The concept of a fuzzy set was developed by Zadeh [1] in 1965, and it was applied by Rosenfeld [2] to group theory in 1971, which led to the evolution of fuzzy group theory. In 1982, Liu [3] introduced the concept of lattice-valued fuzzy (L)-subgroups. This pioneered studies on L-algebraic substructures. Recently, various concepts of classical group theory, such as characteristic subgroups, normalizer of a subgroup, nilpotent subgroups, solvable subgroups, normal closure of a subgroup, and maximal subgroups, have been studied [4–9] within the framework of an L-setting and have been shown to be compatible. Thus, a coherent and systematic theory has emerged with this development.

In classical group theory, pronormal subgroups are closely related to normal and subnormal subgroups. Pronormality together with subnormality is equivalent to normality. Pronormal fuzzy subgroups were introduced by Abou-Zaid [10] using level subsets. However, his definition and the corresponding study on pronormal fuzzy subgroups failed to provide any information on their deeper structures. Hence, a new approach for pronormal L-subgroups is required. The concept of the conjugate of an L-subgroup of an L-group was introduced in [11] using an L-point. In the present study, this concept was utilized to develop pronormal L-subgroups of an L-group.

Section 2 presents a preliminary discussion. Section 3 provides the definition of pronormal L-subgroups of an L-group using the concept of conjugate L-subgroups developed in [11]. An example is provided to show the existence of pronormal L-subgroups. Subsequently, the relationship between the classical property of pronormality of level subsets and that of pronormal L-subgroups is presented. The image of a pronormal L-subgroup under group homomorphism is revealed to be a pronormal L-subgroup, provided that the parent group, μ, possesses the sup-property. Section 4 discusses the important relationships between normality, subnormality, normalizer, and maximality and pronormality defined in L-group theory. First, every normal L-subgroup of an L-group μ is presented to be a pronormal L-subgroup of μ. The pronormality of the normalizer of a pronormal L-subgroup is discussed. The maximal L-subgroup of an L-group is shown to be a pronormal L-subgroup. The concept of a subnormal L-subgroup of an L-group was introduced in [5] and studied in detail in [12]. In this study, this concept was used to establish that an L-subgroup of an L-group μ that is both subnormal and pronormal is a normal L-subgroup of μ. This result was then applied to prove that every pronormal L-subgroup of a nilpotent L-group μ with the same tip and tail as μ was normal in μ.

Throughout this paper, L = ⟨L,≤,∨, ∧⟩ denotes a complete and completely distributive lattice where “≤” denotes the partial ordering on L and “∨” and “∧” denote, respectively, the join (supremum) and meet (infimum) of the elements of L. The maximal and minimal elements of L are denoted as 1 and 0, respectively. The concept of a completely distributive lattice can be found in any standard text on this subject [13].

In 1965, Zadeh [1] introduced the concept of a fuzzy subset of a set. In 1967, Goguen [14] extended this concept to L-fuzzy sets. In this section, we present the basic definitions and results associated with L-subsets that are used throughout this study. These definitions can be found in Chapter 1 of [15].

Let X be a nonempty set. An L-subset of X is a function of X in L. The set of L-subsets of X is called the L-power set of X and is denoted as L^X. For μ ∈ L^X, set {μ(x) | x ∈ X} is called the image of μ and denoted as 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 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 nonempty index set, the union, ∪i∈Iμi, and intersection, ∩i∈Iμi of {μ_i | i ∈ I}, are defined as follows:

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

for each x ∈ X. If μ ∈ L^X and a ∈ L, the level subset, μ_a, of μ is defined as follows:

μa={x∈X∣μ(x)≥a}.

For μ, ν ∈ L^X, if μ ⊆ ν, μ_a ⊆ ν_a can be easily verified 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.

a_x is an L-point or L-singleton. a_x is an L-point of μ if and only if μ(x) ≥ a, and a_xcanbeexpressed ∈ μ.

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

μ∘η(x)=⋁x=yz{μ(y)∧η(z)}.

