An Application of Maximality to Nilpotent and Finitely Generated -Subgroups of an -Group

Iffat Jahan; Ananya Manas

doi:10.5391/IJFIS.2023.23.1.20

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 20-33

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.20

An Application of Maximality to Nilpotent and Finitely Generated -Subgroups of an -Group

Iffat Jahan¹ and Ananya Manas²

¹Department of Mathematics, Ramjas College, University of Delhi, Delhi, India
²Department of Mathematics, University of Delhi, Delhi, India

Correspondence to :
Iffat Jahan (ij.umar@yahoo.com)

Received: November 25, 2020; Revised: September 3, 2022; Accepted: February 16, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

This study is a continuation of the study on the maximal and Frattini L subgroups of an L-group. The normality of the maximal L subgroups of a nilpotent L group was explored. Subsequently, the concept of a finitely generated L-subgroup is introduced, and its relation with the maximal condition on L subgroups is established. Thereafter, several results of the notions of the Frattini L-subgroup and finitely generated L-subgroups have been investigated.

Keywords: L-algebra, L-subgroup, Generated L-subgroup, Normal L-subgroup, Maximal L-subgroup, Frattini L-subgroup, Finitely generated L-subgroup

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

In 1965, Zadeh’s pioneering paper [1] established the foundations of employing fuzzy logic to approximate reasoning. Since its inception, fuzzy sets and fuzzy logic have been applied in various fields such as information theory, linear programming, and pattern recognition. In 1971, Rosenfeld [2] introduced fuzzy groups using fuzzy sets. Following this, various group theoretic concepts have been extended to the fuzzy setting, for example, the notion of a set product of two fuzzy subgroups and the notion of normality of a fuzzy subgroup developed by Liu [3]. All these studies were conducted within the framework of a fuzzy setting. Moreover, in these studies, the parent structure was considered ordinary algebra (i.e., group, ring, and semigroup). Therefore, most researchers have defined and studied the notion of a fuzzy subgroup of an ordinary group. However, this approach does not permit the transference of several properties of algebraic structures into the fuzzy setting. This shortcoming can be easily overcome if the parent structure is considered a fuzzy group rather than a classical group. Significantly few researchers, such as Martinez [4] and Malik and Mordeson [5], have studied the properties of an L-subalgebra of an L-algebra. Prajapati and Ajmal [6, 7] defined and explored the notion of maximal L-ideals of a fuzzy ring. However, these studies have been inadequate for L-groups. Recently, various concepts of classical group theory have been extended to L-settings, particularly considering their compatibility [8–12]. Moreover, in [13], the authors investigated the maximal and Frattini L-subgroups of an L-group. The notion of non-generators of an L-group was also introduced, and a relationship between the Frattini L-subgroup and the set of non-generators of an L-group was established. This study is a continuation of these topics.

The rest of this paper is organized as follows. In Section 3, we begin by demonstrating that every maximal L-subgroup of a nilpotent L-group μ having the same tip and tail as μ is normal in μ. To establish this result, the concepts of nilpotent L-subgroups [10], the normalizer of an L-subgroup of an L-group [11], and the normal closure of an L-subgroup of an L-group [9] are used. This section demonstrates the compatibility of these concepts. In Section 4, the maximal condition on L-subgroups is introduced. An L-group satisfies the maximal condition on L-subgroups if and only if every proper ascending chain of its L-subgroups is finite. Thereafter, the notion of a finitely generated L-subgroup was established through L-points. A relationship between the maximal condition on L-subgroups and the finitely generated L-subgroups is also established, similar to their classical counterparts. Section 5 shows that if L is an upper well-ordered lattice and μ is a normal L-group, then Frattini L-subgroup Φ(μ) is normal in μ. Thereafter, we demonstrate that every finitely generated L-group, where L is an upper well-ordered lattice, has a maximum L-subgroup. Moreover, several relations between Frattini L-subgroups, finitely generated L-subgroups, and the set product of L-subgroups are established.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

In this study, L = 〈L,≤, ∨, ∧〉 denotes a completely distributive lattice where ’≤’ denotes the partial ordering on L, and ’∨’ and ’∧’ denote the join (supremum) and meet (infimum) of the elements of L, respectively. The maximal and minimal elements of L are denoted by 1 and 0, respectively. The concept of a completely distributive lattice can be found in any standard text on this subject [14].

The notion of a fuzzy subset of a set was introduced by Zadeh [1] in 1965. In 1967, Goguen [15] extended this concept to L-fuzzy sets. In this section, we recall the basic definitions and results associated with L-subsets used throughout the remainder of this paper. These definitions can be found in Chapter 1 of [16].

Let X be a non-empty set. A subset of X with cardinality L is a mapping from X to L. The set of L-subsets of X is called the L-power set of X and is denoted by L^X. For μ ∈ L^X, the set {μ(x) | x ∈ X} is called the image of μ and is denoted by Im μ. The tip and tail of μ are defined as ⋁x∈Xμ(x) and ⋀x∈Xμ(x), respectively. An L-subset μ of X is said to be contained in an L-subset η of X if μ(x) ≤ η(x) for all x ∈ X. This is denoted as μ ⊆ η. For a family {μ_i | i ∈ I} of L-subsets in X, where I is a non-empty index set, the union ∪i∈Iμi and intersection ∩i∈Iμi of {μ_i | i ∈ I} are, respectively, defined by

∪i∈Iμi(x)=⋁i∈Iμ(x) and ∩i∈Iμi(x)=⋀i∈Iμ(x),

for each x ∈ X. If μ ∈ L^X and a ∈ L, then the level subset μ_a and strong level subset μa> of μ are defined as follows:

μa={x∈X∣μ(x)≥a},

and

μa>={x∈X∣μ(x)>a},

respectively. For μ, ν ∈ L^X, it is easy to verify that, if μ ⊆ ν, then μ_a ⊆ ν_a for each a ∈ L.

For a ∈ L and x ∈ X, we define a_x ∈ L^X as follows: for all y ∈ X,

ax(y)={aif y=x,0if y≠x.

In the equation above, a_x is referred to as an L-point or L-singleton. We say that a_x is an L-point of μ if and only if μ(x) ≥ a, and we write a_x ∈ μ.

Let S be a groupoid. The set product μ ○ η of μ, η ∈ L^S is an L-subset of S defined by

μ∘η(x)=⋁x=yz{μ(y)∧η(z)}.

Note that if x cannot be factored as x = yz in S, then μ ○ η(x), being the least upper bound of the empty set, is zero. We can verify that the set product is associative in L^S if S is a semigroup.

Let f be the mapping from set X to set Y. If μ ∈ L^X and ν ∈ L^Y, then the image f(μ) of μ under f and the preimage f⁻¹(ν) of ν under f are L-subsets of Y and X respectively, defined by

f(μ)(y)=⋁x∈f-1(y){μ(x)} and f-1(ν)(x)=ν(f(x)).

If f⁻¹(y) = ϕ, then f(μ)(y) is the least upper bound of the empty set and is zero.

Proposition 2.1 ([16]

Theorem 1.1.12). Let f : X → Y be a mapping.

(i) Let {μ_i}_i_∈_I be a family of L-subsets of X. Then, f(∪i∈Iμi)=∪i∈If(μi) and f(∩i∈Iμi)⊆∩i∈If(μi).
(ii) Let μ ∈ L^X. Then, f⁻¹(f(μ)) ⊇ μ. The equality holds if f is injective.
(iii) Let ν ∈ L^Y. Then, f(f⁻¹(ν)) ⊆ ν. The equality holds if f is surjective.
(iv) Let μ ∈ L^X and ν ∈ L^Y. Then, f(μ) ⊆ ν if and only if μ ⊆ f⁻¹(ν). Moreover, if f is injective, then f⁻¹(ν) ⊆ μ if and only if ν ⊆ f(μ).

In this study, G denotes an ordinary group with the identity element ‘e’, and I denotes a non-empty indexing set. In addition, 1_A denotes the characteristic function of a non-empty set A.

In 1971, Rosenfeld [2] applied the notion of fuzzy sets to groups to introduce a fuzzy subgroup of a group. Liu [3], in 1981, extended the notion of fuzzy subgroups to L-fuzzy subgroups (L-subgroups), which we define below.

Definition 2.2 ([2])

Let μ ∈ L^G. Then, μ is called an L-subgroup of G if for each x, y ∈ G,

(i) μ(xy) ≥ μ(x) ∧ μ(y),
(ii) μ(x⁻¹) = μ(x).

The set of L-subgroups of G is denoted by L(G). Clearly, the tip of the L-subgroup is attained for the identity element of G.

Theorem 2.3 ([17])

Let μ ∈ L^G. Then, μ is an L-subgroup of G if and only if each non-empty level subset μ_a is a subgroup of G.

Theorem 2.4 ([17])

Let f : G → H be the group homomorphism. Let μ ∈ L(G) and ν ∈ L(H). Then, f(μ) ∈ L(H) and f⁻¹(ν) ∈ L(G).

It is well known that the intersection of an arbitrary family of L-subgroups of a group is an L-subgroup of a given group.

Definition 2.5 ([2])

Let μ ∈ L^G. Then, the L-subgroup of G generated by μ is defined as the smallest L-subgroup of G that contains μ. It is denoted by 〈μ〉, that is,

〈μ〉=∩{μi∈L(G)∣μ⊆μi}.

In 1982, Liu [18] introduced the concept of a normal fuzzy subgroup of a group. We define the normal L-subgroup of group G as follows:

Definition 2.6 ([18])

Let μ ∈ L(G). Then, μ is called a normal L-subgroup of G if, for all x, y ∈ G, μ(xy) = μ(yx). The set of normal L-subgroups of G is denoted by NL(G).

Theorem 2.7 ([16]

Theorem 1.3.3). Let μ ∈ L(G). Then, μ ∈ NL(G) if and only if each non-empty level subset μ_a is a normal subgroup of G.

Let η, μ ∈ L^G such that η ⊆ μ. Then, η is said to be an L-subset of μ. The set of all L-subsets of μ is denoted by L^μ. Moreover, if η, μ ∈ L(G) such that η ⊆ μ, then η is considered an L-subgroup of μ. The set of all L-subgroups of μ is denoted by L(μ).

Henceforth, μ denotes an L-subgroup of G which is considered as the parent L-group. The variable μ is an L-subgroup of G if and only if μ is an L-subgroup of 1_G.

Definition 2.8 ([12])

Let η ∈ L(μ) such that η is nonconstant and η ≠ μ. Then, η is said to be a proper L-subgroup of μ.

Clearly, η is a proper L-subgroup of μ if and only if η has distinct tip and tail and η ≠ μ.

Definition 2.9 ([10])

Let η ∈ L(μ). Let a₀ and t₀ denote the tips and tails of η, respectively. We define the trivial L-subgroup of η as follows:

ηt0a0(x)={a0if x=e,t0if x≠e.

Theorem 2.10 ([10]

Theorem 2.1). Let η ∈ L^μ. Then,

(i) η ∈ L(μ) if and only if each non-empty level subset η_a is a subgroup of μ_a.
(ii) η ∈ L(μ) if and only if each non-empty strong level subset ηa> is a subgroup of μa>, provided L is a chain.

In 1981, the normal fuzzy subgroup of a fuzzy group was introduced by Wu [19]. To develop this concept, Wu [19] preferred the L-setting. Below, we recall the notion of a normal L-subgroup of an L-group.

Definition 2.11 ([19])

Let η ∈ L(μ). Then, we say that η is a normal L-subgroup of μ if

η(yxy-1)≥η(x)∧μ(y) for all x, y∈G.

The set of normal L-subgroups of μ is denoted by NL(μ). If η ∈ NL(μ), we write η _ μ.

In this case, the arbitrary intersection of a family of normal L-subgroups of an L-group μ is again a normal L-subgroup of μ.

Theorem 2.12 ([20])

Let η ∈ L(μ). Then, η ∈ NL(μ) if and only if each non-empty level subset η_a is a normal subgroup of μ_a.

Definition 2.13 ([2])

Let μ ∈ L^X. Then, μ is said to possess sup-propery if, for each A ⊆ X, there exists a₀ ∈ A such that ⋁a∈Aμ(a)=μ(a0).

Finally, recall the following form [8, 12].

Theorem 2.14 ([8]

Theorem 3.1). Let η ∈ L^μ. Let a0=⋁x∈G{η(x)}, and define an L-subset η̂ of G by

η^ (x)=⋁a≤a0{a∣x∈〈ηa〉},

Then, η̂ ∈ L(μ) and η̂ = 〈η〉.

