A Study on Fuzzy Ideals of 𝑁-Groups

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.

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 a near-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. Let 𝒢 be a group, and let 𝑀(𝒢) be the set of all mappings from 𝒢 into 𝒢. We define + and ⋅ on 𝑀(𝒢) by (𝑥)(𝑥)(𝑥),𝑥)𝑔(𝑥).𝑓+𝑔∶=𝑓+𝑔𝑓⋅𝑔(∶=𝑓 (1) Then, (𝑀(𝒢),+,⋅) is a near-ring.

Just in the same way as 𝑅-modules or vector spaces are used in ring theory, 𝑁-groups are used in near-ring theory.

By an 𝑁-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. Let 𝑁 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.

An ideal 𝐼 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 use 𝑁 to denote a zero-symmetric near-ring and 𝐺 to denote an 𝑁-group.

For any fuzzy subset 𝐴 of 𝐺, 𝐼𝑚𝐴={𝐴(𝑥)∣𝑥∈𝐺} denotes the image of 𝐴. For any subset 𝐼 of 𝐺, 𝜒_𝐼 denotes the characteristic function of 𝐼.

Definition 4 (see [2]). A fuzzy subset 𝐴 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 all 𝑥,𝑦∈𝐺.

Definition 5. For a fuzzy subset 𝐴 of 𝐺, 𝑡∈(0,1], the subset 𝐴_𝑡={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 𝐴𝑦=𝑡(≠0)if0𝑦=𝑥,if𝑦≠𝑥 (2) is said to be a fuzzy point denoted by 𝑥_𝑡. Here 𝑥 is called the support point, and 𝑡 is called its value. A fuzzy point 𝑥_𝑡 is said to belong to (resp., quasi coincident with) a fuzzy set 𝐴 written as 𝑥_𝑡𝜖𝐴 (resp., 𝑥_𝑡𝑞𝐴) if 𝐴(𝑥)≥𝑡 (resp., 𝐴(𝑥)+𝑡>1). If 𝑥_𝑡∈𝐴 or 𝑥_𝑡𝑞𝐴, then we write 𝑥_𝑡∈∨𝑞𝐴. The symbols 𝑥_𝑡∈𝐴, 𝑥_𝑡𝑞𝐴,𝑥_𝑡∈∨𝑞𝐴 mean that 𝑥_𝑡∈𝐴, 𝑥_𝑡𝑞𝐴,𝑥_𝑡∈∨𝑞𝐴 do not hold, respectively.

Definition 6 (see [7, 12]). A fuzzy subset of a group 𝐺 is said to be an (𝜖,𝜖∨𝑞)-fuzzy subgroup of 𝐺 if for all 𝑥,𝑦∈𝐺 and 𝑡,𝑟∈(0,1], (i)𝑥_𝑡,𝑦_𝑟∈𝐴⇒(𝑥+𝑦)_{min{𝑡,𝑟}}∈∨𝑞𝐴, (ii)𝑥_𝑡∈𝐴⇒(−𝑥)_𝑡∈∨𝑞𝐴.

Remark 7 (see [7]). The conditions (i) and (ii) of Definition 6 are respectively equivalent to(i)𝐴(𝑥+𝑦)≥min{𝐴(𝑥),𝐴(𝑦),0.5}, (ii)𝐴(−𝑥)≥min{𝐴(𝑥),0.5},  for all 𝑥,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 if 𝐴 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 that 𝐴 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], 𝑥_𝑡∈𝐴⟹𝑦+𝑥−𝑦_𝑡∈∨𝑞𝐴. (3)

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 Definition 10 can be replaced by any one of the above conditions in Remark 8.

Definition 12 (see [18]). Let 𝐴 be a fuzzy subset of an 𝑁-group 𝐺. It is called a fuzzy 𝑁-subgroup of 𝐺 if it satisfies the following conditions: (i)𝐴(𝑥+𝑦)≥min{𝐴(𝑥),𝐴(𝑦)}, (ii)𝐴(𝑛𝑥)≥𝐴(𝑥),  for all 𝑥,𝑦∈𝐺, 𝑛∈𝑁.