Note that if x cannot be factored as x = yz in S, μ ◦ η(x), which is the least upper bound of an empty set, is zero. The set product can be verified to be associated with L^S if S is a semigroup.

Let f be a mapping from set X to set Y. If μ ∈ L^X and ν ∈ L^Y, 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 as

f(μ) (y)=⋁x∈f-1(y){μ(x)},

and

f-1(ν)(x)=ν(f(x)).

As previously, if f⁻¹(y) = ∅, f(μ)(y), which is the least upper bound of the empty set, becomes zero.

Throughout this paper, G denotes an ordinary group with the identity element, “e,” and I denotes a nonempty indexing set. In addition, 1_A denotes the characteristic function of a nonempty set A.

In 1971, Rosenfeld [2] applied the concept of fuzzy sets to groups to introduce fuzzy subgroups of a group. In 1981, Liu [3] extended the concept of fuzzy subgroups to L-fuzzy subgroups (L-subgroups), which we define as follows:

Definition 2.1 ([2])

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

• (1) μ(xy) ≥ μ(x) ∧ μ(y),

• (2) μ(x⁻¹) = μ(x).

The set of L-subgroups of G is denoted as L(G). The tip of an L-subgroup is attained for the identity element of G.

Theorem 2.2 ([15, Lemma 1.2.5])

If μ ∈ L^G, μ is an L-subgroup of G if and only if each nonempty level subset μ_a is a subgroup of G.

The intersection of an arbitrary family of L-subgroups of a group is an L-subgroup of this group.

Definition 2.3 ([2])

If μ ∈ L^G, an L-subgroup of G generated by μ is defined as the smallest L-subgroup of G with μ. This is denoted as ⟨μ⟩, i.e.,

⟨μ⟩=∩{μi∈L(G)∣μ⊆μi}.

Let η, μ ∈ L^G such that η ⊆ μ. Thus, η is an L-subset of μ. The set of all L-subsets of μ is denoted as L^μ. Moreover, if η, μ ∈ L(G) such that η ⊆ μ, η is an L-subgroup of μ. The set of all L-subgroups of μ is denoted as L(μ).

In the following, μ denotes an L-subgroup of G, which is considered the parent L-group. μ is an L-subgroup of G if and only if μ is an L-subgroup of 1_G.

Definition 2.4 ([8])

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

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

Definition 2.5 ([6])

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

ηt0a0(x)={a0if x=e,t0if x≠e.

Theorem 2.6 ([6, Theorem 2.1])

If η ∈ L^μ, η ∈ L(μ) if and only if each nonempty level subset η_a is a subgroup of μ_a.

In 1981, Wu [16] introduced normal fuzzy subgroups of a fuzzy group. To develop this concept, Wu [16] preferred the L-setting. The concept of a normal L-subgroup of an L-group is defined as follows:

Definition 2.7 ([16])

If η ∈ L(μ), 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 as NL(μ). If η ∈ NL(μ), η ◁ μ.

Note that as previously, an arbitrary intersection of the family of normal L-subgroups of an L-group μ is a normal L-subgroup of μ.

Theorem 2.8 ([?])

If η ∈ L(μ), η ∈ NL(μ) if and only if each nonempty level subset η_a is a normal subgroup of μ_a.

Definition 2.9 ([2])

If μ ∈ L^X, μ possesses sup-property if for each A ⊆ X there exists a₀ ∈ A such that ∨a∈Aμ(a)=μ(a0).

Finally, consider the following form [17]:

Theorem 2.10 [17, Theorem 3.1]

Let η ∈ L^μ. Let a0=∨x∈G{η (x)} and define an L-subset η̂ of G as

η^ (x)=∨a≤a0{a∣x∈⟨ηa⟩}.

Thus, η̂ ∈ L(μ) and η̂ = ⟨η⟩.

Theorem 2.11 [17, Theorem 3.7]

Let η ∈ L^μ possess sup-property. If a0=∨x∈G{η (x)}, for all b ≤ a₀, ⟨η_b⟩ = ⟨η⟩_b.

