Development of L-Group Theory

Abstract

In this work, we present a systematic and successful development of L-group theory. A universal construction of a generated L-subgroup has been provided by using level subsets of given L-subsets. This construction allows us to define and study commutator L-subgroups, normalizer of an L-subgroup, nilpotent L-subgroups, solvable L-subgroups, normal closure of an L-subgroup. All these concepts and their inter-relationships have been presented. Here we mention that in this work we also exhibit a characterization of solvable L-subgroup with the help of a series of L-subgroups such that at each level, the factor groups of level subgroups of their consecutive members are Abelian. This allows us to introduce the notion of a supersolvable L-subgroup by using the factors of level subgroups at each level of a subinvariant series of an L-subgroup. Also, by using successive normal closures, we transfinitely define a series called the normal closure series of the L-subgroup. It has been shown that it is the fastest descending normal series containing given L-subgroup. This sets the ground for the development of subnormality in L-group theory. In the last, we study the notion of subnormal L-subgroups.

1. Introduction

After Rosenfeld [1] introduced his notion of fuzzy subgroups of an ordinary group, there was a great activity in the investigations of fuzzy algebraic structures. Various aspects of fuzzy algebra were explored. Specially in the field of fuzzy group theory, several researchers contributed towards development [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. However, the researchers were not coherent and their studies suffered from one kind of incompatibility or the other. In the year 1981, Liu [21] introduced his notion of normality of a fuzzy subgroup in an ordinary group. Soon after that Wu [22] came up with an idea and just gave a hint of pursuing studies of fuzzy subgroups of a fuzzy group by providing the definition of a fuzzy normal subgroup in a fuzzy group. Afterwards, this idea was further taken up only by Martinez [23, 24]. Following this approach, the theory of $L$-subgroups is developed in a systematic and consistent manner in our papers [10, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

In fact, $L$-group theory came into existence as an answer to the problems posed by fuzzy group theory and the theories of other fuzzy algebraic structures. The development of fuzzy group theory was hampered mainly because of two reasons. Firstly, it was due to some inherent problems that the analogs of some of the concepts and results of classical group theory were not even formulated in this theory (for details, see Section 7, Discussion and Analysis). Moreover, various types of series such as derived series, descending central series, normal closure series etc. could not be formulated. Secondly, since most of the concepts studied in fuzzy algebraic structures were generically defined, Tom Head [35] proposed his well known metatheorem (which is based on the concept of Rep function) to extend the results of classical algebra to fuzzy setting. Therefore, the results of fuzzy group theory became simple instances of an application of metatheorem. Throughout the development of $L$-group theory, the above two drawbacks of fuzzy group theory have been very well taken care of (for details, see Section 7, Discussion and Analysis) and we are in a position to put forward a theory parallel to classical group theory. Thus a consistent theory came into existence. The subject matter discussed in this chapter, in particular; the join problem of $L$-subgroups provides sufficient testimony to its success [10].

As an application and motivation, here we mention that if we replace the lattice$L$, in our work by the closed unit interval$\left[0,1\right]$, then we retrieve the corresponding version for fuzzy group theory. Also, as an application of this theory we mention that if we replace the lattice L by the two elements set$\left\{0,1\right\}$, then the results of classical group theory follow as simple corollaries of the corresponding results of$L$-group theory. Moreover, this development of$L$-group theory is beyond the purview of metatheorem, contrary to the development of fuzzy group theory.

Section 2 provides a list of all the basic definitions and results regarding $L$-subsets and $L$-algebraic substructures which are required for the development of the subsequent sections. For the sake of completeness, few definitions from lattice theory have also been incorporated. In papers [10, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], various concepts of $L$-group theory have been explored.

Section 3 introduces the concept of a normalizer of an $L$-subgroup of an $L$-group. In Subsection 3.1, this notion of normalizer is very carefully formulated by using the concept of a coset by an $L$-point of an $L$-group. The normalizer of an $L$-subgroup in an $L$-group, formulated in this work, is an $L$-subgroup of the given $L$-group. This concept of classical group theory was left untouched during the evolution of fuzzy group theory. Although, Mukherjee and Bhattachrya [36, 37] tried to introduce a notion of normalizer of a fuzzy subgroup of an ordinary group, but it turned out to be a crisp subgroup of the given ordinary group. This idea was followed by several researchers [16, 38] in the past, but they were unable to obtain the results, presented in this work, due the fact that they carried out their researches within the framework where the parent structure was an ordinary group. We have shown in our work that for a normal $L$-subgroup of an $L$-group, its left and right $L$-cosets are identical. Also, it has been proved that each $L$-subset of an $L$-group commutes with every normal $L$-subgroup of the given $L$-group. In the end of this subsection, we state certain properties and the nature of this normalizer under the action of a group homomorphism. In Subsection 3.2, a universal construction of a generated $L$-subgroup by an $L$-set has been provided and its relationship with level subsets is investigated. This construction along with the construction of commutator $L$-subsets, studied in Section 4, allows us to define a commutator $L$-subgroup. In Subsection 3.3, again by replacing the parent structure of an ordinary group by an $L$-subgroup, we formulate the concept of a characteristic $L$-subgroup of an $L$-group. After obtaining the level subset characterizations of this concept, we establish some group theoretic analogs. Then, we construct various types of lattices and sublattices of characteristic $L$-subgroups. Finally, in this section we are able to establish that a characteristic $L$-subgroup of a normal $L$-subgroup is normal. Also, the well known property of transitivity of characteristic subgroup of classical group theory is extended to the $L$-setting. However, the same could not be even formulated in the works of earlier researchers [14, 39, 40, 41].

Section 4 starts with the notion of a commutator $L$-subset and commutator $L$-subgroup of an $L$-group. It is worthwhile to mention here that, earlier Gupta and Sarma [8] extended the notion of commutator subgroups in fuzzy setting which was utilized to formulate the concepts of descending central chain and derived chain in their further studies [9, 10]. However, the above mentioned studies have been carried out within the framework where the underlying group is an ordinary group. Therefore, they lost certain compatibility with other fuzzy algebraic notions. Here, we obtain some group theoretic analogs of commutator subgroups, we state a property of infimums of the set product of two $L$-subsets which is used in the further development of the subject matter. The whole development is justified by the level subset and strong level subset characterizations of commutator $L$-subgroups. Further in this section, we introduce the concept of a descending central chain of an $L$-subgroup by making the use of the notion of commutators. Then this, in turn, is used to define the notion of nilpotent $L$-subgroups of an $L$-group. Here the concept of the trivial $L$-subgroup of an $L$-subgroup comes into play. The members of the descending central chain are normal $L$-subgroups in their preceding ones in the sense of Wu [22]. Then, we present some peculiarities of $L$-setting which will be discussed in the end of this chapter. The level subset and strong level subset characterizations of these notions justify these extensions. The concept of the central chain of an $L$-subgroup is also introduced with the help of the trivial $L$-subgroup of the given $L$-group which is followed by analogs of some well known results of classical group theory. In the end of this section, we establish a necessary and sufficient condition for the set product of two trivial $L$-subgroups to be a trivial $L$-subgroup (see Theorem 1.81). Finally, this result has been used very effectively to establish a sufficient condition for the set product of two nilpotent $L$-subgroups to be nilpotent [31]. It has also been shown that the notion of a normalizer of an $L$-subgroup, which has been introduced in Section 3, is compatible with the notion of nilpotent $L$-subgroups. That is, nilpotent $L$-subgroup satisfies normalizer condition [34]. On the other hand, Kim [13] also defined an ascending series of crisp subgroups of an ordinary group to introduce his concept of nilpotent fuzzy subgroups. However, this could not lead to any substantial progress.

Section 5 deals with solvability and supersolvability of $L$-subgroups. Some more researchers [38, 42, 43] also discussed the notion solvability in the fuzzy setting. In the studies carried out by these authors the concept of normality introduced by Liu [21] is used. Consequently, the parent structure in their studies is an ordinary group, not a fuzzy group. For this purpose we introduce the concepts of derived series and solvable series with the help of the notion of commutator $L$-subgroups. This is possible because we use the normality in the sense of Wu [22] rather than Liu [21]. We discuss some results pertaining to the members of the derived central chain which are peculiarities of $L$-setting(Theorem 1.84, Theorem 1.85). All these concepts are justified by their level subset and strong level subset characterizations. Moreover, solvability is also characterized in terms of solvable series and some group theoretic analogs are obtained. Finally, the concept of central series is used to establish the inter connection of nilpotency and solvability of $L$-subgroups. We also discuss the behavior of homomorphic and inverse homomorphic images of solvable $L$-subgroups. Next, we define normal and subinvariant $L$-subgroups of an $L$-subgroup with Abelian factors. In case when the lattice $L$ is a dense chain, we characterize solvability of an $L$-subgroup with the help of a normal series with Abelian factors or subinvariant series with Abelian factors. This characterization motivated us to introduce the notion of a supersolvable $L$-subgroup by using the factors of level subgroups at each level of a subinvariant series of an $L$-subgroup. Also, commutator $L$-subgroup of a supersolvable $L$-subgroup is shown to be nilpotent. In the last, we extend Zassenhaus Theorem to $L$-setting and utilize it to establish a version of Schreier Refinement Theorem.