Remark 13. If 𝐺 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 subset 𝐴 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 all 𝑥,𝑦∈𝐺, 𝑛∈𝑁.

Definition 15 (see [14]). A fuzzy set 𝐴 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) 𝑥_𝑡,𝑦_𝑟∈𝐴⇒(𝑥+𝑦)_{min{𝑡,𝑟}}∈∨𝑞𝐴,  (b) 𝑥_𝑡∈𝐴⇒(−𝑥)_𝑡∈∨𝑞𝐴,(ii)𝑥_𝑡,𝑦_𝑟∈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)𝑥_𝑡∈𝐴⇒(𝑦+𝑥−𝑦)∈∨𝑞𝐴, (iv)𝑦_𝑟∈𝐴,𝑥∈𝑁⇒(𝑦𝑥)_𝑟∈∨𝑞𝐴, (v)𝑎_𝑡∈𝐴⇒(𝑦(𝑥+𝑎)−𝑦𝑥)_𝑡∈∨𝑞𝐴, for all 𝑥,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 subset 𝐴 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)𝑥_𝑡,𝑦_𝑟∈𝐴⇒(𝑥+𝑦)_{min{𝑡,𝑟}}∈∨𝑞𝐴, (ii)𝑥_𝑡∈𝐴⇒(−𝑥)_𝑡∈∨𝑞𝐴, (iii)𝑥_𝑡∈𝐴, 𝑛∈𝑁⇒(𝑛𝑥)_𝑡∈∨𝑞𝐴.

Lemma 17. Let 𝐴 be a fuzzy subset of 𝐺 and 𝑡,𝑟∈(0,1]. Then,(i)𝑥_𝑡,𝑦_𝑟∈𝐴⇒(𝑥+𝑦)_{min{𝑡,𝑟}}𝜖∨𝑞𝐴⇔𝐴(𝑥+𝑦)≥min{𝐴(𝑥),𝐴(𝑦),0.5}, (ii)𝑥_𝑡∈𝐴⇒(−𝑥)_𝑡∈∨𝑞𝐴⇔𝐴(−𝑥)≥min{𝐴(𝑥),0.5}, for all 𝑥∈𝐺, (iii)𝑥_𝑡∈𝐴⇒(𝑛𝑥)_𝑡𝜖∨𝑞𝐴⇔𝐴(𝑛𝑥)≥min{𝐴(𝑥),0.5}, for all 𝑥∈𝐺, 𝑛∈𝑁.

Proof. (i) Let 𝑥,𝑦∈𝐺. Consider the case (a): min{𝐴(𝑥),𝐴(𝑦)}<0.5.
Assume that 𝐴(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 𝑥_𝑡∈𝐴, 𝑦_𝑡∈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 𝑥_𝑡,𝑦_𝑡∈𝐴 but (𝑥+𝑦)_𝑡∈∨𝑞𝐴.
Conversely, let 𝑥_𝑡,𝑦_𝑟∈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) Let 𝑥∈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, 𝑥_𝑟∈𝐴 but (−𝑥)_𝑟∈∨𝑞𝐴 which contradicts the hypothesis. So, 𝐴(−𝑥)≥min{𝐴(𝑥),0.5} for all 𝑥∈𝐺.
Conversely, let 𝑥_𝑡∈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) Let 𝑥∈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, 𝑥_𝑟∈𝐴, but (𝑛𝑥)_𝑟∈∨𝑞𝐴 as 𝐴(𝑛𝑥)<𝑟 and 𝐴(𝑛𝑥)+𝑟≤1.
Conversely let 𝑥_𝑡∈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. Let 𝐴 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 all 𝑥,𝑦∈𝐺, 𝑛∈𝑁.

Proof. It follows from the previous lemma.

