Nilpotent L -subgroups satisfy the normalizer condition

Abstract

In this paper, we study the inter-connections of the notions of commutator L-subgroups, nilpotent L-subgroups, conjugate L-subgroups and normalizer of L-subgroups like their classical counterparts. Throughout the development of this paper, the parent group is an L-group rather than an ordinary group. Our main result in this work is that every nilpotent L-subgroup of an L-group satisfies the normalizer condition.

Keywords

normal closure normalizer

1 Introduction

It is well known that, in 1971, Rosenfeld applied the notion of fuzzy sets to algebra by formulating the notions of fuzzy subgroupoids and fuzzy subgroups of a group in his pioneering paper [47]. This laid the foundation for the studies of fuzzy algebraic structures.

The study of fuzzy group theory was, infact, initiated by Foster [31] in his paper on fuzzy topological groups in the year 1979 wherein the notion of a fuzzy coset of a fuzzy subgroup in an ordinary group relative to a crisp point is defined. In 1981, Das [26] discovered an important inter-connection of the fuzzy subgroup property of a fuzzy set with that of subgroup property of its level subsets. In the same year, Wu [50] introduced the concept of normality of a fuzzy subgroup in a fuzzy group. However, this notion could not find a favor of researchers working in this area for a long time. Soon afterwards, Liu [40] came up with the ideas of set product of fuzzy subgroupoids and invariant fuzzy subgroup of an ordinary group. The properties of the set product were later explored by Ajmal in [2]. After few years, Mukherjee and Bhattachrya [24, 45] investigated the properties of fuzzy subgroups using the notions of fuzzy cosets by Foster [31] and invariant(normal) fuzzy subgroups by Liu [40]. They also introduced the concepts of conjugate fuzzy subgroups by a crisp point and normalizer of a fuzzy subgroup of an ordinary group. The normalizer in their sense was a crisp subgroup of the parent ordinary group. The concept of generated fuzzy subroups is discussed in detail by Ajmal et al. [3 , 49]. During mid 80’s to mid 90’s, various researchers from all over the world flung themselves into the investigation of the properties of fuzzy subgroups, such as, Abu Osman [1], Ajmal [4 –6], Ajmal and Thomas [15, 16], Akgul [17], Alkhamees [20], Anthony and Sherwood [21], Asaad and Zaid [22], Bhakat and Das [23], Chen and Gu [25], Eroglu [28], Felip [30], Gupta and Sarma [35], Kim [38], Kumar [39], Murali and Makamba [42], Mashinchi and Mukaidono [43], Ray [46], Saxena [48], Yu [52] etc. These researchers extended severalconcepts from classical group theory to fuzzy setting. Despite their efforts, a consistent development of the properties of fuzzy subgroups was not possible which can be put forward as a theory parallel to classical group theory. In fact, this was due to the dominance of crispness in their works, which did not allow such a development. Moreover, the progress in this discipline could not sustain the impact of the development of Tom Head’s well known metatheorem [36]. After almost three decades, the development of this area came to an standstill.On the other hand, Mordeson and Malik [44] developed the subject of fuzzy rings to a certain degree of respectability which is comparable with its classical counterpart. However, in the studies of fuzzy groups such an effort is lacking. This provided us sufficient motivation for the development of L-group theory.

The theory of L-algebraic substructures, whose basic concepts were introduced by Liu [41], is pursued in a systematic manner quite recently in a series of papers [8 –12]. In these papers, various concepts of L-group theory such as the normalizer of an L-subgroup, nilpotent L-subgroup, solvable L-subgroup, normal closure of an L-subgroup are introduced and investigated. The notion of a normal closure of an L-subgroup leads to the formulation of subnormal L-subgroup of an L-group. Throughout this work the consistency and the compatibility of all these above mentioned concepts have been exihibited. In this work L is a completely distributive lattice. If the lattice L is replaced by the closed unit interval, then we can reterieve fuzzy group theory which was the objective of all the researchers whose names have been mentioned earlier.

In [10], the normality of an L-subgroup of an L-group due to Wu [50] has been characterized by an L-point. Then, this characterization has been applied to formulate the notion of normalizer of an L-subgroup of an L-group. This notion satisfies most of the desirable features of the notion of normalizer of classical group theory. In [9], firstly the concept of commutator of L-subsets has been introduced. This concept has been utilized to formulate the notion of nilpotent and solvable L-subgroups of an L-group. There were few attempts in the past to introduce the idea of nilpotency in fuzzy group theory. However, non of them could provide a satisfactory development of this concept. An unsuccessful attempt in this direction has been made by Kim [38]. Afterwards, Gupta and Sarma [35] provided a better notion of nilpotency but it has its own shortcomings. A third place where the term nilpotency has been found, is in the works of Akram [18, 19] but that is in the context of nilpotent fuzzy Lie ideals of Lie algebra.

This paper is focussed mainly on two important results (Theorems 4.13, 4.14). The significance of these results lies in the fact that, they reflect a high degree of compatibility among various concepts used in L-group theory such as commutator L-subgroup [11], generated L-subgroup [12], nilpotent L-subgroup [9] and normalizer of an L-subgroup [10]. The normalizer condition is well known in classical group theory. A group is said to satisfy the normalizer condition if every proper subgroup of a group G is properly contained in its normalizer. In classical group theory, this result follows immediately as a consequence of the inter-connection of commutator subgroup of two subgroups with the normalizer of one of these subgroups. In the present work, we establish a similar inter-connection of commutator L-subgroup with that of normalizer. Then, this is utilized to establish our main result that a nilpotent L-subgroup of an L-group satisfies the normalizer condition.

2 Preliminaries

Throughout this paper, the system 〈L, ≤ , ∨ , ∧ 〉 denotes a completely distributive lattice where ≤ denotes the partial ordering of L, the join (sup) and the meet (inf) of the elements of L are denoted by ∨ and ∧ respectively. Also, we write 1 and 0 for the maximal and the minimal elements of L, respectively. Moreover, our work is carried out by using the definition of L-subset as formulated by Goguen [32]. The definition of a completely distributive lattice is well known in the literature and can be found in any standard text on the subject.