Section 6 evolves the concept of conjugacy in $L$-group theory. Firstly, the conjugate of an $L$-subgroup by an $L$-subgroup has been defined. The normal closure of an $L$-subgroup is defined as the $L$-subgroup generated by its conjugate by the whole parent $L$-group. Then, by using successive normal closures, we transfinitely define a series called the normal closure series of the given $L$-subgroup. Earlier an attempt has been made in [5] to define normal closure of a fuzzy group in an ordinary group. In fact, this is the fuzzy subgroup generated by the union of all the conjugates of the given fuzzy subgroup by crisp points. The concept of this conjugacy comes from [36]. But this idea was not found suitable enough to formulate the successive normal closures and hence it was not found suitable to be applied in the development of subnormal fuzzy subgroups introduced by the same author in [44]. Here, it has been shown that the normal closure series, defined in this work, is the fastest descending normal series containing given $L$-subgroup. This sets the ground for the development of subnormality in $L$-group theory [10, 33]. During the course of the development of subnormality, it has also been proved that every $L$-subgroup of a nilpotent $L$-subgroup is a subnormal $L$-subgroup. Finally in order to show the reach of $L$-group theory, we tackle the well known join problem of subnormal subgroups in $L$-setting and solve it to the same degree of success as that of classical group theory.

2. Preliminaries

Throughout our work, the system $⟨L,\le ,\wedge ,\vee ⟩$ denotes a complete and completely distributive lattice where’$\le$’ denotes the partial ordering of $L$, the join(sup) and the meet(inf) of the elements of $L$ are denoted by ${}^{\text{'}}{\vee }^{\prime }$ and ${}^{\text{'}}{\wedge }^{\prime }$, respectively. We shall denote the maximal and the minimal elements of $L$ by 1 and 0 respectively. Moreover, $I$ denotes a non-empty indexing set.

The definition of a completely distributive lattice is well known in literature [45]. Let $\left\{J_i:i\in I\right\}$ 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 $f\left(i\right)\in J_i$ for each $i\in I$. Then, we say that $L$ is a completely distributive lattice, if

$$\bigwedge \left\{\underset{i\in I}{\bigvee }{J}_i\right\}=\underset{f\in F}{\bigvee }\left\{\underset{i\in I}{\bigwedge }f\left(i\right)\right\}.$$E1

The above law is known as the completely distributive law. Moreover, a lattice $L$ is said to be infinitely meet distributive if for every subset $\left\{a_i:i\in I\right\}$ of $L$ we have

$$a\bigwedge \left\{\underset{i\in I}{\bigvee }{a}_i\right\}=\underset{i\in I}{\bigvee }\left\{a\bigwedge a_i\right\},$$E2

provided $L$ is join complete. The above law is known as infinitely meet distributive law. The definition of infinitely join distributive lattice is dual of the above definition, i.e., a lattice $L$ is said to be infinitely join distributive if for every subset $\left\{a_i:i\in I\right\}$ of $L$ we have

$$a\bigvee \left\{\underset{i\in I}{\bigwedge }{a}_i\right\}=\underset{i\in I}{\bigwedge }\left\{a\bigvee a_i\right\},$$E3

provided $L$ is meet complete. The above law is known as infinitely join distributive law. Both the above laws follow from the definition of a completely distributive lattice. The dual of a completely distributive law is valid in a completely distributive lattice whereas the infinitely meet and join distributive laws are independent of each other. In this section, we recall some definitions and results which will be used in the sequel. An $L$-subset of $X$ is a function from $X$ into $L$. The set of all $L$-subsets of $X$ is called the $L$-power set of $X$ and is denoted by $L^X$.

For an ordinary subset $A$ of $X$, its characteristic function defined by:

$$1_A\left(x\right)=\left\{\begin{array}{c}1,\text{if}x\in A,\\ 0,\text{if}x\notin A;\end{array}\right.$$E4

is an $L$-subset of $X$ representing $A$.

Let $\mu \in L^X$. The set $\left\{\mu \left(x\right):x\in X\right\}$ is called the image of $\mu$ and is denoted by $\mu \left(X\right)$ or $\mathit{Im\mu }$. Let $\underset{x\in X}{sup}\mu \left(x\right)=a_0$ and $\underset{x\in X}{inf}\mu \left(x\right)=t_0$. We call $a_0$ to be the tip of $\mu$ and $t_0$ to be the tail of $\mu$. We denote the tip and tail of $\mu$ by $sup\mu$ and $inf\mu$ respectively. Let $a\in L$. Then, ${\mu }_a=\left\{x\in X:\mu \left(x\right)\ge a\right\}$ is called $a$-level set or $a$-cut of $\mu$. Moreover, ${\mu }_a^{>}=\left\{x\in X:\mu \left(x\right)>a\right\}$ is called $a$-strong level set (or $a$-strong cut) of $\mu$. Note that ${\mu }_a=\varphi$, if $a>a_0$ and ${\mu }_a^{>}=\varphi ,$ if $a\ge a_0$. Moreover, ${\mu }_a=X$ if $a\le t_0$ and ${\mu }_a^{>}=X$ if $a<t_0$. Let $Y\subseteq X$. Then, we define $a_Y\in L^X$ as follows:

$$a_Y\left(x\right)=\left\{\begin{array}{ll}a,& \text{if}x\in Y,\\ 0,& \text{if}x\in X\backslash Y.\end{array}\right.$$E5

In particular, if $Y$ is singleton say $\left\{y\right\}$, then $a_{\left\{y\right\}}$ is called $L$-point or $L$-singleton and is denoted by $a_y$. We say that the $L$-point $a_y\in \mu$ if $\mu \left(x\right)\ge a$. The union $\underset{i\in I}{\cup }{\mu }_i$ and the intersection $\underset{i\in I}{\cap }{\mu }_i$ of any family $\left\{{\mu }_i:i\in I\right\}$ of $L$-subsets of $X$ are, respectively, defined by:

$$\left(\bigcup _{i\in I}{\mu }_i\right)\left(x\right)=\underset{i\in I}{\bigvee }{\mu }_i\left(x\right)\text{and}\left(\bigcap _{i\in I}{\mu }_i\right)\left(x\right)=\underset{i\in I}{\bigwedge }{\mu }_i\left(x\right),$$E6

for each $x\in X$. Let $\eta ,\mu \in L^X$. Then, $\eta$ is said to be contained in $\mu$, if we have $\eta \left(x\right)\le \mu \left(x\right)$ for each $x\in X$ and is written as $\eta \subseteq \mu$ or $\mu \supseteq \eta$.

Theorem 1.1 Let $\eta ,\theta \in L^X$. Then

1. $\eta \subseteq \theta$ if and only if ${\eta }_a\subseteq {\theta }_a\text{for\hspace{0.17em}each}a\in L$,

2. $\eta \subseteq \theta$ if and only if ${\eta }_a^{>}\subseteq {\theta }_a^{>}\text{for\hspace{0.17em}each}\in L\sim \left\{1\right\}$, provided $L$ is a chain.

Theorem 1.2 Let ${\left\{{\mu }_i\right\}}_{i\in I}\subseteq L^{\mu }$. Then

1. ${\left(\bigcap _{i\in I}{\mu }_i\right)}_a=\bigcap _{i\in I}{\left({\mu }_i\right)}_a\text{for}\mathrm{all}\in L,$

2. ${\left(\bigcup _{i\in I}{\mu }_i\right)}_a\subseteq \bigcup _{i\in I}{\left({\mu }_i\right)}_a\text{for}\mathrm{all}\in L,$

3. ${\left(\bigcup _{i\in I}{\mu }_i\right)}_a^{>}=\bigcup _{i\in I}{\left({\mu }_i\right)}_a^{>}\text{for}\mathrm{all}a\in L\sim \left\{1\right\}$, provided $L$ is a chain;

4. ${\left(\bigcap _{i\in I}{\mu }_i\right)}_a^{>}\subseteq \bigcap _{i\in I}{\left({\mu }_i\right)}_a^{>}\text{for}\mathrm{all}a\in L\sim \left\{1\right\}$, provided $L$ is a chain.