Definition 19. A fuzzy subset 𝐴 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)𝑥_𝑡∈𝐴⇒(𝑦+𝑥−𝑦)_𝑡∈𝑣𝑞𝐴, (ii)𝑎_𝑡∈𝐴⇒(𝑛(𝑥+𝑎)−𝑛𝑥)_𝑡∈𝑣𝑞𝐴,  for any 𝑛∈𝑁, 𝑥,𝑎∈𝐺.

Lemma 20. Let 𝐴 be a fuzzy subset of 𝐺 and 𝑡,𝑟∈(0,1]. Then,(i)𝑥_𝑡∈𝐴⇒(𝑦+𝑥−𝑦)_𝑡∈𝑣𝑞𝐴⇔𝐴(𝑦+𝑥−𝑦)≥min{𝐴(𝑥),0.5}, (ii)𝑎_𝑡∈𝐴⇒(𝑛(𝑥+𝑎)−𝑛𝑥)_𝑡∈𝑣𝑞𝐴⇔𝐴(𝑛(𝑥+𝑎)−𝑛𝑥)≥min{𝐴(𝑥),0.5}.

Proof. (i) Assume that 𝐴(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⇒𝑥_𝑡∈𝐴 or 𝑥_0.5∈𝐴. But (𝑦+𝑥−𝑦)_𝑡∈𝑣𝑞𝐴 or (𝑦+𝑥−𝑦)_0.5∈𝑣𝑞𝐴, respectively, which contradicts the hypothesis.
Conversely, assume that 𝑥_𝑡∈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 that 𝐴(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 that 𝐴(𝑛(𝑥+𝑎)−𝑛𝑥)≥min{𝐴(𝑎),0.5} for all 𝑎,𝑥∈𝐺, 𝑛∈𝑁. Let 𝑎_𝑡∈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. Let 𝐴 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 Lemma 20.

By definition, a fuzzy ideal of 𝐺 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. Consider 𝐺=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 {𝐴_𝑖,𝑖∈𝐽} be any family of (𝜖,𝜖∨𝑞)-fuzzy ideals of 𝐺. Then, 𝐴=⋂𝐴_𝑖 is an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺.

Proof. It is straightforward.

Theorem 24. A nonempty subset 𝐼 of 𝐺 is an ideal of 𝐺 if and only if 𝜒_𝐼 is an (∈,∈∨𝑞)-fuzzy ideal of 𝐺.

Proof. If 𝐼 is an ideal of 𝐺, it is clear from [18, Proposition 2.11] that 𝜒_𝐼 is fuzzy ideal of 𝐺. Since every fuzzy ideal is (𝜖,𝜖∨𝑞)-fuzzy ideal, 𝜒_𝐼 is (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺.
Conversely, let 𝜒_𝐼 be an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺. Let 𝑥,𝑦∈𝐼, 𝜒_𝐼(𝑥−𝑦)≥min{𝜒_𝐼(𝑥),𝜒_𝐼(𝑦),0.5}=0.5. So, 𝜒_𝐼(𝑥−𝑦)=1⇒𝑥−𝑦∈𝐼. Let 𝑛∈𝑁, 𝑥∈𝐼, 𝜒_𝐼(𝑛𝑥)≥min{𝜒_𝐼(𝑥),0.5}=0.5⇒𝜒_𝐼(𝑛𝑥)=1⇒𝑛𝑥∈𝐼𝑦∈𝐺, 𝑥∈𝐼, 𝜒_𝐼(𝑦+𝑥−𝑦)≥min{𝜒_𝐼(𝑥),0.5}=0.5⇒𝜒_𝐼(𝑦+𝑥−𝑦)=1⇒𝑦+𝑥−𝑦∈𝐼𝑛∈𝑁, 𝑥∈𝐼, 𝑦∈𝐺, 𝜒_𝐼(𝑛(𝑦+𝑥)−𝑛𝑦)≥min{𝜒_𝐼(𝑥),0.5}=0.5⇒𝜒_𝐼(𝑛(𝑦+𝑥)−𝑛𝑦)=1⇒𝑛(𝑦+𝑥)−𝑛𝑦∈𝐼. Then, 𝐼 is an ideal of 𝐺.