Theorem 2.15 ([8]

Theorem 3.7). Let η ∈ L^μ and possess the sup-property. If a0=⋁x∈G{η(x)}, then for all b ≤ a₀, 〈η_b〉 = 〈η〉_b.

Theorem 2.16 ([12]

Lemma 3.27). Let f : G → H be a group homomorphism, and let μ ∈ L(G) and ν ∈ L(H). Then, for all η ∈ L^μ, 〈f(η)〉 = f(〈η〉) and for all θ ∈ L^ν, 〈f⁻¹(θ)〉 = f⁻¹(〈θ〉).

3. Maximal L-Subgroups of a Nilpotent L-Group

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

In this section, we demonstrate that if μ is a nilpotent L-group, then the maximal L-subgroups of μ having the same tip and tail as μ are normal in μ (Theorem 3.15). To demonstrate this result, the concepts of nilpotent L-subgroups, normalizer of an L-subgroup of an L-group, and normal closure of an L-subgroup in an L-group are used. We begin by recalling the notion of cosets and the normalizer of an L-subgroup from [11].

Definition 3.1 ([11])

Let η ∈ L(μ) and a_x be an L-point of μ. The left (right) coset of η in μ with respect to a_x is defined as the set product a_x ○ η (η ○ a_x). From the definition of set product of two L-subsets, it can be easily seen that for all z ∈ G,

(ax∘η)(z)=a∧η(x-1z) and (η∘ax)(z)=a∧η(zx-1).

Definition 3.2 ([11])

Let η ∈ L(μ). The normalizer of η in μ, denoted by N(η), is the L-subgroup defined as follows:

N (η)=∪{ax∈μ∣ax∘η=η∘ax}.

N(η) is the largest L-subgroup of μ such that η is a normal L-subgroup of N(η). It was also established in [11] that η ∈ NL(μ) if and only if N(μ) = μ. Below, we provide an L-point characterization for an L-subgroup to be contained in the normalizer.

Lemma 3.3

Let η and θ ∈ L(μ). If a_x ○ b_y ○ a_x_^–1 ∈ η for all a_x ∈ θ and b_y ∈ η, then θ ⊆ N(η).

Proof

Suppose a_x ○ b_y ○ a_x_^–1 ∈ η for all a_x ∈ θ and b_y ∈ η. First, we show that for all x, y ∈ G

(1)η(xyx-1)≥η(y)∧θ(x).

Let x, y ∈ G and let a = θ(x) and b = η(y). Then, a_x ∈ θ and b_y ∈ η. Hence, by hypothesis,

(a∧b)xyx-1=ax∘by∘ax-1∈η,

which implies that

η(xyx-1)≥a∧b=η(y)∧θ(x).

This proves (1). Next, to show that θ ⊆ N(η), we claim that for all a_x ∈ θ,

(2)ax∘η=η∘ax.

Therefore, we set z ∈ G and a_x ∈ θ. Then,

(ax∘η)(z) =a∧η(x-1z) =a∧η(x-1(zx-1)x) ≥a∧θ(x)∧η(zx-1) (by (1)) =a∧η(zx-1) (since ax∈θ) =(η∘ax)(z).

Similarly, it can be shown that (η ○ a_x)(z) ≥ (a_x ○ η)(z), which proves the claim. In this case, we write A = {a_x ∈ μ : η ○ a_x = a_x ○ η}. From (2), it follows that

ax∈A for all ax∈θ.

Finally, let x ∈ G, and observe

θ(x)=∪ax∈θ{ax}⊆∪ax∈A{ax}=N(η)(x).

The equation above completes the proof.

The notion of a nilpotent L-subgroup was developed by Ajmal and Jahan [10]. For this purpose, the definition of the commutator of two L-subgroups was modified, and this modified definition was used to develop the notion of the descending central chain of an L-subgroup. We recall the following concepts as follows:

Definition 3.4 ([10])

Let η and θ ∈ L^μ. The commutator of η and θ is the L-subset (η, θ) of μ defined as follows:

(η,θ)(x)={∨{η(y)∧θ(z)}if x=[y,z] for some y,z∈G,inf η∧inf θif x≠[y,z] for any y,z∈G.

The commutator L-subgroup of η, θ ∈ L^μ, denoted by [η, θ], is defined to be the L-subgroup of μ generated by (η, θ).

Definition 3.5 ([10])

Let η ∈ L(μ). Take Z₀(η) = η and for each i ≥ 0, define Z_i₊₁(η) = [Z_i(η), η]. Then, the chain

η=Z0(η)⊇Z1(η)⊇…⊇Zi(η)⊇…

of L-subgroups of μ is called the descending central chain of η.

Definition 3.6 ([10])

Let η ∈ L(μ) with tip a₀ and tail t₀ and a₀ ≠ t₀. If the descending central chain

η=Z0(η)⊇Z1(η)⊇…⊇Zi(η)⊇…

terminates to the trivial L-subgroup ηt0a0 in a finite number of steps, then η is called a nilpotent L-subgroup of μ. Moreover, η is said to be nilpotent for class c if c is the smallest non-negative integer such that Zc(η)=ηt0a0.

In this case, we define the successive normalizers of η as follows:

η0=η and ηi+1=N(ηi) for all i≥0.

Then, according to the definition of normalizer (see Definition 3.2), η_i₊₁ is the largest L-subgroup of μ containing η_i such that η_i ◁ η_i₊₁. Consequently,

(3)η=η0⊆η1⊆…⊆ηi⊆ηi+1⊆…

is an ascending chain of L-subgroups of μ starting from η such that each η_i is a normal L-subgroup of η_i₊₁. We call (3) the ascending chain of normalizers of η in μ.

Lemma 3.7

Let μ ∈ L(G) be a nilpotent L-group and η be an L-subgroup of μ having the same tip and tail as μ. Then, the ascending chain of normalizers of η in μ is finite and terminates at μ.

Proof

Let μ be a nilpotent L-group and let η be an L-subgroup of μ having the same tip and tail as μ. Let a₀ and t₀ denote the tip and tail of μ, respectively. Let

μ=Z0(μ)⊇Z1(μ)⊇…⊇Zc(μ)=μt0a0

be the descending central chain of μ, where

Zi(μ)=[Zi-1(μ),μ] for all i≥1.

Successive normalizers of η are defined as follows:

η0=η and ηi+1=N(ηi) for all i≥0.

Then,

(4)η=η0⊆η1⊆…⊆ηi⊆ηi+1⊆…

is the ascending chain of normalizers of η in μ. We claim that

Zc-i(μ)⊆ηi for all i≥0.

We apply induction on i. When i = 0,

Zc(μ)=μt0a0=ηt0a0⊆η=η0.

Hence, this claim is true for i = 0. Suppose that the claim holds for some i ≥ 0, that is, Z_c_–_i(μ) ⊆ η_i. We shall demonstrate that

Zc-i-1(μ)⊆ηi+1.

Therefore, let a_x ∈ Z_c_–_i_–1(μ). Then, for all b_y ∈ η_i,

ax∘by∘ax-1=(a∧b)xyx-1=(a∧b)[x,y]y=(a∧b)[x,y]∘by∈[Zc-i-1(μ),μ]∘ηi (since ax∈Zc-i-1(μ),by∈μ)=Zc-i(μ)∘ηi (by definition,[Zc-i-1(μ),μ]=Zc-i(μ))⊆ηi (by the induction hypoythesis,Zc-i⊆ηi).

Therefore, by Lemma 3.3,

Zc-i-1(μ)⊆N(ηi)=ηi+1.

Hence, this claim holds true. In particular, when i = c, μ = Z₀⊆ η_c. Thus, η_c = μ, which proves the result.

We now recall the definition of the normal closure of an L-subgroup in an L-group from [9]

Definition 3.8 ([9])

Let η ∈ L(μ). The Lsubset μημ⁻¹ of μ is defined as

μημ-1(x)=⋁x=zyz-1{η(y)∧μ(z)} for each x∈G

is called the conjugate of η in μ.

Definition 3.9 ([9])

Let η ∈ L(μ). The normal closure of η in μ, denoted by η_μ, is defined as the L-subgroup of μ generated by the conjugate μημ⁻¹; that is,

ημ=〈μημ-1〉.

Moreover, η^μ is the smallest normal L-subgroup of μ that contains η.

Definition 3.10 ([9])

Let η ∈ L(μ). We inductively define a descending series of L-subgroups of μ as follows:

η(0)=μ and η(i)=ηη(i-1) for all i≥1.

Then, η⁽ⁱ⁾ is the smallest normal L-subgroup of η⁽ⁱ^–1) that contains η. We call η⁽ⁱ⁾ the ith normal closure of η in μ. The series of L-subgroups

μ=η(0)⊇η(1)⊇…⊇η(i-1)⊇η(i)⊇⋯⊇η

is called the normal closure series of η in μ.

Theorem 3.11 ([9])

Let η ∈ L(μ). Then, η ∈ NL(μ) if and only if η^μ = η.

Proposition 3.12

Let η ∈ NL(μ) and θ ∈ L(μ). Then, η ∩ θ ∈ NL(θ).

Lemma 3.13

Let μ ∈ L(G) and η ∈ L(μ). Then, there exists an ascending chain of L-subgroups

η=θ0⊆θ1⊆…⊆θn⊆…⊆μ.

These subgroups terminate at μ in a finite number of steps such that each θ_i is normal in θ_i₊₁ if and only if the normal closure series of η in μ terminates at η in a finite number of steps.

Proof

Suppose

η=θ0⊆θ1⊆…⊆θn=μ

is a finite ascending chain of L-subgroups such that each θ_i is normal in θ_i₊₁. Let η⁽ⁱ⁾ denote the ith normal closure of η in μ. We claim that

η(i)⊆θn-i for all i≥0.

We apply induction on i. When i = 0,

η(0)=μ=θn.

Hence, this claim is true for i = 0. Suppose that the claim holds for some i ≥ 0, that is, η⁽ⁱ⁾⊆ θ_n_–_i. We shall demonstrate that

η(i+1)⊆θn-i-1.

Note that, by the hypothesis, θ_n_–_i_–1 is normal in θ_n_–_i. In addition, by the induction hypothesis, η⁽ⁱ⁾⊆ θ_n_–1. Therefore, by Proposition 3.12, we have

θn-i-1∩η(i)⊲η(i).

Moreover, according to this hypothesis, η ⊆ θ_n_–_i_–1. In addition, by the definition of normal closure (see Definition 3.11), η ⊆ η₍_i₎. This implies,

η⊆θn-i-1∩η(i)⊆η(i).

Furthermore, by the definition of a normal closure, η⁽ⁱ⁺¹⁾ is the smallest normal L-subgroup of η⁽ⁱ⁾ containing η. Therefore,

η(i+1)⊆θn-i-1∩η(i)⊆θn-i-1.

This proves the result by induction. In particular, when i = n,

η⊆η(n)⊆θ0=η.

Thus, η⁽ⁿ⁾ = η and the normal closure series of η in μ terminates at η in n steps.

Conversely, if η⁽ⁿ⁾ = η for some nonnegative integer n, then the series of L-subgroups

η=η(n)⊆η(n-1)⊆…⊆η(0)=μ

is a finite ascending chain such that each η⁽ⁱ⁾ is normal in η⁽ⁱ^–1).

Now, we recall the definition of a maximal L-subgroup of an L-group from [13]:

Definition 3.14

Let μ ∈ L(G). A proper L-subgroup η of μ is said to be a maximal L-subgroup of μ if, whenever η ⊆ θ ⊆ μ for some θ ∈ L(μ), either θ = η or θ = μ.

Theorem 3.15

Let μ ∈ L(G) be a nilpotent L-group. Then, every maximal L-subgroup of μ having the same tip and tail as μ is normal in μ.

Proof

Let η be a maximal L-subgroup of μ having the same tip and tail as μ. Since μ is nilpotent, by Lemma 3.7, the ascending chain of normalizers of η in μ is finite and terminates at μ after finitely many steps. Therefore, by Lemma 3.13, the normal closure series of η in μ is also finite and terminates at η in finitely many steps, for example, m. Let the finite normal closure series be

(5)η=η(m)⊆η(m-1)⊆…⊆η(0)=μ,

where η⁽ⁱ⁾ = η^η⁽ⁱ^–1). Here, we note that η⁽¹⁾ ≠ η⁽⁰⁾. If η⁽¹⁾ = η⁽⁰⁾, then

η(2)=ηη(1)=ηη(0)=η(1)=η(0),

which implies that η = η⁽^m⁾ = η⁽⁰⁾ = μ, contradicting the fact that η ⊊ μ. Therefore, we have

(6)η⊆η(1)⊊η(0)=μ.