Let $\mathrm{f}$ be a function from $X$ into $Y$, and let $\mu \in L^X$ and $\nu \in L^Y$. Then, the image $\mathrm{f}\left(\mu \right)$ of $\mu$ under $\mathrm{f}$ and the pre-image ${\mathrm{f}}^{-1}\left(\eta \right)$ of $\eta$ under $\mathrm{f}$ are $L$-subsets of $Y$ and $X$ respectively defined by:

$$\mathrm{f}\left(\mu \right)\left(y\right)=\underset{x\in {\mathrm{f}}^{-1}\left(y\right)}{\bigvee }\left\{\mu \left(x\right):x\in X\right\}\text{and}{\mathrm{f}}^{-1}\left(\eta \right)\left(x\right)=\eta \left(\mathrm{f}\left(x\right)\right).$$E7

If ${\mathrm{f}}^{-1}\left(y\right)=\varphi$, then $\mathrm{f}\left(\mu \right)\left(y\right)=0$ since the least upper bound of the empty set in $L$ is $0$. The set product $\mu \circ \eta$ of $\mu ,\eta \in L^S$, where $S$ is a groupoid, is an $L$-subset of $S$ defined by

$$\mu \circ \eta \left(x\right)=\underset{x=\mathit{yz}}{sup}\left\{\mu \left(y\right)\wedge \eta \left(z\right)\right\}.$$E8

If $x$ cannot be factored as $x=\mathit{yz}$ in $S$, then $\mu \circ \eta \left(x\right)$ being the least upper bound of the empty set in $L$ is 0.

The set $L^X$ of $L$-subsets of $X$, together with the operations of union and intersection, is a complete lattice with the partial ordering of $L$-set inclusion $\subseteq$. Its maximal and minimal elements are $1_X$ and $0_X$, respectively. Here $1_X$ and $0_X$ are $L$-subsets of $X$ which map each element of $X$ to 1 and 0, respectively. Moreover, the lattice $P\left(X\right)$ of all subsets of $X$ can be isomorphically embedded into the lattice $L^X$.

From now onwards, $G$ will denote an arbitrary group with the identity element $e$. We recall the definitions of an $L$-subgroup and a normal $L$-subgroup of the group $G$.

Definition 1.1 Let $\mu \in L^G$. Then, $\mu$ is called an $L$-subgroup of $G$ if.

1. $\mu \left(\mathit{xy}\right)\ge \mu \left(x\right)\wedge \mu \left(y\right)$ for each $x,y\in G$,

2. $\mu \left(x^{-1}\right)=\mu \left(x\right)$ for each $x\in G$.

The set of all $L$-subgroups of $G$ is denoted by $L\left(G\right)$. From the definition, it is clear that the tip of $\mu$ is attained at the identity element of $G$.

Definition 1.2 Let $\mu \in L\left(G\right)$. Then, $\mu$ is said to to be normal $L$-subgroup of $G$ if $\mu \left(\mathit{xy}\right)=\mu \left(\mathit{yx}\right)$ for each $x,y\in G$.

It is well known that the intersection of an arbitrary family of $L$-subgroups of a group is an $L$-subgroup of the given group. Hence we have the following definition:

Definition 1.3 Let $\mu \in L^G$. Then, the $L$-subgroup of $G$ generated by $\mu$, denoted by $⟨\mu ⟩$, is defined as the smallest $L$-subgroup of $G$ which contains $\mu$, i.e.,

$$⟨\mu ⟩=\cap \left\{\nu :\mu \subseteq \nu ,\nu \in L\left(G\right)\right\}.$$E9

The set $L\left(G\right)$ is a complete lattice under the ordering of $L$-set inclusion where the meet’$\wedge$’ and join’$\vee$’ of an arbitrary family ${\left\{{\eta }_i\right\}}_{i\in I}$ in $L\left(G\right)$ are defined, respectively, by:

$$\underset{i\in I}{\wedge }{\eta }_i=\left(\bigcap _{i\in I}{\eta }_i\right)\text{and}\underset{i\in I}{\bigvee }{\eta }_i=⟨\bigcup _{i\in I}{\eta }_i⟩.$$E10

Let $\eta ,\mu \in L^G$ such that $\eta \subseteq \mu$. Then, $\eta$ is said to be an $L$-subset of $\mu$. The set of all $L$-subsets of $\mu$ is denoted by $L^{\mu }.$ Moreover, if $\eta ,\mu \in L\left(G\right)$ such that $\eta \subseteq \mu$, then $\eta$ is said to be an $L$-subgroup of $\mu$. The set of all $L$-subgroups of $\mu$ is denoted by $L\left(\mu \right)$. Here we mention that the set $L\left(\mu \right)$ of all $L$-subgroups of $\mu$ is a complete sublattice of the lattice $L\left(1_G\right)$.

Next, we provide level subset and strong level subset characterizations of an $L$-subgroup of an $L$-group.

Theorem 1.3 Let $\eta \in L^{\mu }$. Then,

1. $\eta \in L\left(\mu \right)$ if and only if each non-empty level subset ${\eta }_a$ is a subgroup of ${\mu }_a$,

2. $\eta \in L\left(\mu \right)$ if and only if each non-empty strong level subset ${\eta }_a^{>}$ is a subgroup of ${\mu }_a^{>}$ provided $L$ is a chain.

The following results discuss homomorphic image and pre-image of an $L$-subgroup:

Theorem 1.4 Let $\eta \in L\left(\mu \right)$ and $\mathrm{f}:G\to H$ be a group homomorphism. Then, $\mathrm{f}\left(\eta \right)$ is an $L$-subgroup of $\mathrm{f}\left(\mu \right)$.

Theorem 1.5 Let $\eta ,\mu \in L\left(H\right)$ with $\eta \in L\left(\mu \right)$ and $\mathrm{f}:G\to H$ be a group homomorphism. Then, ${\mathrm{f}}^{-1}\left(\eta \right)$ is an $L$-subgroup of ${\mathrm{f}}^{-1}\left(\mu \right)$.

Let $\eta \in L\left(\mu \right)$ be such that $\eta$ is non-constant and $\eta \ne \mu$. Then, $\eta$ is said to be a proper $L$-subgroup of $\mu$. Clearly, $\eta$ is a proper $L$-subgroup of $\mu$ if and only if $\eta$ has distinct tip and tail and $\eta \ne \mu$.

Definition 1.4 Let $\eta \in L\left(\mu \right)$. Then, $\eta$ is said to be a trivial $L$-subgroup of $\mu$ if its chain of level subgroups contains only $\left\{e\right\}$ and $G.$.

Definition 1.5 Let $\eta \in L\left(\mu \right)$. Then, define an $L$-subset ${\eta }_{t_0}^{a_0}$ of $\mu$ as follows:

$${\eta }_{t_0}^{a_0}\left(y\right)=\left\{\begin{array}{c}{a}_0,\text{if}y=e,\\ t_0,\text{if}y\ne e;\end{array}\right.$$E11

where $a_0=\eta \left(e\right)$ and $t_0=inf\eta$. Clearly, ${\eta }_{t_0}^{a_0}$ is a trivial $L$-subgroup of $\mu$ and is called the trivial $L$-subgroup of $\eta$.

From now onwards, we denote $\mu$ as an $L$-subgroup of $G$ and where there is no likelihood of any confusion, we shall not mention the underlying group $G$. We shall call the parent $L$-subgroup $\mu$ to be simply an $L$-group.

Now, we study the notion of normal $L$-subgroups of an $L$-group and its related properties. In the course of development, we obtain the analogs of certain results of classical group theory for normal $L$-subgroups of an $L$-group.

Definition 1.6 Let $\eta \in L^{\mu }$. Then, $\eta$ is said to be a normal $L$-subset of $\mu$ if

$$\eta \left(y^{-1}\mathit{xy}\right)\ge \eta \left(x\right)\wedge \mu \left(y\right)\text{for\hspace{0.17em}each}x,y\in G.$$E12

The set of all normal $L$-subsets of $\mu$ is denoted by NL${}^{\mu }$. Moreover, if $\eta \in L\left(\mu \right)$, then $\eta$ is said to be a normal $L$-subgroup of $\mu$. The set of all normal $L$-subgroups of $\mu$ is denoted by NL($\mu$). The following result follows immediately by the definition of normal $L$-subsets:

Theorem 1.6 Let ${\left\{{\eta }_i\right\}}_{i\in I}\subseteq {\mathit{NL}}^{\mu }$ be any family. Then,

1. $\bigcap _{i\in I}{\eta }_i\in {\mathit{NL}}^{\mu },$

2. $\bigcup _{i\in I}{\eta }_i\in {\mathit{NL}}^{\mu }.$

Next, we provide level subset and strong level subset characterizations of normal $L$-subsets and normal $L$-subgroups of an $L$-group.

Theorem 1.7 Let $\eta \in L^{\mu }$. Then,

1. $\eta \in {\mathit{NL}}^{\mu }$ if and only if each non-empty level subset ${\eta }_a$ is a normal subset of ${\mu }_a$,

2. $\eta \in {\mathit{NL}}^{\mu }$ if and only if each non-empty strong level subset ${\eta }_a^{>}$ is a normal subset of ${\mu }_a^{>}$ provided $L$ is a chain.