Theorem 25. A fuzzy subset 𝐴 of 𝐺 is an (𝜖,𝜖∨𝑞)-fuzzy (subgroup) ideal of 𝐺 if and only if the level subset 𝐴_𝑡 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∈𝐴_𝑡, 𝑛∈𝑁. (i)𝐴(𝑥−𝑦)≥min{𝐴(𝑥),𝐴(𝑦),0.5}≥min{𝑡,0.5}=𝑡⇒𝑥−𝑦∈𝐴_𝑡, (ii)𝐴(𝑛𝑥)≥min{𝐴(𝑎),0.5}≥min{𝑡,0.5}=𝑡⇒𝑛𝑥∈𝐴_𝑡, (iii)𝐴(𝑦+𝑥−𝑦)≥min{𝐴(𝑥),0.5}≥min{𝑡,0.5}=𝑡⇒𝑦+𝑥−𝑦∈𝐴_𝑡, 𝑦∈𝐺(iv)𝐴(𝑛(𝑦+𝑥)−𝑛𝑦)≥min{𝐴(𝑥),0.5}≥min{𝑡,0.5}=𝑡⇒𝑛(𝑦+𝑥)−𝑛𝑦∈𝐴_𝑡, 𝑦∈𝐺, 𝑛∈𝑁. Hence, 𝐴_𝑡 is an ideal of 𝐺. Again, let 𝐴_𝑡 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∈𝐴_𝑡 and 𝑥−𝑦∉𝐴_𝑡, 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∈𝐴_𝑡, but 𝑛𝑥∉𝐴_𝑡, 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. For 𝑡∈(0.5,1), 𝐴 may be an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺, but 𝐴_𝑡 may not be an ideal of 𝐺. Let 𝑡=0.8 in Example 22. Then, 𝐴_𝑡={𝑖,𝜏₁}. 𝐴_𝑡 is not an ideal of 𝕊₃ as it is not a normal subgroup of 𝕊₃.

We are looking for a corresponding result when 𝐴_𝑡 is an ideal of 𝐺 for all 𝑡∈(0.5,1].

Theorem 27. Let 𝐴 be a fuzzy subset of an 𝑁-group 𝐺. Then, 𝐴_𝑡≠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 all 𝑥,𝑦∈𝐺, 𝑛∈𝑁.

Proof. Suppose that 𝐴_𝑡≠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∈𝐴_𝑡 and 𝑡∈(0.5,1]. Since 𝐴_𝑡 is an ideal, 𝑥−𝑦∈𝐴_𝑡. 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∈𝐴_𝑡⇒𝑛𝑥∈𝐴_𝑡⇒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 that 𝐴_𝑡 is an ideal of 𝐺 for all 𝑡∈(0.5,1]. Let 𝑥,𝑦∈𝐴_𝑡. Then, 0.5<𝑡≤min{𝐴(𝑥),𝐴(𝑦)}≤max{𝐴(𝑥−𝑦),0.5}=𝐴(𝑥−𝑦). So, 𝑥−𝑦∈𝐴_𝑡. Let 𝑛∈𝑁, 𝑥∈𝐴_𝑡. Then, 0.5<𝑡≤𝐴(𝑥)≤max{𝐴(𝑥),0.5}=𝐴(𝑥) so 𝑛𝑥∈𝐴_𝑡. For 𝑥∈𝐴_𝑡, 𝑦∈𝐺, 0.5<𝑡≤𝐴(𝑥)≤max{𝐴(𝑦+𝑥−𝑦),0.5}=𝐴(𝑦+𝑥−𝑦)⇒𝑦+𝑥−𝑦∈𝐴_𝑡. Also, if 𝑛∈𝑁, 𝑥∈𝐴_𝑡, 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∈𝐴_𝑡. Then, 𝐴_𝑡 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 of 𝐺 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 𝑥_𝑡∈𝑣𝑞𝐴 or 𝑦_𝑟∈𝑣𝑞𝐴,  (b) (−𝑥)_𝑡∈𝐴 implies 𝑥_𝑡∈𝑣𝑞𝐴,(ii)(𝑛𝑥)_𝑡∈𝐴 implies 𝑥_𝑡∈𝑣𝑞𝐴. 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 𝑥_𝑡∈𝑣𝑞𝐴, (iv)(𝑛(𝑥+𝑦)−𝑛𝑦)_𝑡∈𝐴 implies 𝑥_𝑡∈𝑣𝑞𝐴.