Since η is a maximal L-subgroup of μ, we must have η = η⁽¹⁾ = η^μ. By Theorem 3.11, it follows that η is a normal L-subgroup of μ.

Below, we provide an example of a nilpotent L-group and its maximal normal L-subgroup η:

Example 1

Let M = {l, f, a, b, c, d, u} be the lattice shown in Figure 1. Let 2 be a chain 0 < 1. Then,

M×2={(l,0),(f,0),(a,0),(b,0),(c,0),(d,0),(u,0),(l,1),(f,1),(a,1),(b,1),(c,1),(d,1),(u,1)}.

Let G = S₄ or the group of all permutations of the set {1, 2, 3, 4} with identity element e. Let

D41=〈(24),(1234)〉, D42=〈(12),(1324)〉, D43=〈(23),(1342)〉

denote the dihedral subgroups of G and V₄ = {e, (12)(34), (13)(24), (14)(23)} denote the Klein-4 subgroup of G. We define the L-subset μ of G as follows:

μ(x)={(u,1)if x=e,(d,1)if x∈V4\{e},(a,1)if x∈D41\V4,(b,1)if x∈D42\V4,(c,1)if x∈D43\V4,(f,0)if x∈S4\∪i=13D4i.

Since each non-empty level subset μ_t is a subgroup of G, by Theorem 2.3, μ is an L-subgroup of G. We show that μ is a nilpotent L-group. Since G′ = A₄, it is easy to verify that the commutator (μ, μ) is given by

(μ,μ)(x)={(u,1)if x=e,(a,1)∧(d,1)=(a,1)if x=(13),(24)(b,1)∧(d,1)=(b,1)if x=(12),(34),(c,1)∧(d,1)=(c,1)if x=(14)(23),(f,0)if x∈S4\V4.

Since every non-empty level subset (μ, μ)_t is a subgroup of μ_t, by Theorem 2.10, (μ, μ) is an L-subgroup of μ; hence, Z₁(μ) = [μ, μ] = (μ, μ). Subsequently, we can easily verify that Z₂(μ) = [μ,Z₁(μ)] is given by

Z2(μ)={(u,1)if x=e,(f,0)if x∈S4\{e}.

Hence, Z2(μ)=μt0a0, where a₀ = (u, 0) and t₀ = (f, 0) are the tips and tails of μ, respectively. We conclude that μ is a nilpotent L-group. We now define η ∈ L^μ as follows:

η(x)={(u,1)if x=e,(d,1)if x∈V4\{e},(a,1)if x∈D41\V4,(b,0)if x∈D42\V4,(c,1)if x∈D43\V4,(f,0)if x∈S4\∪i=13D4i.

Since every non-empty level subset η_t is a subgroup of μ_t, by Theorem 2.10, η ∈ L(μ). We demonstrate that η is a maximal L-subgroup of μ.

Suppose there exists θ ∈ L(μ) such that η ⊊ θ ⊆ μ. Since η(x) = μ(x) for all x∈S4\(D42\V4),θ(x)=η(x) for all x∈S4\(D42\V4). Hence, there exists x0∈D42\V4 such that

(b,1)=μ(x0)≥θ(x0)>η(x0)=(b,0),

that is, θ(x₀) = (b, 1). However, θ(x) = (b, 1) = μ(x) for all x∈D42\V4. This result implies that θ = μ. We conclude that η is the maximal L-subgroup of μ. Finally, note that η_t = μ_t for all t ∈ L \ {(b, 1)} and η₍_b_,1) = V₄ is a normal subgroup of μ(b,1)=D42. Hence, η_t is a normal subgroup of μ_t for each t ∈ L. By Theorem 2.12, η is a normal L-subgroup of μ.

4. Finitely Generated L-Subgroups

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

In this section, we introduce the concepts of the maximal condition on L-subgroups and finitely generated L-groups. These notions are closely related to classical group theory. A similar relation between these concepts has been provided in Theorem 4.10.

Definition 4.1

Let μ ∈ L(G). Then, μ is said to satisfy the maximal condition if every non-empty set S of L-subgroups of μ contains a maximal element.

Theorem 4.2

An L-subgroup μ satisfies the maximal condition if and only if each proper ascending chain of L-subgroups of μ is finite.

Proof

Suppose μ satisfies the maximal condition. Let

(7)η1⊊η2⊊…⊊ηn⊊…

be the proper ascending chain of L-subgroups in μ. Let S = {η_i | i ∈ ℕ}. Then, by hypothesis, S contains a maximal element θ, that is, there exists θ ∈ S such that θ is not contained in any other η_i. Since (7) is a proper chain, it must be terminated at θ.

Conversely, suppose that each proper ascending chain of L-subgroups in μ is finite. Let S be a non-empty set of L-subgroups of μ. We demonstrate that S contains an L-subgroup θ that is not contained in any other element of S. Since S is non-empty, there exists η₀ in S. If η₀ is not contained in any other element of S, then we set θ = η₀. Otherwise, there exists η₁ ∈ S such that η₀ ⊊ η₁. Furthermore, if η₁ is not contained in any other element of S, then we set θ = η₁. Otherwise, there exists η₂ ∈ S such that η₁ ⊊ η₂. Continuing in this manner, we obtain the ascending chain

(8)η0⊊η1⊊η2⊊….

By hypothesis, (8) is finite. Hence, there exists θ ∈ S that is not contained in any other element of S, thus, explaining the result.

Theorem 4.3

Let μ ∈ L(G) satisfy the maximal condition, and let η ∈ L(μ). Thus, η also satisfies the maximal condition. In [21], the authors have introduced the notion of finitely generated fuzzy ideals of a ring through fuzzy singletons:

Definition 4.4 ([21])

A fuzzy ideal μ of a ring R is said to be finitely generated if there exists a finite set of fuzzy singletons ϕ such that μ = 〈ϕ〉. Below, we define the finitely generated L-subgroup of a group using the above definition.

Definition 4.5

An L-subgroup μ of G is said to be finitely generated if there exist finitely many L-points a₁_x_₁, . . ., a_nx_{_n} in μ such that μ = 〈a₁_x_₁, . . ., a_nx_{_n} 〉. Recall the definition of a minimal generating set for a fuzzy set as given in [16].

Definition 4.6 ([16])

Let μ ∈ F(G). Let S denote a set of fuzzy singletons such that if x_a, x_b ∈ S, then a = b > 0 and x ≠ 0. We define the fuzzy subset χ(S) of G as follows: for all x ∈ G,

χ(S)(x)={a,if xa∈S,0,otherwise.

We set 〈S〉 = 〈χ(S)〉.

Note that the following condition

if xa,xb∈S, then a=b>0,

becomes redundant in Definition 4.5, as the concept of a finitely generated L-subgroup involves the union of L-points. Hence, Definition 4.5 is a simple version of this definition. Here, we prove the following:

Theorem 4.7

Let H be a subgroup of G. Then, H is finitely generated if and only if the characteristic function 1_H is a finitely generated L-subgroup of G.

Proof

Let H be a finitely generated subgroup of G. Then, there exist finitely many elements x₁, x₂, . . ., x_n in H such that

H=〈x1,x2,…,xn〉.

We claim that

1H=〈1x1,1x2,…,1xn〉.

Since 1_x_₁, 1_x_₂, . . ., 1_x_{_n} ∈ 1_H,

〈1x1,1x2,…,1xn〉⊆1H.

To demonstrate the reverse inclusion, let 1_z ∈ 1_H. Subsequetly, z ∈ H = 〈x₁, x₂, . . ., x_n〉. This implies

z=y1y2…yk, where yi or yi-1∈{x1,x2,…,xn}.

Thus,

1z=1y1y2…yk=1y1∘1y2∘1yk⊆〈1y1,1y2…,1yk〉⊆〈1x1,1x2,…,1xn〉.

Hence, 1_H = 〈1_x_₁, 1_x_₂, . . . 1_x_{_n}〉.

Conversely, suppose that 1_H is finitely generated. Then, by definition, there exist finitely many L-points a₁_x_₁, a₂_x_₂, . . . a_nx_{_n} of 1_H such that

1H=〈a1x1,a2x2,…anxn〉.

Clearly, we can assume that a_i > 0. This implies

1H(xi)≥ai<0 for all i=1,2,…n,

that is,

1H(xi)=1.

Thus, 1_x_₁, 1_x_₂, . . . , 1_x_{_n} ∈ 1_H; therefore,

1H=〈a1x1,a2x2,…anxn〉⊆〈1x1,1x2,…,1xn〉⊆1H.

Thus, 1_H = 〈1_x_₁, 1_x_₂, . . . 1_x_{_n}〉. We claim that H = 〈x₁, x₂, . . ., x_n〉.

Suppose, if possible, that H ≠ 〈x₁, x₂, . . ., x_n〉. Then, there exists z ∈ H such that z ∉ 〈x₁, x₂, . . . x_n〉. Note that since 1H=〈∪i=1n1xi〉 by Theorem 2.14,

1=1H(z)=⋁a≤1{a∣z∈〈(∪i=1n1xi)a〉}=⋁a≤1{a∣z∈〈∪i=1n(1xi)a〉}.

Now, since z ∉ 〈x₁, x₂, . . ., x_n〉, z∉〈∪i=1n(1xi)a〉 for any value of a > 0. It follows that 1 = 1_H(z) = 0, and this is absent. Hence, we conclude that

H=〈x1,x2,…,xn〉.

The following example illustrates Definition 4.5:

Example 2

Let G = D₈, the dihedral group of order eight is given by

D8=〈r,s∣r4=s2=e,rs=sr-1〉.

Let H₁ = {e, r²} be the center of G, H₂ = {e, s} and K = {e, r₂, s, sr²}. Let L be a lattice represented by the following equation: We define μ : G → L as follows.

μ(x)={1if x=e,bif x∈H1\{e},cif x∈H2\{e},aif x∈K\H1∪H2,0if x∈D8\K.

Because each non-empty level subset μ_t is a subgroup of G, by Theorem 2.3, μ ∈ L(G). We demonstrate that μ = 〈b_r_², c_s〉. Let η = b_r_² ∪ c_s. Then, η ∈ L^μ. Hence, 〈η〉 ⊆ μ. Now, tip(η)=⋁x∈Gη(x). Hence, by Theorem 2.14, for all x ∈ G,

(η)(x)=⋁t≤1{t∣x∈〈ηt〉}.

Note that

η0=D8,ηa={r2,s},ηb={r2},ηc={s},η1=∅.

Hence,

〈η0〉=D8,〈ηa〉=K,〈ηb〉=H1,〈ηc〉=H2,〈η1〉={e}.

Thus, 〈η〉 = 〈b_r_², c_s〉 = μ.

Recall that a lattice L is said to be upper well-ordered if every non-empty subset of L contains its supremum. Clearly, if L is upper well-ordered, then each L-subset η satisfies the sup-property. Additionally, every upper well-ordered lattice is a chain.

Lemma 4.8

Let L be an upper well-ordered lattice and μ be a finitely generated L-group. Then, every proper ascending chain of L-subgroups of μ whose set union equals μ is finite.

Proof

Suppose that μ = 〈a₁_x_₁, . . ., a_nx_{_n}〉. Let

η1⊊η2⊊…⊊ηl⊊…

be a proper ascending chain of L subgroups of μ such that ∪η_i = μ. Then,

ajxj∈∪ηi for all j=1,…,n.

This implies

∨ηi(xj)≥aj for all j=1,…,n.

Since L is upper-well ordered, there existsm_j such that η_m_{_j} (x_j) ≥ a_j, that is, a_jx_{_j} ∈ η_m_{_j} . Let m be the maximum value of m₁, . . ., m_n. Then, a_jx_{_j} ∈ η_m for all j = 1, . . ., n. Thus, μ = 〈a₁_x_₁, . . ., a_nx_{_n}〉 ⊆ η_m. Hence, the chain terminates at some k ≤ m.

Lemma 4.9 ([13])

Let L be a chain, and let μ ∈ L(G). Let C be a chain of L-subgroups of μ: Then,

∪η∈Cη∈L(μ).

Theorem 4.10

Let μ ∈ L(G). If μ satisfies the maximal condition, then every L-subgroup of μ is finitely generated. The converse holds if L is an upper well-ordered lattice.

Proof

Suppose that μ satisfies the maximal condition. Let η ∈ L(μ) and a₁_x_₁ ∈ η. If η = 〈a₁_x_₁ 〉, then η is finitely generated, and the proof is done. Hence, suppose that η ≠ 〈a₁_x_₁ 〉. Then, there exists a₂_x_₂ ∈ η such that a₂_x_₂∉ 〈a₁_x_₁ 〉. If η = 〈a₁_x_₁, a₂_x_₂ 〉, then the proof is done. Otherwise, there exists a₃_x_₃ ∈ η such that a₃_x_₃∉ 〈a₁_x_₁, a₂_x_₂ 〉. Continuing this process, we can define the L-subgroups η_i of η by choosing, as long as possible, L-points a₁_x_₁, a₂_x_₂, . . ., of η such that

ηi=〈a1x1,…,aix1〉 and ai+1xi+1∉ηi.