Theorem 1.8 Let $\eta \in L\left(\mu \right)$. Then,

1. $\eta \in \mathit{NL}\left(\mu \right)$ if and only if each non-empty strong level subset ${\eta }_a$ is a normal subgroup of ${\mu }_a$,

2. $\eta \in \mathit{NL}\left(\mu \right)$ if and only if each non-empty strong level subset ${\eta }_a^{>}$ is a normal subgroup of ${\mu }_a^{>}$ provided $L$ is a chain.

The following results deal with the homomorphic image and pre-image of a normal $L$-subgroups:

Theorem 1.9 Let $\eta ,\mu \in L\left(G\right)$ such that $\eta \in \mathrm{NL}\left(\mu \right)$ and $\mathrm{f}:G\to H$ be an onto group homomorphism. Then, $\mathrm{f}\left(\eta \right)$ is a normal $L$-subgroup of $\mathrm{f}\left(\mu \right)$.

Theorem 1.10 Let $H$ be a group and $\mu ,\eta \in L\left(H\right)$ be such that $\eta \in \mathrm{NL}\left(\mu \right)$. Let $\mathrm{f}:G\to H$ be a group homomorphism. Then, ${\mathrm{f}}^{-1}\left(\eta \right)$ is a normal $L$-subgroup of ${\mathrm{f}}^{-1}\left(\mu \right)$.

The notion of sup-property was introduced by A. Rosenfeld [1] in order to extend certain results of classical group theory to fuzzy setting. Thereafter, this technique was employed by researchers in various fields of fuzzy algebraic substructures [46].

Definition 1.7 Let $\eta \in L^{\mu }$. Then, $\eta$ is said to have sup-property if for each non-empty subset $A$ of $G$, there exists $a_0\in A$ such that $\underset{a\in A}{\vee }\eta \left(a\right)=\eta \left(a_0\right)$. The set of all $L$-subsets of $\mu$ possessing sup-property is denoted by $L_s^{\mu }$.

3. Generated $L$-subgroup, normalizer and characteristic $L$-subgroup of an $L$-group

The significance of the notions of normalizers, generated subgroups and characteristic subgroups can be found in any standard text in the classical group theory. In this section, we study these concepts within the framework of $L$-setting.

3.1 Normalizer of an $L$-group

One of the key notions ‘normalizer of a subgroup’ of classical group theory was left untouched during the evolution of fuzzy group theory. The notion of normalizer of an $L$-subgroup of an $L$-group has been introduced in [26] which, in essence, is comparable with its classical counterpart. This subsection commences with the definition of a coset of an $L$-subgroup by an $L$-point.

Definition 1.8 Let $\eta \in L\left(\mu \right)$. Then for $a_x\in \mu$, left (right) coset of $\eta$ in $\mu$ is defined as the set product $a_x\circ \eta$$\left(\eta \circ a_x\right)$.

The following is immediate:

Let $\eta \in L\left(\mu \right)$. Then, for each $a_x\in \mu$ and $z\in G$

$$a_x\circ \eta \left(z\right)=a\wedge \eta \left(x^{-1}z\right),\eta \circ a_x\left(z\right)=a\wedge \eta \left({\mathit{zx}}^{-1}\right);$$E13

i.e. $a_x\circ \eta =\eta \circ a_x\text{for\hspace{0.17em}each}{a}_x\in \eta .$

The characterization of an $L$-subgroup in terms of $L$-points is obtained in a way similar to classical group theory.

Theorem 1.11 Let $\eta \in L^{\mu }$. Then,

$$\eta \in \mathrm{L}\left(\mu \right)\text{if\hspace{0.17em}and\hspace{0.17em}only\hspace{0.17em}if}{a}_x\circ b_{y^{-1}}\in \eta \text{for\hspace{0.17em}each}{a}_x,b_y\in \eta .$$E14

We can characterize the normality of an $L$-subgroup of a given $L$-group in terms of these ‘cosets’ as follows. We observe here that in case of normal $L$-subgroup, the left coset and the right coset by an $L$-point are identical.

Theorem 1.12 Let $\eta \in L\left(\mu \right)$. Then,

$$\eta \in \mathit{NL}\left(\mu \right)\text{if\hspace{0.17em}and\hspace{0.17em}only\hspace{0.17em}if}{a}_x\circ \eta =\eta \circ a_x\text{for\hspace{0.17em}each}L‐\text{point}{a}_x\in \mu .$$E15

Clearly for $\eta \in L\left(\mu \right)$, $\eta \in \mathit{NL}\left(\eta \right)$. Also, we observe that for $a_x,b_y\in \mu$,

$$a_x\circ b_y={\left(a\wedge b\right)}_{\mathit{xy}}.$$E16

Some further characterizations of normal $L$-subgroups are given below:

Theorem 1.13 Let $\eta \in L\left(\mu \right)$. Then, the following are equivalent:

1. $\eta \in \mathit{NL}\left(\mu \right),$

2. $a_x\circ \eta =\eta \circ a_x$ for each $a_x\in \mu$,

3. $a_x\circ \eta \circ a_{x^{-1}}\subseteq \eta$ for each $a_x\in \mu$,

4. $a_x\circ b_y\circ a_{x^{-1}}\in \eta$ for each $a_x\in \mu$ and $b_y\in \eta$.

The following fact is well known:

If $\mu \in L^X$, then $\mu =\underset{x\in X}{\cup }\left\{{\left(\mu \left(x\right)\right)}_x\right\}$.

This immediately yields:

Theorem 1.14 Let $\eta ,\theta \in L^G$. Then, $\eta \circ \theta =\underset{x\in G}{\cup }\left\{{\left(\eta \left(x\right)\right)}_x\circ \theta \right\}=\underset{x\in G}{\cup }\left\{\eta \circ {\left(\theta \left(x\right)\right)}_x\right\}.$

Hence we have that each $L$-subset of $\mu$ commutes with every normal $L$-subgroup of $\mu$.

Theorem 1.15 Let $\eta \in \mathit{NL}\left(\mu \right)$. Then, $\eta \circ \theta =\theta \circ \eta$ for each $\theta \in L^{\mu }$.

The notion of normalizer which was so far introduced and discussed in fuzzy group theory is a crisp subset (subgroup) of the given parent group $G$. Moreover, this normalizer turns out to be the intersection of normalizers of all the level subsets (subgroups) of the fuzzy subgroup in question. Another drawback of this normalizer is that each member of a certain equivalence class of fuzzy subgroups of a fuzzy group has the same normalizer. This phenomenon arises due to the fact that the studies, carried out by these researchers are for the fuzzy subgroups of an ordinary group. Here we demonstrate, how we can introduce the concept of a normalizer which is not an ordinary subgroup but an $L$-subgroup itself and satisfies most of the properties of the notion of the normalizer of an ordinary subgroup of a group. Firstly, we present the construction of this notion in the following:

Theorem 1.16 Let $\eta \in L\left(\mu \right)$. Define an $L$-subset $\delta$ of $G$ as follows:

$$\delta =\bigcup _{a_x\in \mu }\left\{a_x:a_x\circ \eta =\eta \circ a_x\right\}.$$E17

Then, $\delta$ is the largest $L$-subgroup of $\mu$ such that $\eta$ is a normal $L$-subgroup of $\delta$. Here $\delta$ is called the normalizer of $\eta$ and is denoted by $N\left(\eta \right).$ Moreover, it turns out that $N\left(\eta \right)\left(e\right)=\mu \left(e\right)$.

This immediately leads us to the following result:

Let $\eta \in L\left(\mu \right)$. Then, $\eta \in \mathit{NL}\left(\mu \right)$ if and only if $N\left(\eta \right)=\mu$.

On the other hand, if we replace the parent $L$-group $\mu$ by $1_G$, then we have:

Theorem 1.18 Let $\eta \in L\left(G\right)$. Then, f$\left(\eta \right)$ is an normal $L$-subgroup of $G$ if and only if $N\left(\eta \right)=1_G$.

As a consequence, we recover the classical result:

Theorem 1.19 Let $H$ be a subgroup of $G$. Then, $x\in N\left(H\right)$ if and only if $1_H\circ 1_x=1_x\circ 1_H$.

Below we provide some more properties related to the normalizer so defined: Theorem 1.20 Let $\eta ,\theta \in L\left(\mu \right)$. Then

1. $N\left(\eta \right)\cap N\left(\theta \right)\subseteq N\left(\eta \cap \theta \right),$

2. $N\left(\eta \right)\cap N\left(\theta \right)\subseteq N\left(\eta \circ \theta \right)$ provided $\eta \circ \theta \in L\left(\mu \right)$.

Now, the following results reflect the behavior of this normalizer under the action of a group homomorphism. We start with:

Theorem 1.21 Let $f:G\to K$ be a group homomorphism and $x\in G$. If $y\in K$, then the set of all preimages of ‘$\mathit{yf}\left(x\right)$’ is precisely the set of all elements of the form ‘$\mathit{ux}$’ where $f\left(u\right)=y$.

Theorem 1.22 Let $f\left(v\right)$ be a group homomorphism and $\eta ,\theta \in L^{\mu }$. Then, $f\left(\eta \circ \theta \right)=f\left(\eta \right)\circ f\left(\theta \right)$.

