(Ax)=Anx for all x,y∈G, n∈N. The function A¯(Ax):GA→[0,1] defined by A¯(Ax)=A(x) for all Ax∈GA is (ϵ,ϵ∨q)-fuzzy ideal of GA.Proof.
First, we show that the compositions are well defined. Letx,y,c,d∈G, n∈N such that Ax=Ay and Ac=Ad. Then, A(x-y)≥0.5, A(c-d)≥0.5. Now, we have min{A(x+c-d-y),0.5} ≥min{A(-y+x+c-d),0.5}=0.5 which implies that A(x+c-d-y)≥0.5. So, by Lemma 34, we have Ax+c=Ay+d. Again Anx(g)=min{A(g-nx),0.5}≥min{A(g-ny),A(nx-ny),0.5} =min{A(g-ny),A[n(y-y+x)-ny],0.5}≥min{A(g-ny),A(-y+x),0.5}=Any(g). Similarly, we show that Any(g)≥Anx(g). Thus, for Any(g)=Anx(g) for all g∈G. Hence the compositions are well defined. It is now easy to verify that GA is an N-group with null element A0 and negative element A-x. Next, we check that A¯ is (ϵ,ϵ∨q)-fuzzy ideal of GA. Let Ax,Ay∈GA. We have(i)
A¯(Ax-Ay)=A¯(Ax-y)=A(x-y)≥min{A(x),A(y),0.5}=min{A¯(Ax),A¯(Ay),0.5},(ii)
A¯(Anx)=A(nx)≥min{A(x),0.5}=min{A¯(Ax),0.5},(iii)
A¯(Ay+Ax-Ay)=A¯(Ay+x-y)=A(y+x-y)≥min{A(x),0.5}=min{A¯(Ax),0.5},(iv)
A¯[n(Ay+Ax)-nAy]=A¯[An(y+x)-ny]=A[n(y+x)-ny]≥min{A(x),0.5}=min{A¯(Ax),0.5}.Hence, the proof is completed.Lemma 37.
LetH={x∈G∣Ax≥A0}, K={x∈G∣Ax=A0}. Then, K=H=A0.5.Proof.
Letx∈H. Then, for all g∈G, Ax(g)≥A0(g)⇒min{A(g-x),0.5}≥min{A(g),0.5} ⇒min{A(-x),0.5}≥min{A(0),0.5}=0.5⇒A(x)≥0.5. Also, for g∈G, A0(g) =min{A(g),0.5}=min{A(g-x+x),0.5}≥min{A(g-x),A(x),0.5}=min{A(g-x),0.5}=Ax(g) ⇒A0≥Ax. So, H=K. As we have seen, if x∈H then A(x)≥0.5⇒x∈A0.5. Conversely, if x∈A0.5, for any g∈G, A(g-x)≥min{A(g),A(x),0.5}=min{A(g),0.5}⇒min{A(g-x),0.5} ≥min{A(g),0.5}⇒Ax≥A0. Thus, H=A0.5, which means that H is an ideal of G.Theorem 38.
IfA is an (ϵ,ϵ∨q)-fuzzy ideal of G, then the map f:G→GA given by f(x)=Ax is an N-homomorphism with kernel f=A0.5 and so G/A0.5 is isomorphic to GA.Proof.
It is clear thatf is an onto N-homomorphism from G to GA with kernel f={x∈G∣f(x)=f(0)}={x∈G∣Ax=A0}=A0.5.Corollary 39.
LetA be an (ϵ,ϵ∨q)-fuzzy ideal of G, and let B¯ be an (ϵ,ϵ∨q)-fuzzy ideal of GA. Then, B:G→[0,1] defined by B(x)=B¯(Ax) is an (ϵ,ϵ∨q)-fuzzy ideal of G containing A.



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