Then, η₁ ⊊ η₂ ⊊ . . . is a proper ascending chain of L-subgroups of η. By Theorem 4.3, η satisfies the maximal condition. Hence, according to Theorem 4.2, this chain terminates at a finite m. Thus, there does not exist any b_y ∈ η such that b_y ∉ 〈a₁_x_₁, . . ., a_mx_{_m}〉 = η_m. This implies η = η_m; thus, η is finitely generated.

Conversely, suppose that L is upper well-ordered, and let μ ∈ L(G) such that every L-subgroup of μ is finitely generated. Let η₁ ⊊ η₂ ⊊ . . . be a proper ascending chain of L-subgroups of μ. Since L is an upper well-ordered lattice, by Lemma 4.9, η = ∪η_i is an L-subgroup of μ. According to this hypothesis, η is finitely generated. Hence, by Lemma 4.8, the above chain of L-subgroups must be finite. Therefore, μ satisfies the maximal condition.

5. Frattini L-Subgroup of Finitely Generated L-Group

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

In this section, we explore some properties of the maximal and Frattini L-subgroups of the finitely generated L-subgroups. The Frattini L-subgroup of an L-group was investigated by Jahan and Manas [13]. Moreover, the authors defined the nongenerators of an L-group and established a relationship between the non-generators and the Frattini L-subgroup of μ. We recall these below.

Definition 5.1 ([13])

Let μ ∈ L(G). The Frattini L-subgroup Φ(μ) of μ is defined as the intersection of all maximal L-subgroups of μ. If μ has no maximal L-subgroups, we define Φ(μ) = μ.

Definition 5.2 ([13])

Let μ ∈ L(G). An L-point a_x ∈ μ is said to be a non-generator of μ if whenever 〈η, a_x〉 = μ for some η ∈ L^μ, then 〈η〉 = μ.

Theorem 5.3 ([13])

Let μ ∈ L(G) and define λ ∈ L^μ as follows:

λ=∪{ax∣ax is a non-generator of μ}.

Then, λ ∈ L(μ) and λ ⊆ Φ(μ). Moreover, if L is an upper well-ordered lattice, then λ = Φ(μ).

Theorem 5.4 ([13])

Let L be an upper well-ordered lattice and let μ be a normal L-subgroup of G. Then, Φ(μ) is a normal L-subgroup of μ.

Theorem 5.5

Let L be an upper well-ordered lattice and μ ∈ L(G) be a nontrivial finitely generated L-group. Then, μ has a maximal L-subgroup.

Proof

Since μ is finitely generated, there exist finitely many L-points a₁_x_₁, . . . , a_nx_{_n} such that μ = 〈a₁_x_₁, . . . , a_nx_{_n}〉. Let S be the set of all proper L-subgroups of μ. Then, S is non-empty because of the trivial L-subgroup μt0a0∈S, where a₀ = μ(e) and t₀ = inf μ. Let C = {η_i}_i_∈_I be a chain in S. We claim that

∪i∈Iηi∈S.

First, we note that by Lemma 4.9, ∪i∈Iηi∈L(μ). Hence, we only need to show that ∪i∈Iηi is a proper L-subgroup of μ. Suppose that ∪i∈Iηi is not a proper L-subgroup of μ, that is,

μ=〈a1x1,…,anxn〉=∪i∈Iηi.

Then, a₁_x_₁, . . ., a1x1,…,anxn∈∪i∈Iηi. This implies that

⋁i∈iηi(xj)≥aj for each 1≤j≤n.

Since L is upper well-ordered, for each j, there exists an L-subgroup, say η_i_{_j}, in C such that η_i_{_j} (x_j) ≥ a_j. Moreover, because C is a chain without loss of generality, we may assume that

ηi1⊆ηi2⊆∆⊆ηin.

Thus,

μ=〈a1x1,…,anxn〉=∪i∈Iηi=ηin,

which contradicts the assumption that the elements of C are proper L-subgroups of μ. Hence, ∪i∈Iηi is an appropriate L-subgroup of μ. Consequently, ∪i∈Iηi∈S, and we conclude that each chain in S has an upper bound in S. By Zorn’s lemma, S has a maximal element, which is the required maximal L-subgroup of μ.

Theorem 5.6

Let L be an upper well-ordered lattice μ be an L-group with a finitely generated Frattini L-subgroup Φ(μ). Then, the only L-subgroup η of μ such that

η∘Φ(μ)=μ

is η = μ.

Proof

Since Φ(μ) is finitely generated, there exists a finite number of L-points a₁_x_₁, . . ., a_nx_{_n} in Φ(μ) such that

Φ(μ)=〈a1x1,…,anxn〉.

Clearly,

μ=η∘Φ(μ)⊆〈η,Φ(μ)〉=〈η,a1x1,…anxn〉⊆μ.

Now, since L is upper well-ordered, by Theorem 5.3,

Φ(μ)=∪{ax∣ax is a non-generator of μ}.

Therefore, each a_ix_{_i} is a non-generator of μ. This implies

μ= 〈η,a1x1,…,anxn〉= 〈η,a1x1…an-1xn-1〉 ⋮= 〈η〉=η.

Hence, η = μ.

Lemma 5.7

Let L be an upper well-ordered lattice and μ ∈ L(G). Suppose that θ ∈ L(μ), and let a_x be an L-point of μ such that a_x ∉ θ. Then, there exists η ∈ L(μ) such that η is maximal with respect to the conditions θ ⊆ η and a_x ∉ η.

Proof

Consider the set S = {ν ∈ L(μ) | θ ⊆ ν and a_x ∉ ν}. Then, S is non-empty because θ ∈ S. Let C = {θ_i}_i_∈_I be the chain in S. We claim that

∪i∈Iθi∈S.

By Lemma 4.9, ∪i∈Iθi∈L(μ). Additionally, θ⊆∪i∈Iθi. Now, since L is upper well-ordered and a_x ∉ θ_i for all i ∈ I, ax∉∪i∈Iθi. Hence, ∪i∈Iθi∈S such that every chain in S has an upper bound in S. Therefore, according to Zorn’s lemma, S has the maximal element η, which proves the result.

Theorem 5.8

Let L be an upper well-ordered lattice and μ be a finitely generated L-group. Let η be a proper L-subgroup of μ. Then, η is contained in the maximal L-subgroup of μ.

Proof

Let μ = 〈a₁_x_₁, . . ., a_nx_{_n}〉. Since η ⊊ μ, let b₁_y_₁ be the first of a₁_x_₁, . . ., a_nx_{_n} such that b₁_y_₁∉ η. From Lemma 5.7, there exists θ₁ ∈ L(μ) such that θ₁ is maximal with respect to the conditions η ⊆ θ₁ and b₁_y_₁∉ θ₁. Hence, any L-subgroup of μ containing θ₁ also contains b₁_y_₁ . Let 〈θ₁, b₁_y_₁ 〉 = ν₁. If ν₁ = μ, then θ₁ is the required maximal L-subgroup. Otherwise, let b₂_y_₂ be the first of a₁_x_{_n}, . . ., a_nx_{_n} that is not contained in ν₁. Then, there exists θ₂ ∈ L(μ) that is maximal with respect to the conditions that ν₁⊆ θ₂ and b₂_y_₂∉ θ₂. Since μ = 〈a₁_x_₁, . . ., a_nx_{_n}〉, by continuing this process, we obtain L-points b₁_y_₁, b₂_y_₂, . . ., b_iy_{_i} and a finite chain of L-subgroups

θ1⊆θ2⊆…⊆θi

such that 〈θ_i, b_iy_{_i} 〉 = μ. Thus, θ_i is the required maximal L-subgroup of μ that contains η.

Lemma 5.9

Let L be an upper well-ordered lattice and μ be a finitely generated L-group. Let η ∈ NL(μ). Suppose that η, Φ(μ) and μ have the same tip. Then, η ⊆ Φ(μ) if and only if there is no proper subgroup θ of μ such that η ○ θ = μ.

Proof

Let η ⊆ Φ(μ). Let θ be a proper L-subgroup of μ. Then, by Theorem 5.8, there exists a maximal L-subgroup ν of μ such that θ ⊆ ν. Then, η ⊆ Φ(μ) ⊆ ν and θ ⊆ ν, implying that

η∘θ⊆ν⊊μ.

Hence, there does not exist any proper L-subgroup θ of μ such that η ○ θ = μ. Conversely, suppose there is no proper L-subgroup θ of μ such that η ○ θ = μ. If η ⊈ Φ(μ), then there exists a maximal L-subgroup ν of μ such that η ⊈ ν. Since the tip of Φ(μ) equals the tip of μ, tip ν = tip μ. Therefore,

ν⊊η∘ν⊆μ.

Since ν is a maximal L-subgroup of μ, we must have η ○ ν = μ, which contradicts our assumption. Therefore, η ⊆ Φ(μ).

Lemma 5.10

Let η, θ, σ ∈ L(μ) such that σ ⊆ η. Then,

η∩(θ∘σ)=(η∩θ)∘σ.

Proof

Let x ∈ G. Then,

(η∩(θ∘σ))(x)=η(x)∧{⋁y∈G{θ(xy-1)∧σ(y)}=⋁y∈G{η(x)∧θ(xy-1)∧η(y-1)∧σ(y)}≤⋁y∈G{η(xy-1)∧θ(xy-1)∧σ(y)}=⋁y∈G{(η∩θ)(xy-1)∧σ(y)}=(η∩θ)∘σ)(x).

Hence, η ∩ (θ ○ σ) ⊆ η ∩ θ) ○ σ. Conversely, for all y, z ∈ G such that x = yz. Hence,

(η∩θ)(y)∧σ(z)=η(y)∧θ(y)∧σ(z)=η(y)∧θ(y)∧η(z)∧σ(z)≤η(x)∧(θ(y)∧σ(z)).

Taking the supremum over all y and z, we obtain

((η∩θ)∘σ))(x)=⋁x=yz{(η∩θ)(y)∧σ(z)}≤⋁x=yz{η(x)∧{θ(y)∧σ(z)}=η(x)∧(⋁x=yz{θ(y)∧σ(z)})=(η∩(θ∘σ))(x).

Hence,

η∩(θ∘σ)=(η∩θ)∘σ.

Theorem 5.11

Let L be an upper well-ordered lattice and μ be a finitely generated L-group. Let η ∈ L(μ) and σ ∈ NL(μ) such that η, σ, Φ(η) and Φ(μ) have tips equal to tip μ. If σ ⊆ Φ(η), then σ ⊆ Φ(μ).

Proof

Suppose that σ ⊈ Φ(μ). Then, by Lemma 5.9, there exists a proper L-subgroup θ of μ such that μ = θ ○ σ. Since tip μ = tip σ, tip θ = tip μ. Now, since σ ⊆ Φ(η), σ ⊆ η ⊆ μ = θ ○ σ. Therefore, by Lemma 5.10,

η=η∩μ=η∩(θ∘σ)=(η∩θ)∘σ.

Again, since σ ⊆ Φ(η), η ∩ θ = η by Lemma 5.9. Therefore, σ ⊆ η ⊆ θ, which implies that μ = θ ○ σ = θ, contradicting the assumption that θ is a proper L-subgroup of μ. Hence, we must have σ ⊆ Φ(μ).

Acknowledgments

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

This study was supported by a Junior Research Fellowship jointly funded by CSIR and UGC, India (to Ananya Manas), during the development of this paper.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

Fig. 1.

Fig. 2.