Moreover,

Theorem 1.23 Let $f:G\to K$ be a group homomorphism and $\nu \in L\left(K\right)$.If $\theta \in L^{\nu }$, then $f^{-1}\left(\theta \circ b_{f\left(x\right)}\right)=f^{-1}\left(\theta \right)\circ b_x$.

The above results are helpful in establishing the following:

Theorem 1.24 Let $f:G\to K$ be a group homomorphism. Then, for $\mu \in L\left(G\right)$ and $\nu \in L\left(K\right)$

1. $f\left(N\left(\eta \right)\right)\subseteq N\left(f\left(\eta \right)\right)$ for each $\eta \in L\left(\mu \right)$,

2. $f^{-1}\left(N\left(\theta \right)\right)\subseteq N\left(f^{-1}\left(\theta \right)\right))$ for each $\theta \in L\left(\nu \right)$.

3.2 Generated $L$-subgroup of an $L$-group

Firstly, we recall the following results for generating an $L$-subgroup by a given $L$-subset from [25] and study its relationship with other notions of $L$-group theory.

Theorem 1.25 Let $\eta \in L^{\mu }$. Let $a_0=\underset{x\in G}{\vee }\left\{\eta \left(x\right)\right\}$ and define an $L$-subset $\widehat{\eta }$ of $G$ by

$$\widehat{\eta }\left(x\right)=\underset{a\le a_0}{\vee }\left\{a:x\in ⟨{\eta }_a⟩\right\}.$$E18

Then, $\widehat{\eta }\in L\left(\mu \right)$ and $\widehat{\eta }=⟨\eta ⟩$. Moreover, $⟨\eta ⟩\left(e\right)={\vee }_{x\in G}\left\{\eta \left(x\right)\right\}$.

The above theorem is used to establish the following:

Theorem 1.26 Let $\eta \in {\mathit{NL}}^{\mu }$. Then, $⟨\eta ⟩\in \mathit{NL}\left(\mu \right)$.

In the following, we demonstrate the significance of sup-property in the studies of $L$-group theory:

Theorem 1.27 Let $\eta \in L_s^{\mu }$. Then, define an $L$-subset $\widehat{\eta }$ of $G$ by

$$\widehat{\eta }\left(x\right)=\underset{a\in \mathit{Im\eta }}{\vee }\left\{a:x\in ⟨{\eta }_a⟩\right\}.$$E19

Then, $\widehat{\eta }\in L\left(\mu \right)$ and $\widehat{\eta }=⟨\eta ⟩$. Moreover, $\widehat{\eta }$ possesses sup-property and $Im\widehat{\eta }\subseteq Im\eta .$

The following result is an immediate consequence of the above theorem:

Theorem 1.28 Let $\eta \in L_s^{\mu }$. If $a_0=\underset{x\in G}{\vee }\left\{\eta \left(x\right)\right\}$, then for each $b\le a_0$, $⟨{\eta }_b⟩={⟨\eta ⟩}_b$.

The following example will demonstrate that the condition of sup-property is crucial and can not be removed from the above result:

Example 1: Let $Z$ be the group of integers under addition, and let $⟨2^n⟩$ be the subgroup of $Z$ generated by $2^n$, where $n$ is a fixed positive integer. Then the direct product $Z\times Z$ contains subgroups

$$⟨2^r⟩\times ⟨2^s⟩\text{for\hspace{0.17em}\hspace{0.17em}each}r,s=\mathrm{0,1,2},\dots$$E20

Define the following $L$-subset of $Z\times Z$ where $L$ is the closed unit interval ordered by usual ordering of real numbers:

$$\mu \left(x\right)=\left\{\begin{array}{ll}0& \text{if}x\in Z\times Z\sim ⟨2⟩\times Z,\\ \frac{3}{4}& \text{if}x\in ⟨2⟩\times Z.\end{array}\right.$$E21

$$\eta \left(x\right)=\left\{\begin{array}{ll}0& \text{if}x\in x\in Z\times Z\sim ⟨2⟩\times Z,\\ \frac{1}{2}\left(1-\frac{1}{2^n}\right)& \text{if}x\in ⟨2^n⟩\times Z\sim ⟨2^{n+1}⟩\times Z,\text{where}n=\mathrm{1,2,3},\dots \\ 0& \text{if}x\in ⟨0⟩\times Z\sim ⟨0⟩\times ⟨2⟩,\\ \frac{3}{4}\left(1-\frac{1}{4^n}\right)& \text{if}x\in ⟨0⟩\times ⟨2^n⟩\sim ⟨0⟩\times ⟨2^{n+1}⟩,\text{where}n=\mathrm{1,2,3},\dots \\ \frac{3}{4}& \text{if}x=\left(0,0\right).\end{array}\right.$$E22

Here $A\sim B$ means usual set difference.Clearly, $\eta \subseteq \mu$, $\eta \ne \mu$ and $\mu \in L\left(G\right)$. Observe that $\eta$ does not possess sup-property and for $t=\frac{1}{2}$,

$$⟨0⟩\times ⟨2⟩=⟨{\eta }_{\frac{1}{2}}⟩\subset {⟨\eta ⟩}_{\frac{1}{2}}=⟨0⟩\times Z.$$E23

Moreover,

Theorem 1.29 Let $\eta \in L^{\mu }$ and $a_0=\underset{x\in G}{\vee }\eta \left(x\right)$. If $L$ is a chain, then $⟨{\eta }_a^{>}⟩={⟨\eta ⟩}_a^{>}$ for each $a<a_0$.

3.3 Characteristic $L$-subgroup of an $L$-group

The notion of a characteristic subgroup of a group has been extended to the fuzzy setting by many researchers in the past. However in all these attempts, the parent group in question is an ordinary group rather than a fuzzy group. Here in this section, we firstly introduce the notion of a characteristic $L$-subset of an $L$-group. Then, we introduce the notion of a characteristic $L$-subgroup of an $L$-group in a manner similar to that of a normal $L$-subgroup of an $L$-group introduced earlier. After providing its characterization in terms of level subsets, we provide some group theoretic analogs to establish this notion. We also prove that the set of characteristic $L$-subsets (subgroups) is closed under arbitrary intersections (see [27]).

Definition 1.9 Let $\eta \in L^{\mu }$ with tip $a_0$. Then, $\eta$ is said to be a characteristic $L$-subset of $\mu$ if

$$\eta \left(\mathit{Tx}\right)\ge \eta \left(x\right)\text{for\hspace{0.17em}each}T\in A\left({\mu }_a\right)\text{and\hspace{0.17em}for\hspace{0.17em}each}a\le a_0;$$E24

where $A\left({\mu }_a\right)$ is the group of automorphisms of ${\mu }_a$. We denote the set of all characteristic $L$-subsets of $\mu$ by ${\mathit{CL}}^{\mu }$.

It is easy to see that $\mu$ is a characteristic $L$-subset of itself.

Theorem 1.30 Let $\eta \in L^{\mu }$ with tip $a_0$. Then, $\eta \in {\mathit{CL}}^{\mu }$ if and only if ${\eta }_a$ is a characteristic subset of ${\mu }_a\text{for each}a\le a_0$.

The set of all normal $L$-subsets of the $L$-group $\mu$ is denoted by ${\mathit{NL}}^{\mu }$. The following results are immediate:

Theorem 1.31 Let $\eta \in {\mathit{CL}}^{\mu }$. Then, $\eta \in {\mathit{NL}}^{\mu }$.

Theorem 1.32 Let $\eta \in {\mathit{CL}}^{\mu }$ with tip $a_0$. Then,

$$T\left(\eta |{\mu }_a\right)=\eta \mid {\mu }_a\text{for}\mathrm{all}T\in A\left({\mu }_a\right);\text{where}a\le a_0.$$E25

The set of all $L$-subsets of $\mu$ possessing sup-property is denoted by $L_s^{\mu }$. The following result has been discussed in the fuzzy setting in [27]:

Theorem 1.33 Let $\eta ,\theta \in L_s^{\mu }$. Then, $\eta \cup \theta$ and $\eta \cap \theta \in L_s^{\mu }$ provided $L$ is a chain.

Now, since the meet and the join operations in $L^{\mu }$ are defined to be the intersection and the union of $L$-subsets respectively, the set $L_s^{\mu }$ constitutes a sublattice of $L^{\mu }$ provided $L$ is a chain.

It is easy to observe that if $L$ is a chain, then any $L$-subgroup of an $L$-group with finite range possesses sup-property. The situation, however, in the case of infinite range is varied and interesting (see [46]). Also, we see the role of sup-property when $L$ is not a chain. Here we present a generalization of the notion of sup-property in order to obtain certain results. The following characterization of sup-property forms the basis of our generalization:

Theorem 1.34 Let $\eta \in L^{\mu }$. Then, $\eta$ possesses sup-property if and only if each non-empty subset of $\mathit{Im\eta }$ is closed under arbitrary supremum.