Let {J_i : i ∈ I} be any family of subsets of a complete lattice L and F denotes the set of choice functions for J_i, i.e., functions $f : I \to \prod_{i \in I} J_{i}$ such that f (i) ∈ J_i for each i ∈ I. Then, we say that L is a completely distributive lattice, if $⋀ {⋁_{i \in I} J_{i}} = ⋁_{f \in F} {⋀_{i \in I} f (i)} .$

The above law is known as the complete distributive law. Moreover, a lattice L is said to be infinitely meet distributive if for every subset {b_β : β ∈ B} of L, we have $a ⋀ {⋁_{β \in B} b_{β}} = ⋁_{β \in B} {a ⋀ b_{β}},$ provided L is join complete. The above law is known as the infinitely meet distributive law. The definition of infinitely join distributive lattice is dual to the above definition i.e. a lattice L is said to be infinitely join distributive if for every subset {b_β : β ∈ B} of L, we have $a ⋁ {⋀_{β \in B} b_{β}} = ⋀_{β \in B} {a \lor b_{β}},$ provided L is meet complete. The above law is known as the infinitely join distributive law. Clearly, both these laws follow from the definition of a completely distributive lattice. Here, we also mention that the dual of completely distributive law is valid in a completely distributive lattice whereas the infinitely meet and join distributive laws are independent from each other. Next, we recall the following from [8–12 , 51]:

An L-subset of X is a function from X into 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 Imμ and the tip of μ is defined as ⋁_x∈Xμ (x) . Moreover, the tail of μ is defined as ⋀_x∈Xμ (x) . For μ, η ∈ L^X, we say that μ is contained in η if μ (x) ≤ η (x) for all x ∈ X and is denoted by μ ⊆ η. For a family {μ_i : i ∈ I} of L-subsets in X, where I is a nonempty index set, the union ⋃_i∈Iμ_i and the intersection ⋂_i∈Iμ_i of {μ_i : i ∈ I} are, respectively, defined by: $⋃_{i \in I} μ_{i} (x) = ⋁_{i \in I} μ (x) and ⋂_{i \in I} μ_{i} (x) = ⋀_{i \in I} μ (x),$ for each x ∈ X. If μ ∈ L^X and a ∈ L, then the notion of level subset μ_a of μ is defined as: $μ_{a} = {x \in X : μ (x) \geq a} .$

For a ∈ L and x ∈ X, we define a_x ∈ L^X as follows: $a_{x} (y) = {\begin{matrix} a & if y = x, \\ 0 & if y \neq x . \end{matrix}$ a_x is referred 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. The set product μ ∘ η of μ, η ∈ L^S, where S is a groupoid, is an L-subset of S defined by $μ \circ η (x) = ⋁_{x = yz} {μ (y) \land η (z)} .$

Again recall 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. It can be verified easily that the set product is associative in L^S if S is a semigroup.

Throughout this paper G denotes an ordinary group with the identity element ‘e’, and I denotes a nonempty indexing set.

Definition 2.1. Let μ ∈ L^G. Then, μ is called anL-subgroup of G if

μ (xy) ≥ μ (x) ∧ μ (y),

μ (x^-1) = μ (x),

for each x, y ∈ G

The set of L-subgroups of G is denoted by L (G) . Clearly, the tip of an L-subgroup is attained at the identity element e of G. It has been estabished [3] that if μ ∈ L (G), then each non empty level subset μ_a is a subgroup of G and conversely.

Definition 2.2. Let μ ∈ L (G). Then, μ is called a normal L-subgroup of G if μ (xy) = μ (yx) for all x, y ∈ G.

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

Definition 2.3. Let μ ∈ L^G. Then, the L-subgroup of G generated by μ is defined as the smallestL-subgroup of G which contains μ. It is denoted by〈μ〉 i.e. $〈 μ 〉 = \cap {μ_{i} \in L (G) : μ \subseteq μ_{i}} .$

Definition 2.4. [11] Let μ, η ∈ L (G) and η ⊆ μ. Then, we say that η is an L-subgroup of μ. Further if η is non-constant and μ ≠ η, 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 .

Also, η is said to be a trivial L-subgroup of μ if its chain of level subgroups contains only {e} and G. Thus, an L-subgroup may contain several trivial L-subgroups.

Definition 2.5. [11] Let η ∈ L (μ) with a₀ = η (e) and t₀ = inf η such that a₀ ≠ t₀. Then, the trivial L-subgroup of μ, denoted by $η_{t_{0}}^{a_{0}}$ and definedby: $\begin{matrix} η_{t_{0}}^{a_{0}} (y) = {\begin{matrix} a_{0} & if y = e, \\ t_{0} & if y \neq e, \end{matrix} \end{matrix}$ is contained in η and is called the trivial L-subgroup of η.

Henceforth μ denotes an L-subgroup of G and we call the parent L-subgroup simply an L-group. The set of L-subgroups of μ is denoted by L (μ).

Remark 1. If η ∈ L^μ, then it can be easily verified that 〈η〉_μ = 〈η〉, where 〈η〉_μ denotes the L-subgroup of μ generated by η.

We recall the definition of a normal L-subgroup of an L-group.

Definition 2.6. [50] Let η ∈ L (μ). Then, we say that η is a normal L-subgroup of μ if $η ({yxy}^{- 1}) \geq η (x) \land μ (y) for all x, y \in G .$

The set of normal L-subgroups of μ is denoted by NL (μ).

In the end of this section, we provide the following characterizations for L-subgroups:

Theorem 2.7. [11] Let η ∈ L^μ . Then,

η ∈ L (μ) if and only if each non empty level subset η_a is a subgroup of μ_a,

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

Proposition 2.8. [2] Let η ∈ L^μ . Then, η ∈ L (μ) if and only if η ∘ η = η and η (x) = η (x^-1) for allx ∈ G .

3 The Normalizer of an L-subgroup and commutator L-subgroup

In 1983, Wang Jin Liu [41] introduced lattice valued fuzzy subgroups of a group. The notion of normality of a fuzzy subgroup of a fuzzy group was introduced by Wu [50] in 1982 whereas the normality of a fuzzy subgroup of an ordinary group was introduced by Liu [40] in 1981. Based upon this notion of normality, a concept of normalizer of a fuzzy group of an ordinary group was introduced by Mukherjee and Bhttacharya [24, 45] in the year 1986. The researchers working in the area of fuzzy group theory continued using this concept for more than two decades. The problem with this normalizer is that it is a crisp subset (subgroup) of the given parent group rather than a fuzzy subgroup. Consequently, many of the properties of the the notion of normalizer in classical group theory could not be even formulated in fuzzy set up due to its crisp nature. In fact, this normalizer turns out to be the intersection of the normalizers of all the members of the chain of level subgroups of the given fuzzy subgroup. The definition of a normalizer based upon the normality due to Wu was awaited. For an L-subgroup η of an L-group μ which is not normal in μ, it is natural to discuss the nature of the largest L-subgroup of μ which contains η as a normal L-subgroup. The construction of such an L-subgroup is provided in [10] which is made possible by the L-point characterization of normality. ThisL-subgroup possesses most of the the desirable features of a normalizer like its classical counterpart.