References

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Rosenfeld, A (1971). Fuzzy groups. Journal of Mathematical Analysis and Applications. 35, 512-517.
Liu, WJ (1983). Operations on fuzzy ideals. Fuzzy Sets and Systems. 11, 31-39. https://doi.org/10.1016/s0165-0114(83)80067-0
Martinez, L (1995). Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings. Journal of Fuzzy Mathematics. 3, 833-849.
Malik, DS, and Mordeson, JN (1996). R-primary L-representations of L-ideals. Information Sciences. 88, 227-246. https://doi.org/10.1016/0020-0255(95)00166-2
Prajapati, AS, and Ajmal, N (2007). Maximal ideals of L-subrings. Journal of Fuzzy Mathematics. 15, 383-398.
Prajapati, AS, and Ajmal, N (2007). Maximal ideals of L-subrings II. Journal of Fuzzy Mathematics. 15, 399-411.
Ajmal, N, and Jahan, I (2015). Generated L-subgroup of an L- group. Iranian Journal of Fuzzy Systems. 12, 129-136.
Ajmal, N, and Jahan, I (2014). Normal closure of an L-subgroup of an L-group. The Journal of Fuzzy Mathematics. 22, 115-126.
Ajmal, N, and Jahan, I (2014). Nilpotency and theory of L- subgroups of an L-group. Fuzzy Information and Engineering. 6, 1-17. https://doi.org/10.1016/j.fiae.2014.06.001
Ajmal, N, and Jahan, I (2014). An L-point characterization of normality and normalizer of an L-subgroup of an L-group. Fuzzy Information and Engineering. 6, 147-166. https://doi.org/10.1016/j.fiae.2014.08.002
Jahan, I, and Ajmal, N (2015). Solvable L-subgroup of an L- group. Iranian Journal of Fuzzy Systems. 12, 151-166.
Jahan, I, and Manas, A (2020). Maximal and Frattini L- subgroups of an L-group. Journal of Intelligent & Fuzzy Systems. 39, 3995-4007. https://doi.org/10.3233/JIFS-200157
Gratzer, G (1978). General Lattice Theory. Basel, Switzerland: Birkhauser
Goguen, JA (1967). L-fuzzy sets. Journal of Mathematical Analysis and Applications. 18, 145-174.
Mordeson, JN, Bhutani, KR, and Rosenfeld, A (2005). Fuzzy Group Theory. Berlin, Germany: Springer
Ajmal, N (1994). Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups. Fuzzy Sets and Systems. 61, 329-339. https://doi.org/10.1016/0165-0114(94)90175-9
Liu, WJ (1982). Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets and Systems. 8, 133-139. https://doi.org/10.1016/0165-0114(82)90003-3
Wu, WM (1981). Normal fuzzy subgroups. Fuzzy Mathematics. 1, 21-30.
Ajmal, N, and Jahan, I (2012). A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group. Fuzzy Information and Engineering. 4, 123-143. https://doi.org/10.1007/s12543-012-0106-0
Jaballah, A, and Mordeson, JN (1995). Minimal generating systems for fuzzy ideals. Soochow Journal of Mathematics. 21, 183-192.

Biographies

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Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group
Acknowledgments
Conflict of Interest
Figures
References
Biographies

Iffat Jahan received a Ph.D. from the Department of Mathematics, University of Delhi, India, in 2014. She is working as a professor in the Department of Mathematics, Ramjas College, University of Delhi, India. She has authored more than 15 research papers in the area of L-group theory. Her areas of interest and research are group theory, ring theory, lattice theory, and fuzzy sets. E-mail: ij.umar@yahoo.com

Ananya Manas is a doctoral student at the Department of Mathematics, University of Delhi. His areas of research are group theory, lattice theory, and L-groups. E-mail: anayamanas@gmail.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 20-33

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.20

An Application of Maximality to Nilpotent and Finitely Generated -Subgroups of an -Group

Iffat Jahan¹ and Ananya Manas²

¹Department of Mathematics, Ramjas College, University of Delhi, Delhi, India
²Department of Mathematics, University of Delhi, Delhi, India

Correspondence to:Iffat Jahan (ij.umar@yahoo.com)

Received: November 25, 2020; Revised: September 3, 2022; Accepted: February 16, 2023

Abstract

Keywords: L-algebra, L-subgroup, Generated L-subgroup, Normal L-subgroup, Maximal L-subgroup, Frattini L-subgroup, Finitely generated L-subgroup

1. Introduction

2. Preliminaries

Let X be a non-empty set. A subset of X with cardinality L is a mapping from X to L. The set of L-subsets of X is called the L-power set of X and is denoted by L^X. For μ ∈ L^X, the set {μ(x) | x ∈ X} is called the image of μ and is denoted by Im μ. The tip and tail of μ are defined as $\underset{x \in X}{⋁} μ (x)$ and $\underset{x \in X}{⋀} μ (x)$ , respectively. An L-subset μ of X is said to be contained in an L-subset η of X if μ(x) ≤ η(x) for all x ∈ X. This is denoted as μ ⊆ η. For a family {μ_i | i ∈ I} of L-subsets in X, where I is a non-empty index set, the union $\underset{i \in I}{\cup} μ_{i}$ and intersection $\underset{i \in I}{\cap} μ_{i}$ of {μ_i | i ∈ I} are, respectively, defined by

\underset{i \in I}{\cup} μ_{i} (x) = \underset{i \in I}{⋁} μ (x) and \underset{i \in I}{\cap} μ_{i} (x) = \underset{i \in I}{⋀} μ (x),

for each x ∈ X. If μ ∈ L^X and a ∈ L, then the level subset μ_a and strong level subset $μ_{a}^{>}$ of μ are defined as follows:

μ_{a} = {x \in X ∣ μ (x) \geq a},

and

μ_{a}^{>} = {x \in X ∣ μ (x) > a},

respectively. For μ, ν ∈ L^X, it is easy to verify that, if μ ⊆ ν, then μ_a ⊆ ν_a for each a ∈ L.

For a ∈ L and x ∈ X, we define a_x ∈ L^X as follows: for all y ∈ X,

a_{x} (y) = {\begin{array}{l} a & if y = x, \\ 0 & if y \neq x . \end{array}

In the equation above, a_x is referred to as an L-point or L-singleton. We say that a_x is an L-point of μ if and only if μ(x) ≥ a, and we write a_x ∈ μ.

Let S be a groupoid. The set product μ ○ η of μ, η ∈ L^S is an L-subset of S defined by

μ \circ η (x) = \underset{x = y z}{⋁} {μ (y) \land η (z)} .

Let f be the mapping from set X to set Y. If μ ∈ L^X and ν ∈ L^Y, then the image f(μ) of μ under f and the preimage f⁻¹(ν) of ν under f are L-subsets of Y and X respectively, defined by

f (μ) (y) = \underset{x \in f^{- 1} (y)}{⋁} {μ (x)} and f^{- 1} (ν) (x) = ν (f (x)) .

If f⁻¹(y) = ϕ, then f(μ)(y) is the least upper bound of the empty set and is zero.

Proposition 2.1 ([16]

Theorem 1.1.12). Let f : X → Y be a mapping.

(i) Let {μ_i}_i_∈_I be a family of L-subsets of X. Then, $f (\underset{i \in I}{\cup} μ_{i}) = \underset{i \in I}{\cup} f (μ_{i})$ and $f (\underset{i \in I}{\cap} μ_{i}) \subseteq \underset{i \in I}{\cap} f (μ_{i})$ .
(ii) Let μ ∈ L^X. Then, f⁻¹(f(μ)) ⊇ μ. The equality holds if f is injective.
(iii) Let ν ∈ L^Y. Then, f(f⁻¹(ν)) ⊆ ν. The equality holds if f is surjective.
(iv) Let μ ∈ L^X and ν ∈ L^Y. Then, f(μ) ⊆ ν if and only if μ ⊆ f⁻¹(ν). Moreover, if f is injective, then f⁻¹(ν) ⊆ μ if and only if ν ⊆ f(μ).

In this study, G denotes an ordinary group with the identity element ‘e’, and I denotes a non-empty indexing set. In addition, 1_A denotes the characteristic function of a non-empty set A.

Definition 2.2 ([2])

Let μ ∈ L^G. Then, μ is called an L-subgroup of G if for each x, y ∈ G,

(i) μ(xy) ≥ μ(x) ∧ μ(y),
(ii) μ(x⁻¹) = μ(x).

The set of L-subgroups of G is denoted by L(G). Clearly, the tip of the L-subgroup is attained for the identity element of G.

Theorem 2.3 ([17])

Let μ ∈ L^G. Then, μ is an L-subgroup of G if and only if each non-empty level subset μ_a is a subgroup of G.

Theorem 2.4 ([17])

Let f : G → H be the group homomorphism. Let μ ∈ L(G) and ν ∈ L(H). Then, f(μ) ∈ L(H) and f⁻¹(ν) ∈ L(G).

It is well known that the intersection of an arbitrary family of L-subgroups of a group is an L-subgroup of a given group.

Definition 2.5 ([2])

Let μ ∈ L^G. Then, the L-subgroup of G generated by μ is defined as the smallest L-subgroup of G that contains μ. It is denoted by 〈μ〉, that is,

〈 μ 〉 = \cap {μ_{i} \in L (G) ∣ μ \subseteq μ_{i}} .

In 1982, Liu [18] introduced the concept of a normal fuzzy subgroup of a group. We define the normal L-subgroup of group G as follows:

Definition 2.6 ([18])

Let μ ∈ L(G). Then, μ is called a normal L-subgroup of G if, for all x, y ∈ G, μ(xy) = μ(yx). The set of normal L-subgroups of G is denoted by NL(G).

Theorem 2.7 ([16]

Theorem 1.3.3). Let μ ∈ L(G). Then, μ ∈ NL(G) if and only if each non-empty level subset μ_a is a normal subgroup of G.

Henceforth, μ denotes an L-subgroup of G which is considered as the parent L-group. The variable μ is an L-subgroup of G if and only if μ is an L-subgroup of 1_G.

Definition 2.8 ([12])

Let η ∈ L(μ) such that η is nonconstant and η ≠ μ. Then, η is said to be a proper L-subgroup of μ.

Clearly, η is a proper L-subgroup of μ if and only if η has distinct tip and tail and η ≠ μ.

Definition 2.9 ([10])

Let η ∈ L(μ). Let a₀ and t₀ denote the tips and tails of η, respectively. We define the trivial L-subgroup of η as follows:

η_{t_{0}}^{a_{0}} (x) = {\begin{array}{l} a_{0} & if x = e, \\ t_{0} & if x \neq e . \end{array}

Theorem 2.10 ([10]

Theorem 2.1). Let η ∈ L^μ. Then,

(i) η ∈ L(μ) if and only if each non-empty level subset η_a is a subgroup of μ_a.
(ii) η ∈ L(μ) if and only if each non-empty strong level subset $η_{a}^{>}$ is a subgroup of $μ_{a}^{>}$ , provided L is a chain.

Definition 2.11 ([19])

Let η ∈ L(μ). Then, we say that η is a normal L-subgroup of μ if

η (y x y^{- 1}) \geq η (x) \land μ (y) for all x, y \in G .

The set of normal L-subgroups of μ is denoted by NL(μ). If η ∈ NL(μ), we write η _ μ.

In this case, the arbitrary intersection of a family of normal L-subgroups of an L-group μ is again a normal L-subgroup of μ.

Theorem 2.12 ([20])

Let η ∈ L(μ). Then, η ∈ NL(μ) if and only if each non-empty level subset η_a is a normal subgroup of μ_a.

Definition 2.13 ([2])

Let μ ∈ L^X. Then, μ is said to possess sup-propery if, for each A ⊆ X, there exists a₀ ∈ A such that $\underset{a \in A}{⋁} μ (a) = μ (a_{0})$ .

Finally, recall the following form [8, 12].

Theorem 2.14 ([8]

Theorem 3.1). Let η ∈ L^μ. Let $a_{0} = \underset{x \in G}{⋁} {η (x)}$ , and define an L-subset η̂ of G by

\hat{η} (x) = \underset{a \leq a_{0}}{⋁} {a ∣ x \in 〈 η_{a} 〉},

Then, η̂ ∈ L(μ) and η̂ = 〈η〉.

Theorem 2.15 ([8]

Theorem 3.7). Let η ∈ L^μ and possess the sup-property. If $a_{0} = \underset{x \in G}{⋁} {η (x)}$ , then for all b ≤ a₀, 〈η_b〉 = 〈η〉_b.

Theorem 2.16 ([12]

3. Maximal L-Subgroups of a Nilpotent L-Group

Definition 3.1 ([11])

(a_{x} \circ η) (z) = a \land η (x^{- 1} z) and (η \circ a_{x}) (z) = a \land η (z x^{- 1}) .

Definition 3.2 ([11])

Let η ∈ L(μ). The normalizer of η in μ, denoted by N(η), is the L-subgroup defined as follows:

N (η) = \cup {a_{x} \in μ ∣ a_{x} \circ η = η \circ a_{x}} .

Lemma 3.3

Let η and θ ∈ L(μ). If a_x ○ b_y ○ a_x_^–1 ∈ η for all a_x ∈ θ and b_y ∈ η, then θ ⊆ N(η).

Proof

Suppose a_x ○ b_y ○ a_x_^–1 ∈ η for all a_x ∈ θ and b_y ∈ η. First, we show that for all x, y ∈ G

(1)

η (x y x^{- 1}) \geq η (y) \land θ (x) .