The concept of pronormal subgroups in classical group theory utilizes the concept of conjugate subgroups. In fuzzy group theory, conjugate fuzzy subgroups were introduced by Mukherjee and Bhattacharya [18]. In their method, the conjugate of an L-subgroup is developed using a crisp point of the parent group, G, instead of a fuzzy point, and thus, cannot be applied to develop pronormal fuzzy subgroups. The pronormality of fuzzy subgroups was developed by Abou-Zaid [10] using level subsets, which do not reveal any information regarding their structure.

In [11], the authors introduced the conjugate of an L-subgroup of an L-group using an L-point. This definition is highly compatible with other concepts in L-group theory, such as the normal L-subgroups of an L-group, normalizer of an L-subgroup of an L-group [7], and maximal L-subgroup of an L-group [9]. Moreover, this definition eliminates the shortcomings of the conjugate of an L-subgroup introduced in [18] and can be easily utilized to develop pronormal L-subgroups of an L-group.

Definition 3.1 ([11])

Let η be an L-subgroup of μ and a_z be an L-point of μ. The conjugate, η^a_z, of η with respect to a_z is an L-subset of G, and is defined as

ηaz(x)=a∧η(zxz-1) for all x∈G.

Note that for an L-subgroup η of μ and an L-point a_z of μ, the conjugate, η^a_z forms an L-subgroup of μ. Moreover, tip(η^a_z) = a ∧ tip(η), because

ηaz(e)=a∧η(zez-1)=a∧η(e).

Considering the level subset characterization for conjugate L-subgroups in [11], note that for a subgroup H of G and for x ∈ G, H^x denotes the conjugate of H with respect to x.

Theorem 3.2 ([11])

If η, ν ∈ L(μ) and a ∈ L such that tip(ν) = a ∧ tip(η), ν = η^a_z for a_z ∈ μ if and only if ν_t = η_t^z−1 for all t ≤ tip(ν).

In the following, the concept of pronormal L-subgroups is defined.

Definition 3.3

Let μ ∈ L(G). An L-subgroup η of μ is a pronormal L-subgroup of μ if for every L-point a_x ∈ μ, there exists an L-point b_y ∈ ⟨η, η ^a_x ⟩ such that η^b_y = η^a_x.

Here, we note that for L-subgroups η and ν of μ, ⟨η, ν⟩ denotes an L-subgroup of μ generated by η ∪ ν.

Our definition of pronormal L-subgroups is motivated by the following.

Theorem 3.4

If H and K be subgroups of G such that H ⊆ K, H is a pronormal subgroup of K if and only if 1_H is a pronormal L-subgroup of 1_K.

Example 1

Let M = {l, f, a, b, c, d, u} be the lattice as shown in Figure 1. Let G = S₄, the group of all permutations of the set, {1, 2, 3, 4}, with the 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 an L-subset μ of G as follows:

μ(x)={uif x∈V4,dif x∈S4\V4.

Because μ_t is a subgroup of G for all t ≤ u, by Theorem 2.2, μ ∈ L(G). Let η be an L-subset of μ defined as

η(x)={uif x=e,dif x∈V4\{e},aif x∈D41\V4,bif x∈D42\V4,cif x∈D43\V4,fif x∈S4\∪i=13D4i.

Because η_t is a subgroup of μ_t for all t ≤ u, by Theorem 2.6, η is an L-subgroup of μ. η can be easily verified to be a pronormal L-subgroup of μ. For example, considering L-point d₍₁₂₃₎ ∈ μ,

ηd(123)(x) =d∧η((123)x(132)) ={dif x∈V4,aif x∈D43\V4,bif x∈D41\V4,cif x∈D42\V4,fif x∈S4\∪i=13D4i.

Thus,

(η∪ηd(123))(x)={uif x=e,dif x∈∪i=13D4i,fif x∈S4\∪i=13D4i.

Hence,

⟨η∪ηd(123)⟩(x)={uif x=e,dif x∈S4.