We start with the definition of a coset of anL-subgroup by an L-point.

Definition 3.1. [10] Let η ∈ L (μ). Then, for a_x ∈ μ, left (right) coset of η in μ is defined as the set product a_x ∘ η (η ∘ a_x).

Proposition 3.2. [10] Let η ∈ L (μ). Then,

a_x ∘ η (z) = a ∧ η (x^-1z),

η ∘ a_x (z) = a ∧ η (zx^-1) ,

for each a_x ∈ μ and z ∈ G .

Theorem 3.3. [10] Let η ∈ L (μ). Then, $\begin{matrix} η \in NL (μ) if and only if & a_{x} \circ η = η \circ a_{x} \\ for each L - point a_{x} \in μ . \end{matrix}$

Now, we recall the concept of normalizer of an L-subgroup.

Theorem 3.4. [9] Let η ∈ L (μ). Define an L-subset δ of G as follows: $δ = \cup {a_{x} \in μ : a_{x} \circ η = η \circ a_{x}} .$

Then, δ is the largest L-subgroup of μ such that η is a normal L-subgroup of δ. Here δ is called the normalizer of η and is denoted by N_μ (η).

Here, we provide an example of a normalizer of an L-subgroup.

Example 1. Let D₈ = {〈x, y〉 : x² = e = y⁸ ; xy = y^-1x} be the dihedral group of degree 8. If $D_{4} = {〈 x, y^{2} 〉 : x^{2} = e = (y^{2})^{4}; {xy}^{- 2} = y^{- 2} x},$ is the dihedral subgroup of D₈, then define the following L-subsets of D₈: $\begin{matrix} μ (z) & = & {\begin{matrix} \frac{1}{2} & if z \in D_{4}, \\ \frac{1}{4} & if z \in D_{8} \sim D_{4}; \end{matrix} \\ η (z) & = & {\begin{matrix} \frac{1}{3} & if z \in 〈 x 〉, \\ \frac{1}{9} & if z \in D_{4} \sim 〈 x 〉, \\ \frac{1}{27} & if z \in D_{8} \sim D_{4} . \end{matrix} \end{matrix}$

Here A ∼ B means usual set difference and 〈x〉 denotes the subgroup of D₈ generated by ‘x’. Clearly η ⊆ μ, η ≠ μ and η, μ ∈ L (G). Thus η is a proper L-subgroup of μ. Moreover, note that $η_{\frac{1}{3}} = 〈 x 〉$ , and $μ_{\frac{1}{3}} = D_{4}$ , and 〈x〉 is not a normal subgroup of D₄. Hence, by Theorem 2.7(ii), η ∉ NL (μ). If K₄ = {e, x, y⁴, xy⁴} is a Klein four subgroup of D₈, then one can verify easily that the normalizer N_μ (η)of η has the following definition: $N_{μ} (η) (z) = {\begin{matrix} \frac{1}{2} & if z \in K_{4}, \\ \frac{1}{9} & if z \in D_{8} \sim K_{4} . \end{matrix}$

In classical group theory, the concepts of commutator and commutator subgroups play very important role in its advancement. In particular, the concept of nilpotent groups and solvable subgroups are formulated using the notion of commutator subgroups. In [9], we have extended this notion to L-setting. It is worthwhile to mention here that the parent group taken in the definition of commutator L-subgroup is an L-group. Here we recall the following:

Definition 3.5. [9] Let η, θ ∈ L^μ. Then, the commutator of η and θ is an L-subset (η, θ) of G defined as follows: $\begin{matrix} (η, θ) (x) = {\begin{matrix} \lor {η (y) \land θ (z)}, & if x = [y, z] \\ for some y, z \in G, \\ inf η \land inf θ, & if x \neq [y, z] \\ for any y, z \in G . \end{matrix} \end{matrix}$

The commutator L-subgroup of η and θ is defined as the L-subgroup of G generated by (η, θ). It is denoted by [η, θ]. Clearly, inf(η, θ) = inf η ∧ inf θ and [η, θ] ∈ L (μ).

The following example exhibits the notion of commutator L-subgroup:

Example 2. [7] Let G be the quaternian group Q₈ given by: $Q_{8} = {\pm 1, \pm i, \pm j, \pm k}$ where 1 is the identity element of Q₈ and (-1) i = - i = i (-1), (-1) j = - j = j (-1), (-1) k = - k = k (-1), (-1) ² = 1, i² = j² = k² = -1, ij = k, ji = - k, jk = i, kj = - i, ki = j, ik = - j. Let $\begin{matrix} C = {\pm 1}, H_{1} = {\pm 1, \pm i}, \\ H_{2} = {\pm 1, \pm j}, H_{3} = {\pm 1, \pm k}, \end{matrix}$ and the evaluation lattice be given by the diagram:

Fig.1

The evaluation lattice.

Consider the parent L-subgroup of G given by: $μ (x) = {\begin{matrix} u & if x \in C, \\ d & if x \in G \ C . \end{matrix}$

Now define L-subsets η and θ of μ as given below: $\begin{matrix} η (x) & = & {\begin{matrix} d & if x \in C, \\ a & if x \in H_{1} \ C, \\ b & if x \in H_{2} \ C, \\ f & if x \in H_{3} \ C; \end{matrix} \\ θ (x) & = & {\begin{matrix} d & if x \in C, \\ c & if x \in H_{1} \ C, \\ a & if x \in H_{2} \ C, \\ f & if x \in H_{3} \ C; \end{matrix} \end{matrix}$

Note that G′ = {1, - 1}. Then, in view of Definition 3.5, the commutator L-subset (η, θ) of μ is given by $\begin{matrix} (η, θ) (x) = {\begin{matrix} d & if x = 1, \\ a & if x \in C \ {1}, \\ f & if x \in G \ C . \end{matrix} \end{matrix}$

As the level subsets of (η, θ) are subgroups of G and (η, θ) ⊆ μ, in view of Theorem 2.7(i) we have, (η, θ) ∈ L (μ). This implies $\begin{matrix} [η, θ] = (η, θ) . \end{matrix}$

The following results from [9] play a key role in the development of our main result:

Proposition 3.6. [9] Let η, θ ∈ L^μ and η ⊆ θ. Then,

(η, σ) ⊆ (θ, σ) ,

(σ, η) ⊆ (σ, θ) ,

for each σ ∈ L^μ.