Let x, y ∈ G and let a = θ(x) and b = η(y). Then, a_x ∈ θ and b_y ∈ η. Hence, by hypothesis,

{(a \land b)}_{x y x^{- 1}} = a_{x} \circ b_{y} \circ a_{x^{- 1}} \in η,

which implies that

η (x y x^{- 1}) \geq a \land b = η (y) \land θ (x) .

This proves (1). Next, to show that θ ⊆ N(η), we claim that for all a_x ∈ θ,

(2)

a_{x} \circ η = η \circ a_{x} .

Therefore, we set z ∈ G and a_x ∈ θ. Then,

\begin{array}{l} (a_{x} \circ η) (z) & = a \land η (x^{- 1} z) \\ = a \land η (x^{- 1} (z x^{- 1}) x) \\ \geq a \land θ (x) \land η (z x^{- 1}) & (by (1)) \\ = a \land η (z x^{- 1}) & (since a_{x} \in θ) \\ = (η \circ a_{x}) (z) . \end{array}

Similarly, it can be shown that (η ○ a_x)(z) ≥ (a_x ○ η)(z), which proves the claim. In this case, we write A = {a_x ∈ μ : η ○ a_x = a_x ○ η}. From (2), it follows that

a_{x} \in A for all a_{x} \in θ .

Finally, let x ∈ G, and observe

θ (x) = \underset{a_{x} \in θ}{\cup} {a_{x}} \subseteq \underset{a_{x} \in A}{\cup} {a_{x}} = N (η) (x) .

The equation above completes the proof.

Definition 3.4 ([10])

Let η and θ ∈ L^μ. The commutator of η and θ is the L-subset (η, θ) of μ defined as follows:

(η, θ) (x) = {\begin{array}{l} \lor {η (y) \land θ (z)} & if x = [y, z] for some y, z \in G, \\ inf η \land inf θ & if x \neq [y, z] for any y, z \in G . \end{array}

The commutator L-subgroup of η, θ ∈ L^μ, denoted by [η, θ], is defined to be the L-subgroup of μ generated by (η, θ).

Definition 3.5 ([10])

Let η ∈ L(μ). Take Z₀(η) = η and for each i ≥ 0, define Z_i₊₁(η) = [Z_i(η), η]. Then, the chain

η = Z_{0} (η) \supseteq Z_{1} (η) \supseteq \dots \supseteq Z_{i} (η) \supseteq \dots

of L-subgroups of μ is called the descending central chain of η.

Definition 3.6 ([10])

Let η ∈ L(μ) with tip a₀ and tail t₀ and a₀ ≠ t₀. If the descending central chain

η = Z_{0} (η) \supseteq Z_{1} (η) \supseteq \dots \supseteq Z_{i} (η) \supseteq \dots

terminates to the trivial L-subgroup $η_{t_{0}}^{a_{0}}$ in a finite number of steps, then η is called a nilpotent L-subgroup of μ. Moreover, η is said to be nilpotent for class c if c is the smallest non-negative integer such that $Z_{c} (η) = η_{t_{0}}^{a_{0}}$ .

In this case, we define the successive normalizers of η as follows:

η_{0} = η and η_{i + 1} = N (η_{i}) for all i \geq 0.

Then, according to the definition of normalizer (see Definition 3.2), η_i₊₁ is the largest L-subgroup of μ containing η_i such that η_i ◁ η_i₊₁. Consequently,

(3)

η = η_{0} \subseteq η_{1} \subseteq \dots \subseteq η_{i} \subseteq η_{i + 1} \subseteq \dots

is an ascending chain of L-subgroups of μ starting from η such that each η_i is a normal L-subgroup of η_i₊₁. We call (3) the ascending chain of normalizers of η in μ.

Lemma 3.7

Let μ ∈ L(G) be a nilpotent L-group and η be an L-subgroup of μ having the same tip and tail as μ. Then, the ascending chain of normalizers of η in μ is finite and terminates at μ.

Proof

Let μ be a nilpotent L-group and let η be an L-subgroup of μ having the same tip and tail as μ. Let a₀ and t₀ denote the tip and tail of μ, respectively. Let

μ = Z_{0} (μ) \supseteq Z_{1} (μ) \supseteq \dots \supseteq Z_{c} (μ) = μ_{t_{0}}^{a_{0}}

be the descending central chain of μ, where

Z_{i} (μ) = [Z_{i - 1} (μ), μ] for all i \geq 1.

Successive normalizers of η are defined as follows:

η_{0} = η and η_{i + 1} = N (η_{i}) for all i \geq 0.

Then,

(4)

η = η_{0} \subseteq η_{1} \subseteq \dots \subseteq η_{i} \subseteq η_{i + 1} \subseteq \dots

is the ascending chain of normalizers of η in μ. We claim that

Z_{c - i} (μ) \subseteq η_{i} for all i \geq 0.

We apply induction on i. When i = 0,

Z_{c} (μ) = μ_{t_{0}}^{a_{0}} = η_{t_{0}}^{a_{0}} \subseteq η = η_{0} .

Hence, this claim is true for i = 0. Suppose that the claim holds for some i ≥ 0, that is, Z_c_–_i(μ) ⊆ η_i. We shall demonstrate that

Z_{c - i - 1} (μ) \subseteq η_{i + 1} .

Therefore, let a_x ∈ Z_c_–_i_–1(μ). Then, for all b_y ∈ η_i,

\begin{array}{l} a_{x} \circ b_{y} \circ a_{x} - 1 = {(a \land b)}_{x y x^{- 1}} \\ = {(a \land b)}_{[x, y] y} \\ = {(a \land b)}_{[x, y]} \circ b_{y} \in [Z_{c - i - 1} (μ), μ] \circ η_{i} (since a_{x} \in Z_{c - i - 1} (μ), b_{y} \in μ) \\ = Z_{c - i} (μ) \circ η_{i} (by definition, [Z_{c - i - 1} (μ), μ] = Z_{c - i} (μ)) \\ \subseteq η_{i} (by the induction hypoythesis, Z_{c - i} \subseteq η_{i}) . \end{array}

Therefore, by Lemma 3.3,

Z_{c - i - 1} (μ) \subseteq N (η_{i}) = η_{i + 1} .

Hence, this claim holds true. In particular, when i = c, μ = Z₀⊆ η_c. Thus, η_c = μ, which proves the result.

We now recall the definition of the normal closure of an L-subgroup in an L-group from [9]

Definition 3.8 ([9])

Let η ∈ L(μ). The Lsubset μημ⁻¹ of μ is defined as

μ η μ^{- 1} (x) = \underset{x = z y z^{- 1}}{⋁} {η (y) \land μ (z)} for each x \in G

is called the conjugate of η in μ.

Definition 3.9 ([9])

Let η ∈ L(μ). The normal closure of η in μ, denoted by η_μ, is defined as the L-subgroup of μ generated by the conjugate μημ⁻¹; that is,

η^{μ} = 〈 μ η μ^{- 1} 〉 .

Moreover, η^μ is the smallest normal L-subgroup of μ that contains η.

Definition 3.10 ([9])

Let η ∈ L(μ). We inductively define a descending series of L-subgroups of μ as follows:

η^{(0)} = μ and η^{(i)} = η^{η^{(i - 1)}} for all i \geq 1.

Then, η⁽ⁱ⁾ is the smallest normal L-subgroup of η⁽ⁱ^–1) that contains η. We call η⁽ⁱ⁾ the ith normal closure of η in μ. The series of L-subgroups

μ = η^{(0)} \supseteq η^{(1)} \supseteq \dots \supseteq η^{(i - 1)} \supseteq η^{(i)} \supseteq \dots \supseteq η

is called the normal closure series of η in μ.

Theorem 3.11 ([9])

Let η ∈ L(μ). Then, η ∈ NL(μ) if and only if η^μ = η.

Proposition 3.12

Let η ∈ NL(μ) and θ ∈ L(μ). Then, η ∩ θ ∈ NL(θ).

Lemma 3.13

Let μ ∈ L(G) and η ∈ L(μ). Then, there exists an ascending chain of L-subgroups

η = θ_{0} \subseteq θ_{1} \subseteq \dots \subseteq θ_{n} \subseteq \dots \subseteq μ .

Proof

Suppose

η = θ_{0} \subseteq θ_{1} \subseteq \dots \subseteq θ_{n} = μ

is a finite ascending chain of L-subgroups such that each θ_i is normal in θ_i₊₁. Let η⁽ⁱ⁾ denote the ith normal closure of η in μ. We claim that

η^{(i)} \subseteq θ_{n - i} for all i \geq 0.

We apply induction on i. When i = 0,

η^{(0)} = μ = θ_{n} .

Hence, this claim is true for i = 0. Suppose that the claim holds for some i ≥ 0, that is, η⁽ⁱ⁾⊆ θ_n_–_i. We shall demonstrate that

η^{(i + 1)} \subseteq θ_{n - i - 1} .

Note that, by the hypothesis, θ_n_–_i_–1 is normal in θ_n_–_i. In addition, by the induction hypothesis, η⁽ⁱ⁾⊆ θ_n_–1. Therefore, by Proposition 3.12, we have

θ_{n - i - 1} \cap η^{(i)} ⊲ η^{(i)} .

Moreover, according to this hypothesis, η ⊆ θ_n_–_i_–1. In addition, by the definition of normal closure (see Definition 3.11), η ⊆ η₍_i₎. This implies,

η \subseteq θ_{n - i - 1} \cap η^{(i)} \subseteq η^{(i)} .

Furthermore, by the definition of a normal closure, η⁽ⁱ⁺¹⁾ is the smallest normal L-subgroup of η⁽ⁱ⁾ containing η. Therefore,

η^{(i + 1)} \subseteq θ_{n - i - 1} \cap η^{(i)} \subseteq θ_{n - i - 1} .

This proves the result by induction. In particular, when i = n,

η \subseteq η^{(n)} \subseteq θ_{0} = η .

Thus, η⁽ⁿ⁾ = η and the normal closure series of η in μ terminates at η in n steps.

Conversely, if η⁽ⁿ⁾ = η for some nonnegative integer n, then the series of L-subgroups

η = η^{(n)} \subseteq η^{(n - 1)} \subseteq \dots \subseteq η^{(0)} = μ

is a finite ascending chain such that each η⁽ⁱ⁾ is normal in η⁽ⁱ^–1).

Now, we recall the definition of a maximal L-subgroup of an L-group from [13]:

Definition 3.14

Let μ ∈ L(G). A proper L-subgroup η of μ is said to be a maximal L-subgroup of μ if, whenever η ⊆ θ ⊆ μ for some θ ∈ L(μ), either θ = η or θ = μ.

Theorem 3.15

Let μ ∈ L(G) be a nilpotent L-group. Then, every maximal L-subgroup of μ having the same tip and tail as μ is normal in μ.

Proof

(5)

η = η^{(m)} \subseteq η^{(m - 1)} \subseteq \dots \subseteq η^{(0)} = μ,

where η⁽ⁱ⁾ = η^η⁽ⁱ^–1). Here, we note that η⁽¹⁾ ≠ η⁽⁰⁾. If η⁽¹⁾ = η⁽⁰⁾, then

η^{(2)} = η^{η^{(1)}} = η^{η^{(0)}} = η^{(1)} = η^{(0)},

which implies that η = η⁽^m⁾ = η⁽⁰⁾ = μ, contradicting the fact that η ⊊ μ. Therefore, we have

(6)

η \subseteq η^{(1)} ⊊ η^{(0)} = μ .

Since η is a maximal L-subgroup of μ, we must have η = η⁽¹⁾ = η^μ. By Theorem 3.11, it follows that η is a normal L-subgroup of μ.