Definition 1.10 A non-empty subset $X$ of a lattice $L$ is said to be supstar if every non-empty subset $A$ of $X$ contains its supremum. That is, if $supA=a_0$, then $a_0\in A$.

By the definition, it is clear that every subset of a supstar subset is again a supstar subset. Now, define:

Definition 1.11 Let ${\left\{{\eta }_i\right\}}_{i\in I}\subseteq L^{\mu }$ be an arbitrary family. Then, ${\left\{{\eta }_i\right\}}_{i\in I}$ is said to be a supstar family if $\underset{i\in I}{\cup }\mathit{Im}{\eta }_i$ is a supstar subset of $L$. In particular, a pair of $L$-subsets $\eta$ and $\theta$ is said to be jointly supstar if the set $\left\{\eta ,\theta \right\}$ is a supstar family.

The following results are obtained by using the notions of sup-property and supstar family:

Theorem 1.35 Let $\eta \in L^{\mu }$. Then, $\eta$ has sup-property if and only if $\mathit{Im\eta }$ is a supstar subset of $L$.

Theorem 1.36 Let ${\left\{{\eta }_i\right\}}_{i\in I}\subseteq L^{\mu }$ be a supstar family. Then,

1. ${\eta }_i$ possesses sup-property for each $i\in I$,

2. $\bigcup _{i\in \mathrm{\Omega }}{\eta }_i$ possess sup-property, where $\mathrm{\Omega }\subseteq I$.

Moreover, we have:

Theorem 1.37 If ${\left\{{\eta }_i\right\}}_{i\in I}\subseteq L^{\mu }$ is a maximal supstar family, then ${\left\{{\eta }_i\right\}}_{i\in I}$ is a complete lattice under the ordering of $L$-set inclusion.

In order to study the lattice theoretic behavior of characteristic $L$-subsets, have the following:

Theorem 1.38 Let ${\left\{{\eta }_i\right\}}_{i\in I}\subseteq {\mathit{CL}}^{\mu }$. Then,

1. $\bigcap _{i\in I}{\eta }_i\in {\mathit{CL}}^{\mu },$

2. $\bigcup _{i\in I}{\eta }_i\in {\mathit{CL}}^{\mu }$ provided ${\left\{{\eta }_i\right\}}_{i\in I}$ is a supstar family.

In view of Theorem 1.31, ${\mathit{CL}}^{\mu }\subseteq {\mathit{NL}}^{\mu }\subseteq L^{\mu }$. By Theorem 1.6, ${\mathit{NL}}^{\mu }$ is closed under arbitrary unions and intersections. Hence ${\mathit{NL}}^{\mu }$ is a complete sublattice of $L^{\mu }$. Further by Theorem 1.38, ${\mathit{CL}}^{\mu }$ is closed under arbitrary intersections with the greatest element $\mu$. Thus ${\mathit{CL}}^{\mu }$ is a lower complete sublattice of $L^{\mu }$ and is a complete lattice in its own right. Moreover, if both ${\mathit{CL}}^{\mu }$ and ${\mathit{NL}}^{\mu }$ are supstar families and the lattice $L$ is a chain, then by Theorem 1.31 and Theorem 1.36, ${\mathit{CL}}^{\mu }\subseteq {\mathit{NL}}^{\mu }\subseteq L_s^{\mu }$. By Theorem 1.38, ${\mathit{CL}}^{\mu }$ is closed under arbitrary unions with the least element identically zero function. So ${\mathit{CL}}^{\mu }$ is an upper complete sublattices of ${\mathit{NL}}^{\mu }$. Similarly by Theorem 1.6 and Theorem 1.36, ${\mathit{NL}}^{\mu }$ is an upper complete sublattices of $L_s^{\mu }$.

On the other hand, we discuss the behavior of set products of $L$-subsets when the given lattice is a chain or the two $L$-subsets are jointly supstar.

Theorem 1.39 Let $\eta ,\nu \in L^{\mu }$. Then,

1. ${\left(\eta \circ \nu \right)}_a^{>}={\eta }_a^{>}{\nu }_a^{>}$ for each $a\in L\sim \left\{1\right\}$, provided $L$ is a chain,

2. ${\left(\eta \circ \nu \right)}_a={\eta }_a{\nu }_a$ for each $a\in L$, provided $\eta$ and $\nu$ are jointly supstar.

This helps us in establishing the following:

Theorem 1.40 Let $\eta ,\theta \in {\mathit{CL}}^{\mu }$ be jointly supstar. Then, $\eta \circ \theta \in {\mathit{CL}}^{\mu }$.

Below we study the notion of a characteristic $L$-subgroup of an $L$-group and its related properties:

Definition 1.12 Let $\eta \in L\left(\mu \right)$. Then, $\eta$ is said to be a characteristic $L$-subgroup of $\mu$ if $\eta \in {\mathit{CL}}^{\mu }$. We denote by $\mathit{CL}\left(\mu \right)$ the set of all characteristic $L$-subgroups of $\mu$.

Clearly, an $L$-group $\mu$ is a characteristic $L$-subgroup of itself. The following theorem characterizes the notion of a characteristic $L$-subgroup in terms of its level subsets:

Theorem 1.41 Let $\eta \in L\left(\mu \right)$. Then, $\eta \in \mathit{CL}\left(\mu \right)$ if and only if ${\eta }_a$ is a characteristic subgroup of ${\mu }_a\text{foreach}a\le \eta \left(e\right)$.

Moreover, we have the following results:

Theorem 1.42 Let $\eta \in \mathit{CL}\left(\mu \right)$. Then, $\eta \in \mathit{NL}\left(\mu \right)$.

Theorem 1.43 Let $\eta \in \mathit{CL}\left(\mu \right)$. Then,

$$T\left(\eta |{\mu }_a\right)\mid =\eta \mid {\mu }_a\text{for}\mathrm{all}T\in A\left({\mu }_a\right);\text{where}a\le \eta \left(e\right).$$E26

Theorem 1.44 Let $\eta ,\theta \in \mathit{CL}\left(\mu \right)$ be jointly supstar. Then, $\eta \circ \theta \in \mathit{CL}\left(\mu \right)$.

Theorem 1.45 Let $\theta \in \mathit{NL}\left(\mu \right)$ and $\eta \in \mathit{CL}\left(\theta \right)$. Then, $\eta \in \mathit{NL}\left(\mu \right)$.

In classical group theory, it is well known that the property of being a characteristic subgroup is transitive, and a characteristic subgroup of a normal subgroup of a group is a normal subgroup. However, the same could not even be formulated in the work of earlier researchers who have defined the notion of a characteristic fuzzy subgroup of an ordinary group in various ways. However, these pleasing features of classical group theory are retained in our studies.

Theorem 1.46 Let $\theta \in \mathit{CL}\left(\mu \right)$ and $\eta \in \mathit{CL}\left(\theta \right)$. Then, $\eta \in \mathit{CL}\left(\mu \right)$.

Let us denote by ${\mathit{CL}}_s\left(\mu \right)$ the set of all characteristic $L$-subgroups of $\mu$, each member of which possesses sup-property. In classical group theory, the subgroup generated by a characteristic subset of a group is a characteristic subgroup. Below we provide its counterpart in $L$-group theory:

Theorem 1.47 Let $\eta \in {\mathit{CL}}^{\mu }$ and possesses sup-property. Then, $⟨\eta ⟩\in {\mathit{CL}}_s\left(\mu \right)$.

Now, we exhibit that the set of all normal $L$-subgroups and the set of all characteristic $L$-subgroups each member of which possesses sup-property, constitute sublattices of the lattice of $L$-subgroups of a given $L$-group.

Theorem 1.48 The set $\mathit{NL}\left(\mu \right)$ is a complete sublattice of $L\left(\mu \right).$.

In the following results the lattice $L$ is a chain:

Theorem 1.49 The set $L_s\left(\mu \right)$ of all $L$-subgroups of $\mu$ each member of which possesses sup-property, is a sublattice of $L\left(\mu \right)$.

Theorem 1.50 The set ${\mathit{NL}}_s\left(\mu \right)$ of all normal $L$-subgroup of $\mu$ each member of which possesses sup-property, is a sublattice of $L_s\left(\mu \right)$ and hence of $L\left(\mu \right)$.

Theorem 1.51 The set ${\mathit{CL}}_s\left(\mu \right)$ of all characteristic $L$-subgroups of $\mu$ each member of which possesses sup-property, is a sublattice of ${\mathit{NL}}_s\left(\mu \right)$ and hence of $L\left(\mu \right)$.

The following diagram provides the lattice structure of the sublattices of the lattice $L\left(G\right)$ provided the lattice $L$ is a chain (Figure 1):

Figure 1.

The lattice structure of sublattices of the lattice $L\left(G\right)$.