Proof. (i) Let σ ∈ L^μ and x ∈ G . If x is not a commutator, then in view of the fact η ⊆ θ, we have $(η, σ) (x) = inf η \land inf σ \leq inf θ \land inf σ = (θ, σ) (x)$

So assume that x = [y, z] for some y, z ∈ G . Then, $\begin{matrix} (η, σ) (x) & = & ⋁_{x = [y, z]} {η (y) \land σ (z)} \\ \leq & ⋁_{x = [y, z]} {θ (y) \land σ (z)} (as η \subseteq θ) \\ = & (θ, σ) (x) . \end{matrix}$

(ii) Proof is similar to (i). □

Next, we have:

Proposition 3.7. [9] Let η, θ ∈ L^μ and η ⊆ θ. Then, [η, σ] ⊆ [θ, σ] for each σ ∈ L^μ.

Proof. Let σ ∈ L^μ . Then, by Proposition 3.6 $(η, σ) \subseteq (θ, σ) .$

Thus in view of the definition of a generated L-subgroup, we have $\begin{matrix} [η, σ] \subseteq [θ, σ] . □ \end{matrix}$

Here we recall the following:

Theorem 3.8. [12] Let $η \in L^{μ}$ . Let a₀ = ⋁ _x∈G {η (x)} and define an L-subset $\hat{η}$ of G by $\hat{η} (x) = ⋁_{a \leq a_{0}} {a : x \in 〈 η_{a} 〉} .$

Then, $\hat{η} \in L (μ)$ and $\hat{η} = 〈 η 〉$ .

Proposition 3.9. [9] Let η, θ ∈ L^μ. Then, [η, θ] = [θ, η].

Proof. Let x ∈ G and a₀ = ⋁ _x∈G {(η, θ) (x)} . Then, by Theorem 3.8 $[η, θ] (x) = ⋁_{a \leq a_{0}} {a : x \in 〈 (η, θ)_{a} 〉} .$

Now, by the definition of a commutator L-subset, $(η, θ) (y) = (θ, η) (y^{- 1}) for each y \in G .$ (1)

This implies ⋁_y∈G {(θ, η) (y)} = a₀ . Next, we shall show that if a ≤ a₀, then $x \in 〈 (η, θ)_{a} 〉 if and only if x \in 〈 (θ, η)_{a} .$ (2)

So let a ≤ a₀. Then, $\begin{matrix} x \in 〈 (η, θ)_{a} 〉 \\ \Leftrightarrow x = x_{1} x_{2} \dots x_{n} where x_{i} or \\ x_{i}^{- 1} \in (η, θ)_{a} for each i \\ \Leftrightarrow x = x_{1} x_{2} \dots x_{n} where x_{i}^{- 1} or \\ x_{i} \in (θ, η)_{a} for each i (by (1)) \\ \Leftrightarrow x \in 〈 (θ, η)_{a} 〉 . \end{matrix}$

This proves (2). Consequently, by Theorem 3.8 $\begin{matrix} [η, θ] (x) & = & ⋁_{a \leq a_{0}} {a : x \in 〈 (η, θ)_{a} 〉} \\ = & ⋁_{a \leq a_{0}} {a : x \in 〈 (θ, η)_{a} 〉} (by (2)) \\ = & [θ, η] (x) . □ \end{matrix}$

We end this section by extending the well known normalizer condition of classical group theory to L-setting:

Definition 3.10. Let η ∈ L (μ) . Then, η is said to satisfy the normalizer condition if η is a proper L-subgroup of μ, then it is a proper L-subgroup of its normalizer N_μ (η).

Next, we demonstrate the ‘normalizer condition’ by the following example:

Example 3. Let D₈ = {〈x, y〉 : x² = e = y⁸ ; xy = y^-1x} be the dihedral group of degree 8. If C = {e, y⁴} is the centre of D₈ and K₄ = {e, x, y⁴, xy⁴} is a Klein four subgroup of D₄, then define the following L-subsets of D₈: $\begin{matrix} μ (z) & = & {\begin{matrix} \frac{1}{2} & if z \in C, \\ \frac{1}{4} & if z \in K_{4} \sim C, \\ \frac{1}{8} & if z \in D_{8} \sim K_{4} . \end{matrix} \\ η (z) & = & {\begin{matrix} \frac{1}{3} & if z \in C, \\ \frac{1}{6} & if z \in K_{4} \sim C, \\ \frac{1}{27} & if z \in D_{8} \sim K_{4} . \end{matrix} \end{matrix}$

Clearly, η ⊆ μ, η ≠ μ and η, μ ∈ L (G). Thus η is a proper L-subgroup of μ. One can verify easily that the normalizer N_μ (η) of η has the following definition: $\begin{matrix} N_{μ} (η) (z) = {\begin{matrix} \frac{1}{2} & if z \in C, \\ \frac{1}{4} & if z \in K_{4} \sim C, \\ \frac{1}{8} & if z \in D_{4} \sim K_{4}, \\ \frac{1}{27} & if z \in D_{8} \sim D_{4} . \end{matrix} \end{matrix}$

Clearly η ⊆ N_μ (η) and η ≠ N_μ (η) i.e. η is a proper L-subgroup of its normalizer. Thus η satisfies the normalizer condition.

4 Nilpotent L-subgroup, Conjugate L-subgroup and the Normalizer condition

The definition of commutator L-subgroups has been used to define descending central series in [9]. Then, this notion of descending central series in a natural way gives rise to the notion of nilpotentL-subgroup. Recall the following from [9]:

Let η be an L-subgroup of μ.

Take γ₀ (η) = η, γ₁ (η) = [γ₀ (η) , η]. And in general, for each i, we define γ_i (η) = [γ_i-1 (η) , η].

Proposition 4.1. [9] Let η ∈ L (μ). Then, γ_i (η) ⊆ γ_i-1 (η) for each i.

Definition 4.2. [9] Let η ∈ L (μ). Then, the chain $η = γ_{0} (η) \supseteq γ_{1} (η) \supseteq \dots \supseteq γ_{i} (η) \supseteq \dots$ of L-subgroups of μ is called the descending central chain of η. Clearly, the tip of γ_i (η) coincides with η (e) for each i.