Below, we provide an example of a nilpotent L-group and its maximal normal L-subgroup η:

Example 1

Let M = {l, f, a, b, c, d, u} be the lattice shown in Figure 1. Let 2 be a chain 0 < 1. Then,

\begin{array}{r} M \times 2 = {(l, 0), (f, 0), (a, 0), (b, 0), (c, 0), (d, 0), (u, 0), \\ (l, 1), (f, 1), (a, 1), (b, 1), (c, 1), (d, 1), (u, 1)} . \end{array}

Let G = S₄ or the group of all permutations of the set {1, 2, 3, 4} with identity element e. Let

\begin{array}{l} D_{4}^{1} = 〈 (24), (1234) 〉, D_{4}^{2} = 〈 (12), (1324) 〉, \\ D_{4}^{3} = 〈 (23), (1342) 〉 \end{array}

denote the dihedral subgroups of G and V₄ = {e, (12)(34), (13)(24), (14)(23)} denote the Klein-4 subgroup of G. We define the L-subset μ of G as follows:

μ (x) = {\begin{array}{l} (u, 1) & if x = e, \\ (d, 1) & if x \in V_{4} \ {e}, \\ (a, 1) & if x \in D_{4}^{1} \ V_{4}, \\ (b, 1) & if x \in D_{4}^{2} \ V_{4}, \\ (c, 1) & if x \in D_{4}^{3} \ V_{4}, \\ (f, 0) & if x \in S_{4} \ \cup_{i = 1}^{3} D_{4}^{i} . \end{array}

(μ, μ) (x) = {\begin{array}{l} (u, 1) & if x = e, \\ (a, 1) \land (d, 1) = (a, 1) & if x = (13), (24) \\ (b, 1) \land (d, 1) = (b, 1) & if x = (12), (34), \\ (c, 1) \land (d, 1) = (c, 1) & if x = (14) (23), \\ (f, 0) & if x \in S_{4} \ V_{4} . \end{array}

Z_{2} (μ) = {\begin{array}{l} (u, 1) & if x = e, \\ (f, 0) & if x \in S_{4} \ {e} . \end{array}

Hence, $Z_{2} (μ) = μ_{t_{0}}^{a_{0}}$ , where a₀ = (u, 0) and t₀ = (f, 0) are the tips and tails of μ, respectively. We conclude that μ is a nilpotent L-group. We now define η ∈ L^μ as follows:

η (x) = {\begin{array}{l} (u, 1) & if x = e, \\ (d, 1) & if x \in V_{4} \ {e}, \\ (a, 1) & if x \in D_{4}^{1} \ V_{4}, \\ (b, 0) & if x \in D_{4}^{2} \ V_{4}, \\ (c, 1) & if x \in D_{4}^{3} \ V_{4}, \\ (f, 0) & if x \in S_{4} \ \cup_{i = 1}^{3} D_{4}^{i} . \end{array}

Since every non-empty level subset η_t is a subgroup of μ_t, by Theorem 2.10, η ∈ L(μ). We demonstrate that η is a maximal L-subgroup of μ.

Suppose there exists θ ∈ L(μ) such that η ⊊ θ ⊆ μ. Since η(x) = μ(x) for all $x \in S_{4} \ (D_{4}^{2} \ V_{4}), θ (x) = η (x)$ for all $x \in S_{4} \ (D_{4}^{2} \ V_{4})$ . Hence, there exists $x_{0} \in D_{4}^{2} \ V_{4}$ such that

(b, 1) = μ (x_{0}) \geq θ (x_{0}) > η (x_{0}) = (b, 0),

that is, θ(x₀) = (b, 1). However, θ(x) = (b, 1) = μ(x) for all $x \in D_{4}^{2} \ V_{4}$ . This result implies that θ = μ. We conclude that η is the maximal L-subgroup of μ. Finally, note that η_t = μ_t for all t ∈ L \ {(b, 1)} and η₍_b_,1) = V₄ is a normal subgroup of $μ_{(b, 1)} = D_{4}^{2}$ . Hence, η_t is a normal subgroup of μ_t for each t ∈ L. By Theorem 2.12, η is a normal L-subgroup of μ.

4. Finitely Generated L-Subgroups

Definition 4.1

Let μ ∈ L(G). Then, μ is said to satisfy the maximal condition if every non-empty set S of L-subgroups of μ contains a maximal element.

Theorem 4.2

An L-subgroup μ satisfies the maximal condition if and only if each proper ascending chain of L-subgroups of μ is finite.

Proof

Suppose μ satisfies the maximal condition. Let

(7)

η_{1} ⊊ η_{2} ⊊ \dots ⊊ η_{n} ⊊ \dots

(8)

η_{0} ⊊ η_{1} ⊊ η_{2} ⊊ \dots .

By hypothesis, (8) is finite. Hence, there exists θ ∈ S that is not contained in any other element of S, thus, explaining the result.

Theorem 4.3

Definition 4.4 ([21])

Definition 4.5

Definition 4.6 ([16])

Let μ ∈ F(G). Let S denote a set of fuzzy singletons such that if x_a, x_b ∈ S, then a = b > 0 and x ≠ 0. We define the fuzzy subset χ(S) of G as follows: for all x ∈ G,

χ (S) (x) = {\begin{array}{l} a, & if x_{a} \in S, \\ 0, & otherwise. \end{array}

We set 〈S〉 = 〈χ(S)〉.

Note that the following condition

if x_{a}, x_{b} \in S, then a = b > 0,

Theorem 4.7

Let H be a subgroup of G. Then, H is finitely generated if and only if the characteristic function 1_H is a finitely generated L-subgroup of G.

Proof

Let H be a finitely generated subgroup of G. Then, there exist finitely many elements x₁, x₂, . . ., x_n in H such that

H = 〈 x_{1}, x_{2}, \dots, x_{n} 〉 .

We claim that

1_{H} = 〈 1_{x_{1}}, 1_{x_{2}}, \dots, 1_{x_{n}} 〉 .

Since 1_x_₁, 1_x_₂, . . ., 1_x_{_n} ∈ 1_H,

〈 1_{x_{1}}, 1_{x_{2}}, \dots, 1_{x_{n}} 〉 \subseteq 1_{H} .

To demonstrate the reverse inclusion, let 1_z ∈ 1_H. Subsequetly, z ∈ H = 〈x₁, x₂, . . ., x_n〉. This implies

z = y_{1} y_{2} \dots y_{k}, where y_{i} or {y_{i}}^{- 1} \in {x_{1}, x_{2}, \dots, x_{n}} .

Thus,

\begin{array}{l} 1_{z} = 1_{y_{1} y_{2} \dots y_{k}} \\ = 1_{y_{1}} \circ 1_{y_{2}} \circ 1_{y_{k}} \\ \subseteq 〈 1_{y_{1}}, 1_{y_{2}} \dots, 1_{y_{k}} 〉 \\ \subseteq 〈 1_{x_{1}}, 1_{x_{2}}, \dots, 1_{x_{n}} 〉 . \end{array}

Hence, 1_H = 〈1_x_₁, 1_x_₂, . . . 1_x_{_n}〉.

Conversely, suppose that 1_H is finitely generated. Then, by definition, there exist finitely many L-points a₁_x_₁, a₂_x_₂, . . . a_nx_{_n} of 1_H such that

1_{H} = 〈 a_{1_{x_{1}}}, a_{2_{x_{2}}}, \dots a_{n_{x_{n}}} 〉 .

Clearly, we can assume that a_i > 0. This implies

1_{H} (x_{i}) \geq a_{i} < 0 for all i = 1, 2, \dots n,

that is,

1_{H} (x_{i}) = 1.

Thus, 1_x_₁, 1_x_₂, . . . , 1_x_{_n} ∈ 1_H; therefore,

1_{H} = 〈 a_{1_{x_{1}}}, a_{2_{x_{2}}}, \dots a_{n_{x_{n}}} 〉 \subseteq 〈 1_{x_{1}}, 1_{x_{2}}, \dots, 1_{x_{n}} 〉 \subseteq 1_{H} .

Thus, 1_H = 〈1_x_₁, 1_x_₂, . . . 1_x_{_n}〉. We claim that H = 〈x₁, x₂, . . ., x_n〉.

Suppose, if possible, that H ≠ 〈x₁, x₂, . . ., x_n〉. Then, there exists z ∈ H such that z ∉ 〈x₁, x₂, . . . x_n〉. Note that since $1_{H} = 〈 \cup_{i = 1}^{n} 1_{x_{i}} 〉$ by Theorem 2.14,

\begin{array}{l} 1 = 1_{H} (z) = \underset{a \leq 1}{⋁} {a ∣ z \in 〈 {(\cup_{i = 1}^{n} 1_{x_{i}})}_{a} 〉} \\ = \underset{a \leq 1}{⋁} {a ∣ z \in 〈 \cup_{i = 1}^{n} {(1_{x_{i}})}_{a} 〉} . \end{array}

Now, since z ∉ 〈x₁, x₂, . . ., x_n〉, $z \notin 〈 \cup_{i = 1}^{n} {(1_{x_{i}})}_{a} 〉$ for any value of a > 0. It follows that 1 = 1_H(z) = 0, and this is absent. Hence, we conclude that

H = 〈 x_{1}, x_{2}, \dots, x_{n} 〉 .

The following example illustrates Definition 4.5:

Example 2

Let G = D₈, the dihedral group of order eight is given by

D_{8} = 〈 r, s ∣ r^{4} = s^{2} = e, r s = s r^{- 1} 〉 .

Let H₁ = {e, r²} be the center of G, H₂ = {e, s} and K = {e, r₂, s, sr²}. Let L be a lattice represented by the following equation: We define μ : G → L as follows.

μ (x) = {\begin{array}{l} 1 & if x = e, \\ b & if x \in H_{1} \ {e}, \\ c & if x \in H_{2} \ {e}, \\ a & if x \in K \ H_{1} \cup H_{2}, \\ 0 & if x \in D_{8} \ K . \end{array}

Because each non-empty level subset μ_t is a subgroup of G, by Theorem 2.3, μ ∈ L(G). We demonstrate that μ = 〈b_r_², c_s〉. Let η = b_r_² ∪ c_s. Then, η ∈ L^μ. Hence, 〈η〉 ⊆ μ. Now, $tip (η) = \underset{x \in G}{⋁} η (x)$ . Hence, by Theorem 2.14, for all x ∈ G,

(η) (x) = \underset{t \leq 1}{⋁} {t ∣ x \in 〈 η_{t} 〉} .

Note that

η_{0} = D_{8}, η_{a} = {r^{2}, s}, η_{b} = {r^{2}}, η_{c} = {s}, η_{1} = \emptyset .

Hence,

〈 η_{0} 〉 = D_{8}, 〈 η_{a} 〉 = K, 〈 η_{b} 〉 = H_{1}, 〈 η_{c} 〉 = H_{2}, 〈 η_{1} 〉 = {e} .

Thus, 〈η〉 = 〈b_r_², c_s〉 = μ.

Lemma 4.8

Let L be an upper well-ordered lattice and μ be a finitely generated L-group. Then, every proper ascending chain of L-subgroups of μ whose set union equals μ is finite.

Proof

Suppose that μ = 〈a₁_x_₁, . . ., a_nx_{_n}〉. Let

η_{1} ⊊ η_{2} ⊊ \dots ⊊ η_{l} ⊊ \dots

be a proper ascending chain of L subgroups of μ such that ∪η_i = μ. Then,

a_{j_{x_{j}}} \in \cup η_{i} for all j = 1, \dots, n .

This implies

\lor η_{i} (x_{j}) \geq a_{j} for all j = 1, \dots, n .

Lemma 4.9 ([13])

Let L be a chain, and let μ ∈ L(G). Let C be a chain of L-subgroups of μ: Then,

\underset{η \in C}{\cup} η \in L (μ) .

Theorem 4.10

Let μ ∈ L(G). If μ satisfies the maximal condition, then every L-subgroup of μ is finitely generated. The converse holds if L is an upper well-ordered lattice.

Proof

η_{i} = 〈 a_{1_{x_{1}}}, \dots, a_{i_{x_{1}}} 〉 and a_{i + 1_{x_{i + 1}}} \notin η_{i} .

5. Frattini L-Subgroup of Finitely Generated L-Group

Definition 5.1 ([13])

Let μ ∈ L(G). The Frattini L-subgroup Φ(μ) of μ is defined as the intersection of all maximal L-subgroups of μ. If μ has no maximal L-subgroups, we define Φ(μ) = μ.

Definition 5.2 ([13])

Let μ ∈ L(G). An L-point a_x ∈ μ is said to be a non-generator of μ if whenever 〈η, a_x〉 = μ for some η ∈ L^μ, then 〈η〉 = μ.

Theorem 5.3 ([13])

Let μ ∈ L(G) and define λ ∈ L^μ as follows:

λ = \cup {a_{x} ∣ a_{x} is a non-generator of μ} .

Then, λ ∈ L(μ) and λ ⊆ Φ(μ). Moreover, if L is an upper well-ordered lattice, then λ = Φ(μ).

Theorem 5.4 ([13])

Let L be an upper well-ordered lattice and let μ be a normal L-subgroup of G. Then, Φ(μ) is a normal L-subgroup of μ.

Theorem 5.5

Let L be an upper well-ordered lattice and μ ∈ L(G) be a nontrivial finitely generated L-group. Then, μ has a maximal L-subgroup.