Thus, d₍₁₂₃₎ ∈ ⟨η ∪ η^{d₍₁₂₃₎}⟩. Similarly, for L-point u_(12)(34) ∈ μ,

ηu(12)(34)=u∧η((12)(34)x(12)(34))=η.

Therefore, ⟨η, η^u_(12)(34)⟩ = η. Now,

u(12)(34)∉⟨η,ηu(12) (34)⟩.

However, u_e ∈ ⟨η, η^u_(12)(34)⟩ such that η^u_e = η^u_(12)(34). Similarly, pronormality of η is as follows:

Theorem 3.7 discusses the image of a pronormal L-subgroup under group homomorphisms. For this purpose, Lemma 3.5 from [11] is considered.

Lemma 3.5 ([11])

Let f : G → H be a group homomorphism and μ ∈ L(G). Thus, for η ∈ L(μ) and a_z ∈ μ, L-subgroup f(η^a_z) is a conjugate L-subgroup of f(η) in f(μ).

f(ηaz)=f(η)af(z).

Lemma 3.6

Let f : G → H be a group homomorphism and μ ∈ L(G). Thus, for η ∈ L(μ),

f(ηt)⊆f(η)t,

for all t ≤ η(e).

Theorem 3.7

Let f : G → H be a surjective group homomorphism. Let μ ∈ L(G) such that μ possesses sup-property. If η is a pronormal L-subgroup of μ, f(η) is a pronormal L-subgroup of f(μ).

Theorem 3.8

Let L be an upper well-ordered lattice and μ ∈ L(G). If η is a pronormal L-subgroup of μ, η_t is a pronormal subgroup of μ_t for all t ≤ η(e).

In this section, we present the various relationships between pronormal L-subgroups and the concepts of normal L-subgroups, subnormal L-subgroups, normalizer of an L-subgroup of an L-group, and maximal L-subgroup. The results discussed in this section parallel the interactions between these concepts in classical group theory. This section highlights the strengths of the concept of pronormality developed in this study.

In Theorem 4.2, we prove that a normal L-subgroup of an L-group μ is pronormal in μ. First, we consider the following result from [11]:

Lemma 4.1 ([11])

Let η ∈ L(μ). Thus, η is a normal L-subgroup of μ if and only if η^a_z ⊆ η for every L-point a_z ∈ μ. Moreover, if η ∈ NL(μ) and tip(η^a_z) = tip(η), η^a_z = η.

Theorem 4.2

Let η be a normal L-subgroup of μ. Thus, η is a pronormal L-subgroup of μ.

Example 2

Considering L-subgroup η of L-group μ discussed in Example 1, η is a pronormal L-subgroup of μ. Note that for t = a, ηa=D41, which is not a normal subgroup of μ_a = S₄. Hence, from Thorem 2.8, η ∉ NL(μ).

The normalizer of an L-subgroup was explored in detail by Ajmal and Jahan [7]. They defined it using the concept of co-sets of L-subgroups. The normalizer developed was shown to be strongly compatible with the concept of normality in L-group theory.

Theorem 4.6 discusses the pronormality of the normalizer of a pronormal L-subgroup of an L-group. First, we consider the following definitions of co-sets and normalizers from [7]:

Definition 4.3 ([7])

Let η ∈ L(μ), and let a_x be an L-point of μ. The left ( right) co-set of η in μ with respect to a_x is defined as the set product, a_x ◦ η (η ◦ a_x).

The definition of the set product of two L-subsets shows that for all z ∈ G,

(ax∘η) (z)=a∧η(x-1z),

and

(η∘ax) (z)=a∧η(zx-1).

Definition 4.4 ([7])

Let η ∈ L(μ). The normalizer of η in μ, denoted as N(η), is an L-subgroup and is 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). In addition, [7] established that η is a normal L-subgroup of μ if and only if N(η) = μ.

Reference [11] provides a new definition for the normalizer of an L-subgroup using the concept of the conjugate of an L-subgroup. We consider this definition in the following theorem:

Theorem 4.5 ([11])

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

N(η)=∪{az∈μ∣ηaz⊆η}.