If $L_t\left(\mu \right)$ denotes the set of all $L$-subgroups of $\mu$ each member of which has the same tip $t$, where $t\in \mathit{Im}\mu$, then the following result is easy to verify:

Theorem 1.52 The set $L_t\left(\mu \right)$ is a sublattice of $L\left(\mu \right)$.

Moreover, we have:

Theorem 1.53 The lattice $L\left(\mu \right)$ is a disjoint union of its sublattices $L_t\left(\mu \right)$. That is $L\left(\mu \right)=\bigcup _{t\in \mathit{Im}\mu }{L}_t\left(\mu \right)$.

As the intersection of two sublattices is a sublattice of the given lattice, the following results are immediate:

Theorem 1.54 The set $L_{\mathit{nt}}\left(\mu \right)$ of all normal $L$-subgroups of $\mu$ with the same tip $t$, is a sublattice of $L_n\left(\mu \right)$ and hence of $L\left(\mu \right)$.

In the following results the lattice $L$ is a chain:

Theorem 1.55 The set $L_{\mathit{st}}\left(\mu \right)$ of all $L$-subgroups of $\mu$ with the same tip $t$ and each member of which possesses sup-property, is a sublattice of $L_s\left(\mu \right)$ and hence of $L\left(\mu \right)$.

Theorem 1.56 The set $L_{\mathit{nst}}\left(\mu \right)$ of all normal $L$-subgroups of $\mu$ with the same tip $t$ and each member of which possesses sup-property, is a sublattice of $L_{\mathit{nt}}\left(\mu \right)$ and $L_{\mathit{st}}\left(\mu \right)$ and hence of $L\left(\mu \right)$.

The following lattice diagram shows the inter-relationship of the above discussed sublattices, where the lattice $L$ is a chain (Figure 2):

Figure 2.

Inter-relationship of the sublattices of $L\left(\mu \right)$.

Theorem 1.57 The set $L_{\mathit{cst}}\left(\mu \right)$ of all characteristic $L$-subgroups of $\mu$ with the same tip $t$ and each member of which possesses sup-property, is a sublattice of ${\mathit{NL}}_{\mathit{st}}\left(\mu \right)$ and hence of $L\left(\mu \right)$ provided the lattice $L$ is a chain.

Now, let us consider an arbitrary lattice $L$. Let $L^{\ast }\left(\mu \right)$ denote a subclass of the lattice $L\left(\mu \right)$ consisting of all the supstar families. Then, $L^{\ast }\left(\mu \right)$ is a complete sublattice of $L\left(\mu \right)$. Now if ${\mathit{CL}}_{\mathit{st}}^{\ast }$, ${\mathit{CL}}_s^{\ast }\left(\mu \right)$, ${\mathit{NL}}_s^{\ast }\left(\mu \right)$, $L_s^{\ast }\left(\mu \right)$, ${\mathit{NL}}_{\mathit{st}}^{\ast }\left(\mu \right)$ and $L_{\mathit{st}}^{\ast }\left(\mu \right)$ denote the set of all supstar families of $L$-subsets in ${\mathit{CL}}_{\mathit{st}}$, ${\mathit{CL}}_s\left(\mu \right)$, ${\mathit{NL}}_s\left(\mu \right)$, $L_s\left(\mu \right)$, ${\mathit{NL}}_{\mathit{st}}\left(\mu \right)$ and $L_{\mathit{st}}\left(\mu \right)$ respectively, then it follows that

$${\mathit{CL}}_{\mathit{st}}^{\ast }\subseteq {\mathit{NL}}_{\mathit{st}}^{\ast }\subseteq L_{\mathit{st}}^{\ast }\subseteq L_s^{\ast }\left(\mu \right)\text{and}{\mathit{CL}}_{\mathit{st}}^{\ast }\subseteq {\mathit{CL}}_s^{\ast }\subseteq {\mathit{NL}}_s^{\ast }\subseteq L_s^{\ast }\left(\mu \right);$$E27

are sublattices of $L\left(\mu \right)$. Note that $L_s^{\ast }\left(\mu \right)\subseteq L_s\left(\mu \right)$. Moreover, we mention that it is not known whether $\mathit{CL}\left(\mu \right)$ is a sublattice of $\mathit{NL}\left(\mu \right)$. We leave this question as an open problem and below we describe the inter-relationship of above discussed lattices, simply under the set inclusion, where the lattice $L$ is taken to be a chain (Figure 3).

Figure 3.

Inter-relationship of various sublattices of $L\left(\mu \right)$ under ordinary set inclusion.

4. Commutator $L$-subgroup and nilpotent $L$-subgroup of an $L$-group

The class of nilpotent groups constitutes an important class in the studies of group theory. In fact, nilpotent groups are very near to Abelian groups and they are always solvable. They arise in the studies of Galois theory as well as in the classification of groups. In the investigation of nilpotent groups and solvable groups, the notion of commutator and commutator subgroups play very significant role. Therefore, we start with notion of commutator $L$-subgroups.

4.1 Commutator $L$-subgroup of an $L$-group

In this subsection, we study the notion of a commutator in $L$-setting [28, 29] by using the notion of infimums of $L$-subsets. In fact, we also discuss here how infimums of $L$-subsets play an effective role in the development of the theory of $L$-subgroups.

Definition 1.13 Let $\eta ,\theta \in L^{\mu }$. Then, the commutator of $\eta$ and $\theta$ is an $L$-subset $\left(\eta ,\theta \right)$ of $G$ defined as follows:

$$\left(\eta ,\theta \right)\left(x\right)=\left\{\begin{array}{ll}\vee \left\{\eta \left(y\right)\wedge \theta \left(z\right)\right\},& \text{if}x=\left[y,z\right]\text{for\hspace{0.17em}some}y,z\in G,\\ inf\eta \wedge inf\theta ,& \text{if}x\ne \left[y,z\right]\text{for}\mathrm{any}y,z\in G.\end{array}\right.$$E28

The commutator $L$-subgroup of $\eta ,\theta \in L^{\mu }$ is defined as the $L$-subgroup of $G$ generated by $\left(\eta ,\theta \right)$. It is denoted by $\left[\eta ,\theta \right]$. Thus, $\left[\eta ,\theta \right]=⟨\left(\eta ,\theta \right)⟩$. Moreover, $\left[\eta ,\theta \right]\left(e\right)=\eta \left(e\right)\wedge \theta \left(e\right)$.

In general, if ${\left\{{\eta }_i\right\}}_{i=1}^n\subseteq L^{\mu }$, then we write $\left({\eta }_1,{\eta }_2,{\eta }_3,\dots ,{\eta }_n\right)=\left(\left(\left({\eta }_1,{\eta }_2\right),{\eta }_3\right),\dots ,{\eta }_n\right)$ and $\left[{\eta }_1,{\eta }_2,{\eta }_3,\dots ,{\eta }_n\right]=\left[\left[\left[{\eta }_1,{\eta }_2\right],{\eta }_3\right],\dots ,{\eta }_n\right]$. Moreover, it follows that $inf\left({\eta }_1,{\eta }_2,{\eta }_3,\dots ,{\eta }_n\right)=inf{\eta }_1\wedge inf{\eta }_2\wedge \dots \wedge inf{\eta }_n$. Also, $\left[{\eta }_1,{\eta }_2,{\eta }_3,\dots {\eta }_n\right]\left(e\right)={\eta }_1\left(e\right)\wedge {\eta }_2\left(e\right)\wedge \dots \wedge {\eta }_n\left(e\right)$.

In the following, we present some extensions of the results of classical group theory with certain deviations and peculiarities:

Theorem 1.58 Let $\eta ,\theta \in L^{\mu }$ and $\eta \subseteq \theta$. Then, $\left[\eta ,\sigma \right]\subseteq \left[\theta ,\sigma \right]$ for each $\sigma \in L^{\mu }$.

Theorem 1.59 Let $\eta ,\theta \in L^{\mu }$. Then, $\left[\eta ,\theta \right]=\left[\theta ,\eta \right]$.

Theorem 1.60 Let $\eta ,\theta \in L^{\mu }$. Then,

$$inf\eta \wedge inf\theta \le inf\eta \circ \theta \le inf\eta \vee inf\theta .$$E29

Theorem 1.61 Let $\eta ,\theta \in \mathit{NL}\left(\mu \right)$ and $\sigma \in L\left(\mu \right).$ If either $\eta$ and $\theta$ or $\theta$ and $\sigma$ have the same tails, then

$$\left[\sigma \circ \eta ,\theta \right]\subseteq \left[\eta ,\theta \right]\circ \left[\sigma ,\theta \right],$$E30

and moreover, if $\eta \left(e\right)=\sigma \left(e\right)$, then the equality holds.