Proof

Since μ is finitely generated, there exist finitely many L-points a₁_x_₁, . . . , a_nx_{_n} such that μ = 〈a₁_x_₁, . . . , a_nx_{_n}〉. Let S be the set of all proper L-subgroups of μ. Then, S is non-empty because of the trivial L-subgroup $μ_{t_{0}}^{a_{0}} \in S$ , where a₀ = μ(e) and t₀ = inf μ. Let C = {η_i}_i_∈_I be a chain in S. We claim that

\underset{i \in I}{\cup} η_{i} \in S .

First, we note that by Lemma 4.9, $\underset{i \in I}{\cup} η_{i} \in L (μ)$ . Hence, we only need to show that $\underset{i \in I}{\cup} η_{i}$ is a proper L-subgroup of μ. Suppose that $\underset{i \in I}{\cup} η_{i}$ is not a proper L-subgroup of μ, that is,

μ = 〈 a_{1_{x_{1}}}, \dots, a_{n_{x_{n}}} 〉 = \underset{i \in I}{\cup} η_{i} .

Then, a₁_x_₁, . . ., $a_{1_{x_{1}}}, \dots, a_{n_{x_{n}}} \in \underset{i \in I}{\cup} η_{i}$ . This implies that

\underset{i \in i}{⋁} η_{i} (x_{j}) \geq a_{j} for each 1 \leq j \leq n .

η_{i_{1}} \subseteq η_{i_{2}} \subseteq ∆ \subseteq η_{i_{n}} .

Thus,

μ = 〈 a_{1_{x_{1}}}, \dots, a_{n_{x_{n}}} 〉 = \underset{i \in I}{\cup} η_{i} = η_{i_{n}},

which contradicts the assumption that the elements of C are proper L-subgroups of μ. Hence, $\underset{i \in I}{\cup} η_{i}$ is an appropriate L-subgroup of μ. Consequently, $\underset{i \in I}{\cup} η_{i} \in S$ , and we conclude that each chain in S has an upper bound in S. By Zorn’s lemma, S has a maximal element, which is the required maximal L-subgroup of μ.

Theorem 5.6

Let L be an upper well-ordered lattice μ be an L-group with a finitely generated Frattini L-subgroup Φ(μ). Then, the only L-subgroup η of μ such that

η \circ Φ (μ) = μ

is η = μ.

Proof

Since Φ(μ) is finitely generated, there exists a finite number of L-points a₁_x_₁, . . ., a_nx_{_n} in Φ(μ) such that

Φ (μ) = 〈 a_{1_{x_{1}}}, \dots, a_{n_{x_{n}}} 〉 .

Clearly,

\begin{array}{l} μ = η \circ Φ (μ) \\ \subseteq 〈 η, Φ (μ) 〉 \\ = 〈 η, a_{1_{x_{1}}}, \dots a_{n_{x_{n}}} 〉 \\ \subseteq μ . \end{array}

Now, since L is upper well-ordered, by Theorem 5.3,

Φ (μ) = \cup {a_{x} ∣ a_{x} is a non-generator of μ} .

Therefore, each a_ix_{_i} is a non-generator of μ. This implies

\begin{array}{l} μ = & 〈 η, a_{1_{x_{1}}}, \dots, a_{n_{x_{n}}} 〉 \\ = & 〈 η, a_{1_{x_{1}}} \dots a_{n - 1_{x_{n - 1}}} 〉 \\ ⋮ \\ = & 〈 η 〉 = η . \end{array}

Hence, η = μ.

Lemma 5.7

Proof

Consider the set S = {ν ∈ L(μ) | θ ⊆ ν and a_x ∉ ν}. Then, S is non-empty because θ ∈ S. Let C = {θ_i}_i_∈_I be the chain in S. We claim that

\underset{i \in I}{\cup} θ_{i} \in S .

By Lemma 4.9, $\underset{i \in I}{\cup} θ_{i} \in L (μ)$ . Additionally, $θ \subseteq \underset{i \in I}{\cup} θ_{i}$ . Now, since L is upper well-ordered and a_x ∉ θ_i for all i ∈ I, $a_{x} \notin \underset{i \in I}{\cup} θ_{i}$ . Hence, $\underset{i \in I}{\cup} θ_{i} \in S$ such that every chain in S has an upper bound in S. Therefore, according to Zorn’s lemma, S has the maximal element η, which proves the result.

Theorem 5.8

Let L be an upper well-ordered lattice and μ be a finitely generated L-group. Let η be a proper L-subgroup of μ. Then, η is contained in the maximal L-subgroup of μ.

Proof

θ_{1} \subseteq θ_{2} \subseteq \dots \subseteq θ_{i}

such that 〈θ_i, b_iy_{_i} 〉 = μ. Thus, θ_i is the required maximal L-subgroup of μ that contains η.

Lemma 5.9

Proof

Let η ⊆ Φ(μ). Let θ be a proper L-subgroup of μ. Then, by Theorem 5.8, there exists a maximal L-subgroup ν of μ such that θ ⊆ ν. Then, η ⊆ Φ(μ) ⊆ ν and θ ⊆ ν, implying that

η \circ θ \subseteq ν ⊊ μ .

ν ⊊ η \circ ν \subseteq μ .

Since ν is a maximal L-subgroup of μ, we must have η ○ ν = μ, which contradicts our assumption. Therefore, η ⊆ Φ(μ).

Lemma 5.10

Let η, θ, σ ∈ L(μ) such that σ ⊆ η. Then,

η \cap (θ \circ σ) = (η \cap θ) \circ σ .

Proof

Let x ∈ G. Then,

\begin{array}{l} (η \cap (θ \circ σ)) (x) = η (x) \land {\underset{y \in G}{⋁} {θ (x y^{- 1}) \land σ (y)} \\ = \underset{y \in G}{⋁} {η (x) \land θ (x y^{- 1}) \land η (y^{- 1}) \land σ (y)} \\ \leq \underset{y \in G}{⋁} {η (x y^{- 1}) \land θ (x y^{- 1}) \land σ (y)} \\ = \underset{y \in G}{⋁} {(η \cap θ) (x y^{- 1}) \land σ (y)} \\ = (η \cap θ) \circ σ) (x) . \end{array}

Hence, η ∩ (θ ○ σ) ⊆ η ∩ θ) ○ σ. Conversely, for all y, z ∈ G such that x = yz. Hence,

\begin{array}{l} (η \cap θ) (y) \land σ (z) = η (y) \land θ (y) \land σ (z) \\ = η (y) \land θ (y) \land η (z) \land σ (z) \\ \leq η (x) \land (θ (y) \land σ (z)) . \end{array}

Taking the supremum over all y and z, we obtain

\begin{array}{l} ((η \cap θ) \circ σ)) (x) = \underset{x = y z}{⋁} {(η \cap θ) (y) \land σ (z)} \\ \leq \underset{x = y z}{⋁} {η (x) \land {θ (y) \land σ (z)} \\ = η (x) \land (\underset{x = y z}{⋁} {θ (y) \land σ (z)}) \\ = (η \cap (θ \circ σ)) (x) . \end{array}

Hence,

η \cap (θ \circ σ) = (η \cap θ) \circ σ .

Theorem 5.11

Proof

η = η \cap μ = η \cap (θ \circ σ) = (η \cap θ) \circ σ .

Abstract
1. Introduction
2. Preliminaries
3. Maximal L-Subgroups of a Nilpotent L-Group
4. Finitely Generated L-Subgroups
5. Frattini L-Subgroup of Finitely Generated L-Group

References

Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Rosenfeld, A (1971). Fuzzy groups. Journal of Mathematical Analysis and Applications. 35, 512-517.
Liu, WJ (1983). Operations on fuzzy ideals. Fuzzy Sets and Systems. 11, 31-39. https://doi.org/10.1016/s0165-0114(83)80067-0
Martinez, L (1995). Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings. Journal of Fuzzy Mathematics. 3, 833-849.
Malik, DS, and Mordeson, JN (1996). R-primary L-representations of L-ideals. Information Sciences. 88, 227-246. https://doi.org/10.1016/0020-0255(95)00166-2
Prajapati, AS, and Ajmal, N (2007). Maximal ideals of L-subrings. Journal of Fuzzy Mathematics. 15, 383-398.
Prajapati, AS, and Ajmal, N (2007). Maximal ideals of L-subrings II. Journal of Fuzzy Mathematics. 15, 399-411.
Ajmal, N, and Jahan, I (2015). Generated L-subgroup of an L- group. Iranian Journal of Fuzzy Systems. 12, 129-136.
Ajmal, N, and Jahan, I (2014). Normal closure of an L-subgroup of an L-group. The Journal of Fuzzy Mathematics. 22, 115-126.
Ajmal, N, and Jahan, I (2014). Nilpotency and theory of L- subgroups of an L-group. Fuzzy Information and Engineering. 6, 1-17. https://doi.org/10.1016/j.fiae.2014.06.001
Ajmal, N, and Jahan, I (2014). An L-point characterization of normality and normalizer of an L-subgroup of an L-group. Fuzzy Information and Engineering. 6, 147-166. https://doi.org/10.1016/j.fiae.2014.08.002
Jahan, I, and Ajmal, N (2015). Solvable L-subgroup of an L- group. Iranian Journal of Fuzzy Systems. 12, 151-166.
Jahan, I, and Manas, A (2020). Maximal and Frattini L- subgroups of an L-group. Journal of Intelligent & Fuzzy Systems. 39, 3995-4007. https://doi.org/10.3233/JIFS-200157
Gratzer, G (1978). General Lattice Theory. Basel, Switzerland: Birkhauser
Goguen, JA (1967). L-fuzzy sets. Journal of Mathematical Analysis and Applications. 18, 145-174.
Mordeson, JN, Bhutani, KR, and Rosenfeld, A (2005). Fuzzy Group Theory. Berlin, Germany: Springer
Ajmal, N (1994). Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups. Fuzzy Sets and Systems. 61, 329-339. https://doi.org/10.1016/0165-0114(94)90175-9
Liu, WJ (1982). Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets and Systems. 8, 133-139. https://doi.org/10.1016/0165-0114(82)90003-3
Wu, WM (1981). Normal fuzzy subgroups. Fuzzy Mathematics. 1, 21-30.
Ajmal, N, and Jahan, I (2012). A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group. Fuzzy Information and Engineering. 4, 123-143. https://doi.org/10.1007/s12543-012-0106-0
Jaballah, A, and Mordeson, JN (1995). Minimal generating systems for fuzzy ideals. Soochow Journal of Mathematics. 21, 183-192.

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SCOPUS 1

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

An Application of Maximality to Nilpotent and Finitely Generated -Subgroups of an -Group

Abstract

1. Introduction

2. Preliminaries

Proposition 2.1 ([16]

Definition 2.2 ([2])

Theorem 2.3 ([17])

Theorem 2.4 ([17])

Definition 2.5 ([2])

Definition 2.6 ([18])

Theorem 2.7 ([16]

Definition 2.8 ([12])

Definition 2.9 ([10])

Theorem 2.10 ([10]

Definition 2.11 ([19])

Theorem 2.12 ([20])

Definition 2.13 ([2])

Theorem 2.14 ([8]

Theorem 2.15 ([8]

Theorem 2.16 ([12]

3. Maximal L-Subgroups of a Nilpotent L-Group

Definition 3.1 ([11])

Definition 3.2 ([11])

Lemma 3.3

Definition 3.4 ([10])

Definition 3.5 ([10])

Definition 3.6 ([10])

Lemma 3.7

Definition 3.8 ([9])

Definition 3.9 ([9])

Definition 3.10 ([9])

Theorem 3.11 ([9])

Proposition 3.12

Lemma 3.13

Definition 3.14

Theorem 3.15

Example 1

4. Finitely Generated L-Subgroups

Definition 4.1

Theorem 4.2

Theorem 4.3

Definition 4.4 ([21])

Definition 4.5

Definition 4.6 ([16])

Theorem 4.7

Example 2

Lemma 4.8

Lemma 4.9 ([13])

Theorem 4.10

Proof

5. Frattini L-Subgroup of Finitely Generated L-Group

Definition 5.1 ([13])

Definition 5.2 ([13])

Theorem 5.3 ([13])

Theorem 5.4 ([13])

Theorem 5.5

Theorem 5.6

Lemma 5.7

Theorem 5.8

Lemma 5.9

Lemma 5.10

Theorem 5.11

Acknowledgments

Conflict of Interest

Figures

References

Biographies

Article

Original Article

An Application of Maximality to Nilpotent and Finitely Generated -Subgroups of an -Group

Abstract

1. Introduction

2. Preliminaries

Proposition 2.1 ([16]

Definition 2.2 ([2])

Theorem 2.3 ([17])

Theorem 2.4 ([17])

Definition 2.5 ([2])

International Journal
of Fuzzy Logic and
Intelligent Systems