Moreover, the following results for the members of a descending central series can be established easily:

Proposition 4.3. [9] Let η ∈ L (μ) . Then, γ_i (η) and η have the same tails.

Proposition 4.4. [9] Let η ∈ NL (μ) . Then, γ_i (η) ∈ NL (μ).

It is worthwhile to note that as η is a normalL-subgroup of itself, γ_i (η) is a normal L-subgroup of η for each i.

Definition 4.5. [9] Let η ∈ L (μ) with the tip a₀ and the tail t₀ and a₀ ≠ t₀. If the descending central chain $\begin{matrix} η = γ_{0} (η) \supseteq γ_{1} (η) \supseteq \dots \supseteq γ_{i} (η) \supseteq \dots \end{matrix}$ terminates finitely to the trivial L-subgroup $η_{t_{0}}^{a_{0}}$ , then η is known as a nilpotent L-subgroup of μ. More precisely, η is said to be nilpotent of class c if c is the least non-negative integer such that $γ_{c} (η) = η_{t_{0}}^{a_{0}}$ . In this case, the series $η = γ_{0} (η) \supseteq γ_{1} (η) \supseteq \dots \supseteq γ_{c} (η) = η_{t_{0}}^{a_{0}}$ is called the descending central series of η. If η is a nilpotent L-subgroup of μ, then we simply write η is nilpotent.

Below we provide an example of a nilpotentL-subgroup of an L-group:

Example 4. [7] Let G = Q₈ and the evaluation lattice be given by the diagram:

Fig.2

The evaluation lattice.

Consider the parent L-subgroup of G given by: $μ (x) = {\begin{matrix} u & if x \in C, \\ d & if x \in G \ C . \end{matrix} .$

Now define an L-subset η of μ as given below: $η (x) = {\begin{matrix} u & if x \in C, \\ d & if x \in H_{1} \ C, \\ a & if x \in H_{2} \ C, \\ b & if x \in H_{3} \ C; \end{matrix}$ where $\begin{matrix} C & = & {\pm 1}, H_{1} = {\pm 1, \pm i}, \\ H_{2} & = & {\pm 1, \pm j}, H_{3} = {\pm 1, \pm k} . \end{matrix}$

Since each non emplty level subsets of η are subgroups of G and η ⊆ μ, by Theorem 2.7(i), η ∈ L (μ). Now, η is a nilpotent L-subgroup of μ in view of Definition 4.5. We demonstrate this as follows:

Note that G′ = {1, - 1}. In order to obtain the members of descending central series of η, we set γ₀ (η) = η. Now, by Definition 3.5, we calculate the commutator L-subset (η, η) of μ to obtain $(η, η) (x) = {\begin{matrix} u & if x = 1, \\ d & if x \in C \ {1}, \\ f & if x \in G \ C . \end{matrix}$

Again in view of Theorem 2.7(i), (η, η) ∈ L (μ). Hence we have $γ_{1} (η) = [η, η] = (η, η) .$

Next, we calculate the commutator ((η, η) , η): $((η, η), η) (x) = {\begin{matrix} u & if x = 1, \\ f & if x \in G \ {1} . \end{matrix}$

Again by the reasons as given above $γ_{2} (η) = [[η, η], η] = ((η, η), η) .$

Observe that γ₂ (η) is not only an L-subgroup, it is the trivial L-subgroup of η and so the descending central series terminates at γ₂ (η), i.e. $η = γ_{0} (η) \supseteq γ_{1} (η) \supseteq γ_{2} (η) = η_{f}^{u} .$

Consequently η is a nilpotent L-subgroup of μ having the nilpotent length 2.

The notion of normal closure is an important concept in classical group theory. In [8], the concept of normal closure of an L-subgroup in an L-group has been defined. For this purpose, the concept of conjugacy of an L-subgroup by an L-group is introduced and effectively utilized. This notion satisfies most of the desirable properties related to this concept. Its compatibility with the notion of commutator L-subgroup of an L-group is established like its counterpart in classical group theory. Next, recall

Definition 4.6. [8] Let η ∈ L (μ). Define an L-subset μημ^-1 of G as follows: $μ η μ^{- 1} (x) = ⋁_{x = zy z^{- 1}} {η (y) \land μ (z)} for each x \in G .$

The L-subset μημ^-1 is called the conjugate of η in μ. It is easy to verify that η ⊆ μημ^-1 ⊆ μ. The subgroup generated by the conjugate μημ^-1 is defined as the normal closure of η in μ. It is denoted by η^μ. It has been proved in [8] that, for η ∈ L (μ), η^μ is the least normal L-subgroup of μ containing η.

Likewise, we can define conjugate of an L-subset by another L-subset of an L-group.

Definition 4.7. Let η, θ ∈ L^μ. Define an L-subset θηθ^-1 of G as follows: $θ η θ^{- 1} (x) = ⋁_{x = zy z^{- 1}} {η (y) \land θ (z)} for each x \in G .$

We call θηθ^-1 the conjugate of η by θ. Clearly θηθ^-1 ⊆ μ. Hence 〈θηθ^-1〉 ∈ L (μ) and we write 〈θηθ^-1〉 = η^θ .

Here we exihibit the above definition with the help of the following example:

Example 5. Let D₈ = {〈x, y〉 : x² = e = y⁸ ; xy = y^-1x} be the dihedral group of degree 8 and $D_{4} = {〈 x, y^{2} 〉 : x^{2} = e = (y^{2})^{4}, x y^{- 2} = y^{- 2} x},$ be a dihedral subgroup of D₈. Let C = {e, y⁴} be the centre and K₄ = {e, x, y⁴, xy⁴} be a Klein four subgroup of D₈. Then, define the following L-subsets of D₈: $\begin{matrix} μ (z) & = & {\begin{matrix} \frac{1}{2} & if z \in D_{4}, \\ \frac{1}{4} & if z \in D_{8} \sim D_{4}; \end{matrix} \\ η (z) & = & {\begin{matrix} \frac{1}{3} & if z \in C, \\ \frac{1}{6} & if z \in K_{4} \sim C, \\ \frac{1}{9} & if z \in D_{4} \sim K_{4}, \\ \frac{1}{12} & if z \in D_{8} \sim D_{4}; \end{matrix} \end{matrix}$ and $\begin{matrix} θ (z) & = {\begin{matrix} \frac{1}{4} & if z \in 〈 x 〉, \\ \frac{1}{8} & if z \in D_{4} \sim 〈 x 〉, \\ \frac{1}{12} & if z \in D_{8} \sim D_{4} . \end{matrix} \end{matrix}$