Theorem 29. A fuzzy subset 𝐴 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 𝑥_𝑡∈𝑣𝑞𝐴 or 𝑦_𝑡∈𝑣𝑞𝐴. But 𝑥_𝑡∈𝐴 and 𝑦_𝑡∈𝐴. Here 𝑥_𝑡𝑞𝐴 or 𝑦_𝑡𝑞𝐴 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 𝑥_𝑡∈𝐴 or 𝑦_𝑟∈𝐴. Thus, 𝑥_𝑡∈𝑣𝑞𝐴 or 𝑦_𝑟∈𝑣𝑞𝐴.
Again if 𝐴(𝑥+𝑦)<min{𝐴(𝑥),𝐴(𝑦)}, then by (1)a0.5≥min𝐴(𝑥),𝐴𝑦>𝐴𝑥+𝑦. (4) Suppose that 𝑥_𝑡∈𝐴 or 𝑦_𝑟∈𝐴 then 𝑡≤𝐴(𝑥)≤0.5 or 𝑟≤𝐴(𝑦)≤0.5. It follows that either 𝑥_𝑡𝑞𝐴 or 𝑦_𝑡𝑞𝐴, and thus 𝑥_𝑡∈𝑣𝑞𝐴 or 𝑦_𝑟∈𝑣𝑞𝐴.
(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 𝑥_𝑡𝑞𝐴. 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 𝑥_𝑡∈𝑣𝑞𝐴. 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 𝑥_𝑡𝑞𝐴 which means that 𝑥_𝑡∈𝑣𝑞𝐴. Similarly, we can prove the remaining parts.

Theorem 30. A fuzzy subset 𝐴 of 𝐺 is an (𝜖,𝜖𝑣𝑞)-fuzzy ideal if and only if 𝐴_𝑡(≠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. Let 𝐺 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 that 𝐴 is an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺. For any 𝑥,𝑦∈𝐺^′; it follows that 𝑓(𝐴)𝑥+𝑦=sup_{(𝑧)𝑥+𝑦=𝑓}{𝐴(𝑧)}≥sup_{𝑓(𝑢)(𝑣)=𝑥,𝑓=𝑦}{𝐴(𝑢+𝑣)}≥sup_{𝑓(𝑢)(𝑣)=𝑥,𝑓=𝑦}{min{𝐴(𝑢),𝐴(𝑣),0.5}}=minsup_{𝑓(𝑢)=𝑥}{𝐴(𝑢)},sup_{𝑓(𝑣)=𝑦}{𝐴(𝑣𝑓(𝐴𝑥)(𝐴)𝑦.)},0.5=min)(,𝑓,0.5 (5) Also, 𝑓(𝐴)(−𝑥)=sup_{𝑓(𝑧)=−𝑥}{𝐴(𝑧)}=sup_{𝑓()−𝑧=𝑥}{𝐴(𝑧)}=sup_{𝑓(𝑧)=𝑥}{𝐴(−𝑧)}≥sup_{𝑓(𝑧)=𝑥}{min{𝐴(𝑢),0.5}}=minsup_{𝑓(𝑧)=𝑥}{𝐴(𝑧𝑓(𝐴𝑥).)},0.5=min)(,0.5 (6) Again, 𝑓(𝐴)(𝑛𝑥)=sup_{𝑓(𝑧)=𝑛𝑥}{𝐴(𝑧)}≥sup_{𝑓(𝑢)=𝑥}{𝐴(𝑛𝑢)}≥sup_{𝑓(𝑢)=𝑥}{min{𝐴(𝑢),0.5}}=minsup_{𝑓(𝑢)=𝑥}{𝐴(𝑢𝑓(𝐴𝑥),𝑓(𝐴))},0.5=min)(,0.5𝑦+𝑥−𝑦=sup_{𝑓(𝑧)=𝑦+𝑥−𝑦}{𝐴(𝑧)}≥sup_{𝑓(𝑣)(𝑢)=𝑥,𝑓=𝑦}{𝐴(𝑢+𝑣−𝑢)}≥sup_{𝑓(𝑣)=𝑥}{min{𝐴(𝑣),0.5}}=minsup_{𝑓(𝑣)=𝑥}{𝐴(𝑣𝑓(𝐴𝑥),𝑓(𝐴𝑛)},0.5=min)(,0.5)𝑦+𝑥−𝑛𝑦=sup_{𝑓(𝑧)=𝑛(𝑦+𝑥)−𝑛𝑦}{𝐴(𝑧)}≥sup_{𝑓(𝑣)(𝑢)=𝑥,𝑓=𝑦}{𝐴(𝑛(𝑢+𝑣)−𝑛𝑢)}≥sup_{𝑓(𝑣)(𝑢)=𝑥,𝑓=𝑦}{{𝐴(𝑢)𝑓(𝐴𝑥).min,0.5}}=min)(,0.5 (7) 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. Let 𝐴 be (∈,∈∨𝑞)-fuzzy subgroup of 𝐺. For any 𝑥∈𝐺, let 𝐴_𝑥 be defined by 𝐴_𝑥(𝑔)=min{𝐴(𝑔−𝑥),0.5} for all 𝑔∈𝐺. This fuzzy subset 𝐴_𝑥 is called the (∈,∈∨𝑞)-fuzzy left coset of 𝐺 determined by 𝐴 and 𝑥.