Theorem 4.6

Let η be a pronormal L-subgroup of μ that satisfies tip(η) = tip(μ). Let N(η) denote the normalizer of η in μ. Thus, N(η) is a pronormal L-subgroup of μ.

Lemma 4.7 ([11])

Let η, ν ∈ L(μ) and a_z be an L-point of μ. Thus,

(η∘ν)az=ηaz∘νaz.

Theorem 4.8

Let η be a normal L-subgroup of μ and ν be a pronormal L-subgroup of μ such that tip(η) = tip(ν). Thus, η ◦ ν is a pronormal L-subgroup of μ.

Definition 4.9 ([9])

We assume that μ ∈ L(G). A proper L-subgroup η of μ is a maximal L-subgroup of μ if whenever η ⊆ θ ⊆ μ for some θ ∈ L(μ), either θ = η or θ = μ.

Theorem 4.10

If η be a maximal L-subgroup of μ, η is a pronormal L-subgroup of μ.

Definition 4.11 ([5])

If η ∈ L(μ), L-subset μημ⁻¹ of μ is defined as

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

which is the conjugate of η in μ. The normal closure of η in μ, denoted as η^μ, is defined as an L-subgroup of μ generated by the conjugate, μημ⁻¹, i.e.,

ημ=⟨μημ-1⟩.

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

Definition 4.12 ([5])

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

η0=μ and ηi=ηηi-1 for all i≥1.

Thus, η_i is the smallest normal L-subgroup of η_i−1 that contains η, which is called the i^th normal closure of η in μ. The series of L-subgroups

μ=η0⊇η1⊇…⊇ηi-1⊇ηi⊇…

is the normal closure series of η in μ. Moreover, if there exists a non-negative integer m such that

η=ηm⊲ηm-1⊲…⊲η0=μ,

η is a subnormal L-subgroup of μ with defect m.

Clearly, m = 0 if η = μ and m = 1 if η ∈ NL(μ) and η ≠ μ.

Here, we prove the following:

Lemma 4.13

Let η ∈ L(μ) and a_z be an L-point of μ. Subsequently, η^a_z is included in the normal closure of η in μ.

Lemma 4.14

Let η and ν be L-subgroups of μ such that η ⊆ ν. If η is a pronormal L-subgroup of μ, η is a pronormal L-subgroup of ν.

Theorem 4.15

Let η ∈ L(μ). If η is both a pronormal and subnormal L-subgroup of μ, η is a normal L-subgroup of μ.

Definition 4.18 ([6])

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 with trivial L-subgroup ηt0a0 in a finite number of steps, η is called a nilpotent L-subgroup of μ. Moreover, η is nilpotent for class c if c is the smallest nonnegative integer such that Zc(η)=ηt0a0.

Here, we define the successive normalizers of η as follows:

η0=ηandηi+1=N(ηi)for all i≥0.

According to the definition of a normalizer (see Definition 4.4), η_i+1 is the the largest L-subgroup of μ containing η_i such that η_i ◁ η_i+1. Consequently,

(1)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 use (1) the ascending chain of normalizers of η in μ.

Lemma 4.19 ([19])

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

Lemma 4.20 ([19])

Let μ ∈ L(G) and η ∈ L(μ), Subsequently, an ascending chain of L-subgroups

η=θ0⊆θ1⊆…⊆θn⊆…⊆μ

terminating 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.

Based on Lemmas 4.19 and 4.20, we have the following:

Theorem 4.21

Let μ ∈ L(G) be a nilpotent L-group and η be an L-subgroup of μ with the same tip and tail as μ. Thus, η is a subnormal L-subgroup of μ.

Thus, Theorems 4.2, 4.15, and 4.21 yield the following result:

Corollary 4.22

Let μ ∈ L(G) be a nilpotent L-group and η be an L-subgroup of μ with the same tip and tail as μ. Thus, η is a normal L-subgroup of μ if and only if η is a pronormal L-subgroup of μ.