Here A ∼ B means usual set difference and 〈x〉 denotes subgroup of D₈ generated by ‘x’. Clearly η, θ ⊆ μ. It can be verified easily that the conjugate ‘θηθ^-1’ is defined by the following level subsets: $\begin{matrix} (θ η θ^{- 1})_{\frac{1}{4}} = C, (θ η θ^{- 1})_{\frac{1}{6}} = K_{4}, \\ (θ η θ^{- 1})_{\frac{1}{9}} = D_{4} and (θ η θ^{- 1})_{\frac{1}{12}} = D_{8} . \end{matrix}$

Note that all non empty level subsets of θηθ^-1 are subgroups of D₈ and θηθ^-1 ⊆ μ. Thus by Theorem 2.7(i), θηθ^-1 is an L-subgroup of μ. Hence $η^{θ} = θ η θ^{- 1} .$

The following results are trivial but fundamental in nature:

Proposition 4.8. [8] Let η, θ ∈ L (μ). Then,

η ⊆ θηθ^-1 provided η (e) ⩽ θ (e) ,

⋁_x∈G {θηθ^-1 (x)} = θ (e) ∧ η (e) .

Proof. (i) Let η (e) ≤ θ (e) and x ∈ G . Then, $\begin{matrix} θ η θ^{- 1} (x) & = & ⋁_{x = zy z^{- 1}} {η (y) \land θ (z)} \\ \geq & η (x) \land θ (e) \\ \geq & η (x) \land η (e) \\ = & η (x) . \end{matrix}$

(ii) In view of the definition of θηθ^-1, we have $\begin{matrix} η (e) \land θ (e) & = & θ η θ^{- 1} (e) \leq ⋁_{x \in G} {θ η θ^{- 1} (x)} \\ \leq & η (e) \land θ (e) . \end{matrix}$

Hence $\begin{matrix} ⋁_{x \in G} {θ η θ^{- 1} (x)} = θ (e) \land η (e) . □ \end{matrix}$

The following lemma is trivial in nature but instrumental in establishing our main result:

Lemma 4.9. Let η, θ ∈ L (μ). Then,

(η, (θ (x)) _x) and ((θ (x)) _x, η) ⊆ [η, θ],

(η, (θ (x)) _x
^-1) and ((θ (x)) _x
^-1, η) ⊆ [η, θ],

for each x ∈ G .

Proof. Let x ∈ G .

(i) By the definition of an L-point (θ (x)) _x ⊆ θ . Hence by Proposition 3.6 $(η, (θ (x))_{x}) \subseteq (η, θ) \subseteq [η, θ] .$

Also, by Proposition 3.9 [η, θ] = [θ, η]. Hence by the arguments as above, we have $((θ (x))_{x}, η) \subseteq [θ, η] = [η, θ] .$

(ii) As θ ∈ L (μ) , we have $(θ (x^{- 1}))_{x^{- 1}} = (θ (x))_{x^{- 1}} = (θ (x))_{x} .$

Thus the result can be established as in part (i). □

The following lemma exhibits the nature of the commutator L-subset of two L-points:

Lemma 4.10. Let a, b ∈ L and x, y ∈ G . Let [x, y] = [l, m] for some l, m ∈ G. Then, $(a_{l}, b_{m}) ([x, y]) = a \land b .$

Proof. By the definition of a commutator L-subset $\begin{matrix} (a_{l}, b_{m}) ([x, y]) = ⋁_{[x, y] = [u, v]} {a_{l} (u) \land b_{m} (v)} . \end{matrix}$

By the definition of an L-point $\begin{matrix} a_{l} (u) = {\begin{matrix} a, & if u = l, \\ 0, & if u \neq l, \end{matrix} \end{matrix}$ and $\begin{matrix} b_{m} (v) = {\begin{matrix} b, & if v = m, \\ 0, & if v \neq m . \end{matrix} \end{matrix}$

Thus $\begin{matrix} a_{l} (u) \land b_{m} (v) \\ = {\begin{matrix} a \land b & if u = l and v = m, \\ 0, & if either u \neq l or v \neq m . \end{matrix} \end{matrix}$

As by the hypothesis [x, y] = [l, m] , it follows that $⋁_{[x, y] = [u, v]} {a_{l} (u) \land b_{m} (v)} = a \land b .$

This implies $\begin{matrix} (a_{l}, b_{m}) ([x, y]) = a \land b . □ \end{matrix}$

The following result establishes the inter-relationship between the notions of conjugate L-subsets (see the Definition 4.7), commutator of L-subsets [9] and the set product of L-subsets:

Lemma 4.11. Let η, θ ∈ L (μ) . Then $\begin{matrix} θ η θ^{- 1} \subseteq (θ, η) \circ η . \end{matrix}$

Proof. Let x ∈ G. Define the following subset of G × G by: $P (x) = {(u, v) \in G \times G : x = uv} .$

Next, consider $\begin{matrix} θ η θ^{- 1} (x) \\ = ⋁_{x = zy z^{- 1}} {η (y) \land θ (z)} \\ = ⋁_{([z, y], y) \in P (x)} {η (y) \land {η (y) \land θ (z)})} \\ \leq ⋁_{([z, y], y) \in P (x)} {η (y) \land (θ, η) ([z, y])} \\ \leq ⋁_{(u, v) \in P (x)} {(θ, η) (v) \land η (u)} \\ = (θ, η) \circ η (x) . □ \end{matrix}$

Similarly, we can prove

Lemma 4.12. Let η, θ ∈ L (μ) . Then, $θ η θ^{- 1} \subseteq η \circ (η, θ) .$

In the following, during the process of establishing that ‘the normalizer condition is satisfied by a nilpotent L-subgroup of an L-group’, we study the analogy of the inter-relationship of the concept of commutator with that of normalizer which is existing in classical group theory:

Theorem 4.13. Let η, θ ∈ L (μ) . Then, $[η, θ] \subseteq η if and only if θ \subseteq N_{μ} (η) .$

Proof. Necessity. Let x ∈ G. Define the following subset of G × G by: $P (x) = {(u, v) \in G \times G : x = uv} .$

Consider $\begin{matrix} (θ, η) (x) \\ = ⋁_{x = [y, z]} {θ (y) \land η (z)} \\ \leq ⋁_{x = [y, z]} {N_{μ} (η) (y) \land η (z)} (as θ \subseteq N_{μ} (η)) \\ = ⋁_{({yzy}^{- 1}, z^{- 1}) \in P (x)} {{N_{μ} (η) (y) \land η (z)} \land η (z^{- 1})} \\ \leq ⋁_{({yzy}^{- 1}, z^{- 1}) \in P (x)} {η ({yzy}^{- 1}) \land η (z^{- 1})} (as η \\ is a normal L - subgroup of N_{μ} (η)) \\ \leq η \circ η (x) (by the definition of set product) \\ = η (x) . (By Proposition 2.8) \end{matrix}$

As η ∈ L (μ) , in view of Proposition 3.9, we have $[η, θ] = [θ, η] = 〈 (θ, η) 〉 \subseteq η .$

This proves the necessity part.