Remark 33. Let 𝐴 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 𝐴_𝑥.

Lemma 34. Let 𝐴 be an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺. Then, 𝐴_𝑥=𝐴_𝑦 if an only if 𝐴(𝑥−𝑦)≥0.5.

Proof. Assume that 𝐴(𝑥−𝑦)≥0.5 and 𝑔∈𝐺. 𝐴_𝑥(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}=𝐴_𝑦(𝑔) which implies that 𝐴_𝑥≥𝐴_𝑦. Similarly, we can verify that 𝐴_𝑥≤𝐴_𝑦. Conversely we assume that 𝐴_𝑥=𝐴_𝑦. Then, 𝐴_𝑦(𝑥)=𝐴_𝑥(𝑥)⇒min{𝐴(𝑥−𝑦),0.5}=0.5⇒𝐴(𝑥−𝑦)≥0.5.

Proposition 35. Every fuzzy coset 𝐴_𝑥 is constant on every coset of 𝐺₀={𝑥∈𝐺∣𝐴(𝑥)=𝐴(0)}.

Proof. Let 𝑦+𝑦₀∈𝑦+𝐺₀. Now, we have 𝐴_𝑥(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}=𝐴_𝑥(𝑦). Also 𝐴_𝑥(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}=𝐴_𝑥(𝑦+𝑦₀). Thus 𝐴_𝑥(𝑦+𝑦₀)=𝐴_𝑥(𝑦) for all 𝑦₀∈𝐺₀.

Theorem 36. For any (𝜖,𝜖∨𝑞)-fuzzy ideal 𝐴 of 𝐺, 𝐺_𝐴 the set of all fuzzy cosets of 𝐴 in 𝐺 is an 𝑁-group under the addition and scalar multiplication defined by 𝐴_𝑥+𝐴_𝑦=𝐴_𝑥+𝑦, 𝑛(𝐴_𝑥)=𝐴_𝑛𝑥 for all 𝑥,𝑦∈𝐺, 𝑛∈𝑁. The function 𝐴(𝐴_𝑥)∶𝐺_𝐴→[0,1] defined by 𝐴(𝐴_𝑥)=𝐴(𝑥) for all 𝐴_𝑥∈𝐺_𝐴 is (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺_𝐴.