Sufficiency. In order to show that θ ⊆ N_μ (η) , we shall demonstrate that (θ (x)) _x ⊆ N_μ (η) for all x ∈ G. To this end, we prove that $η \circ (θ (x))_{x} = (θ (x))_{x} \circ η for all x \in G .$

Thus in view of Proposition 3.2, we need to establish $η ({zx}^{- 1}) \land θ (x) = θ (x) \land η (x^{- 1} z) | for all x, z \in G .$

So let x, z ∈ G and write c = η (zx^-1) . Then, c_{zx
^-1} ⊆ η. Next,consider the commutator L-subset of L-points c_{zx
^-1} and (θ (x)) _x, that is, (c_{zx
^-1}, (θ (x)) _x). As (θ (x)) _x ⊆ θ, by Proposition 3.6 and 3.7, we have $\begin{matrix} (c_{{zx}^{- 1}}, (θ (x))_{x}) \subseteq [η, θ] . \end{matrix}$

By the hypothesis [η, θ] ⊆ η. Hence $\begin{matrix} (c_{{zx}^{- 1}}, (θ (x))_{x}) \subseteq η . \end{matrix}$

Thus for the commutator [z, x] ∈ G, we have $(c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) \leq η ([z, x]) .$

Hence

$\begin{matrix} (c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) \\ = (c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) \land η ([z, x]) . \end{matrix}$ (3)

Note that [z, x] = [zx^-1, x]. Thus by Lemma 4.10

$\begin{matrix} (c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) \\ = (c_{{zx}^{- 1}}, (θ (x))_{x}) ([{zx}^{- 1}, x]) = c \land θ (x) . \end{matrix}$ (4)

Replacing (c_{zx
^-1}, (θ (x)) _x) ([z, x]) on the R.H.S. of (3) by c ∧ θ (x), we get $(c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) = c \land θ (x) \land η ([z, x]) .$ (5)

Next, we claim that $\begin{matrix} θ (x) \land η ([z, x]) \\ = (θ (x))_{x^{- 1}} η ((θ (x))_{x^{- 1}})^{- 1} (x^{- 1} [z, x] x) . \end{matrix}$

By the definition of a conjugate L-subset, we have $\begin{matrix} (θ (x))_{x^{- 1}} η ((θ (x))_{x^{- 1}})^{- 1} ((x^{- 1} [z, x] x) \\ = ⋁_{x^{- 1} [z, x] x = {vuv}^{- 1}} {η (u) \land (θ (x))_{x^{- 1}} (v)} . \end{matrix}$

Now for x^-1 [z, x] x = vuv^-1, by the definition of L-point we have $\begin{matrix} η (u) \land (θ (x))_{x^{- 1}} (v) \\ = {\begin{matrix} η ([z, x]) \land θ (x) & if v = x^{- 1}, \\ 0, & if v \neq x^{- 1} . \end{matrix} \end{matrix}$

Therefore $\begin{matrix} ⋁_{x^{- 1} [z, x] x = {vuv}^{- 1}} {η (u) \land (θ (x))_{x^{- 1}} (v)} \\ = θ (x) \land η ([z, x]) . \end{matrix}$

Hence $\begin{matrix} (θ (x))_{x^{- 1}} η (θ (x))_{x^{- 1}})^{- 1} (x^{- 1} [z, x] x) \\ = θ (x) \land η ([z, x]) . \end{matrix}$

This establishes the claim. Further, $\begin{matrix} θ (x) \land η ([z, x]) \\ = (θ (x))_{x^{- 1}} η ((θ (x))_{x^{- 1}})^{- 1} (x^{- 1} [z, x] x) \\ \leq η \circ (η, (θ (x))_{x^{- 1}}) (x^{- 1} [z, x] x) \\ (by Lemma 4.12) \\ = η \circ (η, (θ (x))_{x^{- 1}}) ([x^{- 1}, z]) \\ (as x^{- 1} [z, x] x = [x^{- 1}, z]) \\ \leq η \circ [η, θ] ([x^{- 1}, z]) \\ (as by Lemma 4.9 (η, (θ (x))_{x^{- 1}}) \subseteq [η, θ]) \\ \leq η \circ η ([x^{- 1}, z]) \\ (by the hypothesis, [η, θ] \subseteq η) \\ = η ([x^{- 1}, z]) . (By Proposition 2.8) \end{matrix}$

Now replacing θ (x) ∧ η ([z, x]) in (5) by η ([x^-1, z]), we get the inequality $\begin{matrix} (c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) \\ \leq c \land η ([x^{- 1}, z]) \\ = η ({zx}^{- 1}) \land η ([x^{- 1}, z]) (as c = η ({zx}^{- 1})) \\ \leq η \circ η ([x^{- 1}, z] {zx}^{- 1}) \\ ((by the definition of the set product)) \\ = η \circ η (x^{- 1} z) (as [x^{- 1}, z] {zx}^{- 1} = x^{- 1} z) \\ = η (x^{- 1} z) . (By Proposition 2.8) \end{matrix}$

Now putting c = η (zx^-1) in (4), we get $\begin{matrix} (c_{{zx}^{- 1}}, (θ (x))_{x}) ([z, x]) = η ({zx}^{- 1}) \land θ (x), \end{matrix}$ and hence the above inequality yields that $η ({zx}^{- 1}) \land θ (x) \leq η (x^{- 1} z),$ which in turn implies that $η ({zx}^{- 1}) \land θ (x) \leq η (x^{- 1} z) \land θ (x) .$

To prove the reverse inequality, write d = η (x^-1z). Now consider the commutator L-subset of the L-points (θ (x)) _x and d_{x
^-1
z}, that is, ((θ (x)) _x, d_{x
^-1
z}). As [z, x^-1] = [x, x^-1z], we have