Proof. First, we show that the compositions are well defined. Let 𝑥,𝑦,𝑐,𝑑∈𝐺, 𝑛∈𝑁 such that 𝐴_𝑥=𝐴_𝑦 and 𝐴_𝑐=𝐴_𝑑. 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 𝐴_𝑥+𝑐=𝐴_𝑦+𝑑. Again 𝐴_𝑛𝑥(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}=𝐴_𝑛𝑦(𝑔). Similarly, we show that 𝐴_𝑛𝑦(𝑔)≥𝐴_𝑛𝑥(𝑔). Thus, for 𝐴_𝑛𝑦(𝑔)=𝐴_𝑛𝑥(𝑔) for all 𝑔∈𝐺. Hence the compositions are well defined. It is now easy to verify that 𝐺_𝐴 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 𝐺_𝐴. Let 𝐴_𝑥,𝐴_𝑦∈𝐺_𝐴. We have(i)𝐴(𝐴_𝑥−𝐴_𝑦)=𝐴(𝐴_𝑥−𝑦)=𝐴(𝑥−𝑦)≥min{𝐴(𝑥),𝐴(𝑦),0.5}=min{𝐴(𝐴_𝑥),𝐴(𝐴_𝑦),0.5}, (ii)𝐴(𝐴_𝑛𝑥)=𝐴(𝑛𝑥)≥min{𝐴(𝑥),0.5}=min{𝐴(𝐴_𝑥),0.5}, (iii)𝐴(𝐴_𝑦+𝐴_𝑥−𝐴𝑦)=𝐴(𝐴_{𝑦+𝑥−𝑦})=𝐴(𝑦+𝑥−𝑦)≥min{𝐴(𝑥),0.5}=min{𝐴(𝐴_𝑥),0.5}, (iv)𝐴[𝑛(𝐴_𝑦+𝐴_𝑥)−𝑛𝐴_𝑦]=𝐴[𝐴_{𝑛(𝑦+𝑥)−𝑛𝑦}]=𝐴[𝑛(𝑦+𝑥)−𝑛𝑦]≥min{𝐴(𝑥),0.5}=min{𝐴(𝐴_𝑥),0.5}.Hence, the proof is completed.

Lemma 37. Let 𝐻={𝑥∈𝐺∣𝐴_𝑥≥𝐴₀}, 𝐾={𝑥∈𝐺∣𝐴_𝑥=𝐴₀}. Then, 𝐾=𝐻=𝐴_0.5.

Proof. Let 𝑥∈𝐻. Then, for all 𝑔∈𝐺, 𝐴_𝑥(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}=𝐴_𝑥(𝑔) ⇒𝐴₀≥𝐴_𝑥. So, 𝐻=𝐾. As we have seen, if 𝑥∈𝐻 then 𝐴(𝑥)≥0.5⇒𝑥∈𝐴_0.5. Conversely, if 𝑥∈𝐴_0.5, 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}⇒𝐴_𝑥≥𝐴₀. Thus, 𝐻=𝐴_0.5, which means that 𝐻 is an ideal of 𝐺.

Theorem 38. If 𝐴 is an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺, then the map 𝑓∶𝐺→𝐺_𝐴 given by 𝑓(𝑥)=𝐴_𝑥 is an 𝑁-homomorphism with kernel 𝑓=𝐴_0.5 and so 𝐺/𝐴_0.5 is isomorphic to 𝐺_𝐴.

Proof. It is clear that 𝑓 is an onto 𝑁-homomorphism from 𝐺 to 𝐺_𝐴 with kernel 𝑓={𝑥∈𝐺∣𝑓(𝑥)=𝑓(0)}={𝑥∈𝐺∣𝐴_𝑥=𝐴₀}=𝐴_0.5.

Corollary 39. Let 𝐴 be an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺, and let 𝐵 be an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺_𝐴. Then, 𝐵∶𝐺→[0,1] defined by 𝐵(𝑥)=𝐵(𝐴_𝑥) is an (𝜖,𝜖∨𝑞)-fuzzy ideal of 𝐺 containing 𝐴.