$\begin{matrix} ((θ (x))_{x}, d_{x^{- 1} z}) ([z, x^{- 1}]) \\ = ((θ (x))_{x}, d_{x^{- 1} z}) ([x, x^{- 1} z]) \\ = θ (x) \land d (by Lemma 4.10) \\ = θ (x) \land η (x^{- 1} z) . (As d = η (x^{- 1} z)) \end{matrix}$ (6)

Next,consider the conjugate L-subset of η by the L-point (θ (x)) _x and calculate its value at zx^-1. That is $\begin{matrix} (θ (x))_{x} η ((θ (x))_{x})^{- 1} ({zx}^{- 1}) \\ = (θ (x))_{x} η ((θ (x))_{x})^{- 1} (x (x^{- 1} z) x^{- 1}) \\ = θ (x) \land η (x^{- 1} z) . \end{matrix}$

Consequently by (6), we have $\begin{matrix} ((θ (x))_{x}, d_{x^{- 1} z}) ([z, x^{- 1}]) \\ = (θ (x))_{x} η ((θ (x))_{x})^{- 1} ({zx}^{- 1}) \\ \leq ((θ (x))_{x}, η) \circ η ({zx}^{- 1}) (by Lemma 4.11) \\ \leq [η, θ] \circ η ({zx}^{- 1}) \\ (by Lemma 4.9 ((θ (x))_{x}, η) \subseteq [η, θ]) \\ \leq η \circ η ({zx}^{- 1}) (by the hypothesis [η, θ] \subseteq η) \\ = η ({zx}^{- 1}) . (By Proposition 2.8) \end{matrix}$

Thus in view of (6), we have $η (x^{- 1} z) \land θ (x) \leq η ({zx}^{- 1}) .$

This implies $η (x^{- 1} z) \land θ (x) \leq η ({zx}^{- 1}) \land θ (x) .$

This establishes the desired equality and consequently the Sufficiency part is proved. □

Theorem 4.14. Let η be a nilpotent L-subgroup of μ having distinct tip a₀ and tail t₀. Then, η satisfies the normalizer condition.

Proof. As η is a nilpotent L-subgroup of μ, there exists a positive integer c for the descending central series for η such that $η = γ_{0} (η) \supseteq γ_{1} (η) \supseteq \dots \supseteq γ_{c} (η) = η_{t_{0}}^{a_{0}} .$

Now to establish that η satisfies the normalizer condition, let θ be a proper L-subgroup of η. As θ is a proper L-subgroup of η, there exists a positive integer i such that $γ_{i + 1} (η) \subseteq θ and γ_{i} (η) ⊈ θ .$

Now in view of Proposition 3.8 $[γ_{i} (η), θ] \subseteq [γ_{i} (η), η] = γ_{i + 1} (η) \subseteq θ .$

Hence by Theorem 4.13 $γ_{i} (η) \subseteq N_{μ} (θ) .$

As θ ⊆ N_μ (θ) and γ_i (η) ⊈ θ, it follows that θ is a proper L-subgroup of N_μ (θ) . This proves the result. □

5 Conclusion

The motivation for our work on L-group theory has been sufficiently discussed in the earlier part of the paper. Throughout the development of the theory of L-subgroups, the parent structure to be chosen is an L-group rather that an ordinary group. It is due to this consideration that the normalizer defined in our work is an L-subgroup of an L-group rather than a crisp subgroup of an ordinary group which has been the focus of attention in all the earlier works in this area. It is worthwhile to mention here that most of the development in L-group theory has been made possible because of the fact that we use the notion of normality of an L-subgroup in an L-group due to Wu rather than the notion of normality in the sense of Liu. In view of this development, we suggest the researchers working in various branches of fuzzy algebra to investigate the properties of L-subalgebras of an L-algebra instead of a classical algebra. For such studies, the infimum of L-subalgebras, as is demonstrated in our various papers [9, 11], may play a very significant role.

As an application and motivation we have already mentioned that if we replace the lattice L, in our work by the closed unit interval [0, 1], then we retrieve the corresponding version of fuzzy group theory. Moreover, as an application of this theory we also mention that if we replace the lattice L by the two elements set {0, 1}, then the results of classical group theory follow as simple corollaries of the corresponding results of L-group theory. This way, the L-group theory provides us a new language and a new tool for the study of the classical group theory. The classical group theory has been founded on abstract sets and therefore the language used for its development is formal set theory. On the other hand, L-group theory expresses itself through the language of functions. The functions which are lattice valued.

After the development of Tom Head’s indigenous metatheorem, in 1995, there was almost no development for almost one decade in the studies of fuzzy algebraic structures. Infact, in the area of fuzzy algebra, most of the notions and concepts considered are generically defined. That is, for a fuzzy subalgebra of a given algebra, its non empty strong level subsets do satisfy the corresponding classical properties. This observation inspired Tom Head to devise the concept of Rep function which in turn is used to define projection closed classes of fuzzy subsets. For such classes, it has been exihibited that the results regarding fuzzy algebraic structures are just simple instances of the metatheorem and the subdirect product theorem. On the other hand, when the evaluation lattice is replaced by a completely distributive lattice L, the strong level subsets of an L-subalgebra do not form the corresponding classical subalgebra and hence the class of that L-subalgebras is not projection closed. Consequenly, the metatheorem is not applicable in L-setting. Therefore, there are more reasons for the researchers pursuing studies in these areas to switch over to L-setting and investigate properties of L-subalgebras of an L-algebra.

Recently, J. Zhan in a series of papers [53–55] developed soft rough hemirings, hybrid soft set models, Z-soft rough fuzzy ideals of hemirings, soft fuzzy rough set model and applied it to decision making. In another paper, Pan and Zhan [56] have studied rough fuzzy groups and soft rough fuzzy groups and applied these notions to decesion making. The theory of L-subgroups so far developed can also find its application in novel soft rough sets. Similarly, the concepts of L-group theory such as rough L-subgroups, soft rough fuzzy normal L-subgroups of L-subgroups can be developed to yield better results in decision making. On the other hand, Farooque, Ali and Akram [29] developed the idea of m-polar fuzzy subgroups. The concepts of L-group theory can be easily carried over to m-polar fuzzy subgroups to develop a parallel theory.

Footnotes

Acknowledgements

The third author was supported by Emeritus Fellowship of UGC, India during the course of development of this paper.

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