Theorem 1.62 Let $\eta ,\theta \in \mathit{NL}\left(\mu \right)$. Then,

$$\left[\eta ,\theta \right]\in \mathit{NL}\left(\mu \right)\text{and}\left[\eta ,\theta \right]\subseteq \eta \cap \theta .$$E31

Theorem 1.63 Let $\eta ,\theta \in L^{\mu }$ and $\mathrm{f}:G\to K$ be a group homomorphism. Then, $\mathrm{f}\left(⟨\eta ⟩\right)=⟨\mathrm{f}\left(\eta \right)⟩$ and $\mathrm{f}\left(\left[\eta ,\theta \right]\right)=⟨\mathrm{f}\left(\left(\eta ,\theta \right)\right)⟩.$

Theorem 1.64 Let $\eta ,\theta \in {\mathit{CL}}^{\mu }$ be jointly supstar. Then, $\left[\eta ,\theta \right]\in {\mathit{CL}}_s\left(\mu \right)$.

Theorem 1.65 Let $\eta ,\theta \in L^{\mu }$ and $\mathrm{f}:G\to K$ be a homomorphism. Let $inf\eta =inf\mathrm{f}\left(\eta \right)$ and $inf\theta =inf\mathrm{f}\left(\theta \right)$. Then, $\left[\mathrm{f}\left(\eta \right),\mathrm{f}\left(\theta \right)\right]=\mathrm{f}\left(\left[\eta ,\theta \right]\right)$.

Theorem 1.66 Let $\mathrm{f}:G\to H$ be a homomorphism and $\nu \in L\left(H\right)$. Let $\lambda ,\sigma \in L^{\nu }$ and $inf\lambda =inf{\mathrm{f}}^{-1}\left(\lambda \right),inf\sigma =inf{\mathrm{f}}^{-1}\left(\sigma \right)$. Then, $\left[{\mathrm{f}}^{-1}\left(\lambda \right),{\mathrm{f}}^{-1}\left(\sigma \right)\right]\subseteq {\mathrm{f}}^{-1}\left(\left[\lambda ,\sigma \right]\right)$.

Theorem 1.67 Let $\eta ,\theta \in L\left(\mu \right)$ be such that $\eta$ and $\theta$ are jointly supstar. Then, the commutator $\left(\eta ,\theta \right)$ and hence the commutator $L$-subgroup $\left[\eta ,\theta \right]$ possess sup-property.

Note that, $a\le inf\eta$ if and only if ${\eta }_a=G$. Moreover, if $a\le inf\eta \wedge inf\theta$, then the level subsets ${\eta }_a,{\theta }_a$ and ${\left(\eta ,\theta \right)}_a$ coincide with $G$ and hence ${\left(\eta ,\theta \right)}_a\ne \left({\eta }_a,{\theta }_a\right)$.

Theorem 1.68 Let $\eta ,\theta \in L\left(\mu \right)$. If $a_0=\eta \left(e\right)\wedge \theta \left(e\right)$ and $a\nleqq inf\eta \wedge inf\theta$, then

1. ${\left[\eta ,\theta \right]}_a=\left[{\eta }_a,{\theta }_a\right]$ for each $a\le a_0$, provided $\eta$ and $\theta$ are jointly supstar.

2. ${\left[\eta ,\theta \right]}_a^{>}=\left[{\eta }_a^{>},{\theta }_a^{>}\right]$ for each $a<a_0$, provided $L$ is a chain.

In general, let ${\left\{{\eta }_i\right\}}_{i=1}^n\subseteq L\left(\mu \right)$. If $a_0={\eta }_1\left(e\right)\wedge {\eta }_2\left(e\right)\wedge \dots \wedge {\eta }_n\left(e\right)$ and $a\nleqq inf{\eta }_i$ for each $i$, then

1. ${\left[{\eta }_1,{\eta }_2,{\eta }_3,\dots ,{\eta }_n\right]}_a=\left[{\left({\eta }_1\right)}_a,{\left({\eta }_2\right)}_a,{\left({\eta }_3\right)}_a,\dots ,{\left({\eta }_n\right)}_a\right]$ for each $a\le a_0$, provided ${\left\{{\eta }_i\right\}}_{i=1}^n$ is a supstar family.

2. ${\left[{\eta }_1,{\eta }_2,{\eta }_3,\dots ,{\eta }_n\right]}_a^{>}=\left[{\left({\eta }_1\right)}_a^{>},{\left({\eta }_2\right)}_a^{>},{\left({\eta }_3\right)}_a^{>},\dots ,{\left({\eta }_n\right)}_a^{>}\right]$ for each $a<a_0$, provided $L$ is a chain.

4.2 Nilpotent $L$-subgroup of an $L$-group

We begin this subsection with the concept of a descending central chain of an $L$-subgroup by making use of the notion of commutators. Then, this in turn has been used to define nilpotent $L$-subgroups [28]. Throughout this subsection, $G$ would denote a group which is not perfect.

We start with the definition of a descending central chain of an $L$-subgroup $\eta$ of an $L$-subgroup $\mu$.

Take $Z_0\left(\eta \right)=\eta$, $Z_1\left(\eta \right)=\left[Z_0\left(\eta \right),\eta \right]$. And in general, for each $i$, define

$$Z_i\left(\eta \right)=\left[Z_{i-1}\left(\eta \right),\eta \right].$$E32

The following result is an immediate consequence of the above definition:

Theorem 1.69 Let $\eta \in L\left(\mu \right)$. Then for each $i,Z_i\left(\eta \right)\subseteq Z_{i-1}\left(\eta \right)$.

Here we provide the definition of a descending central chain.

Definition 1.14 Let $\eta \in L\left(\mu \right)$. Then, the chain

$$\eta =Z_0\left(\eta \right)\supseteq Z_1\left(\eta \right)\supseteq \cdots \supseteq Z_i\left(\eta \right)\supseteq \cdots$$E33

of $L$-subgroups of $\mu$ is called the descending central chain of $\eta$.

It is worthwhile to note that as $\eta \in \mathit{NL}\left(\eta \right)$, in view of Theorem 1.62, $Z_i\left(\eta \right)$ is a normal $L$-subgroup of $\eta$ for each $i$. Moreover, if $\eta \in L\left(\mu \right)$, then $Z_i\left(\eta \right)\in \mathit{NL}\left(\mu \right)$.

Now, we are in a position to formulate the definition of a nilpotent $L$-subgroup of an $L$-group.

Definition 1.15 Let $\eta \in L\left(\mu \right)$ with tip $a_0$ and tail $t_0$ and $a_0\ne t_0$. If the descending central chain

$$\eta =Z_0\left(\eta \right)\supseteq Z_1\left(\eta \right)\supseteq \cdots \supseteq Z_i\left(\eta \right)\supseteq \cdots$$E34

terminates finitely to the trivial $L$-subgroup ${\eta }_{t_0}^{a_0}$, then $\eta$ is known as a nilpotent $L$-subgroup of $\mu$. More precisely, $\eta$ is said to be nilpotent of class $c$ if $c$ is the least non-negative integer such that $Z_c\left(\eta \right)={\eta }_{t_0}^{a_0}$. In this case, the series

$$\eta =Z_0\left(\eta \right)\supseteq Z_1\left(\eta \right)\supseteq \cdots \supseteq Z_c\left(\eta \right)={\eta }_{t_0}^{a_0}$$E35

is called the descending central series of $\eta$. If it is a nilpotent $L$-subgroup of $\mu$, then we simply write $\eta$ is nilpotent. Clearly, the tip and tail of the members $Z_i\left(\eta \right)$ of descending central series coincide with the tip and the tail of the trivial $L$-subgroup ${\eta }_{t_0}^{a_0}$.

Next, we provide some results pertaining to the member ‘$Z_i\left(\eta \right)$’ of the descending central chain which are peculiarities of $L$-setting.

Theorem 1.70 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then for each $i$,

$${\mathit{ImZ}}_i\left(\eta \right)\subseteq \mathit{Im\eta }\cup \left\{inf\eta \right\}.$$E36

Theorem 1.71 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then for each $i$,

1. $Z_i\left(\eta \right)$ possesses sup-property.

2. $Z_i\left(\eta \right)$ and $\eta$ are jointly supstar.

The following result justifies the naturality of the extension of the notion of descending central chain:

Theorem 1.72 Let $H$ be a subgroup of $G$. Then for each $i$, $Z_i\left(1_H\right)=1_{Z_i\left(H\right)}$.

For further justification of these notions, we provide the level subset and strong level subset characterizations of the members of descending central series.

Theorem 1.73 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then for each $a\nleqq inf\eta$ and $a\le \eta \left(e\right)$, $Z_i\left({\eta }_a\right)={\left(Z_i\left(\eta \right)\right)}_a$ for each $i$.

Theorem 1.74 Let $L$ be a chain and $\eta \in L\left(\mu \right)$. Then for each $a$, where $inf\eta \le a<\eta \left(e\right)$, $Z_i\left({\eta }_a^{>}\right)={\left(Z_i\left(\eta \right)\right)}_a^{>}$ for each $i$.

Finally, we obtain the level subset and strong level subset characterizations for nilpotent $L$-subgroups.

Theorem 1.75 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then, $\eta$ is a nilpotent $L$-subgroup of $\mu$ of nilpotent length at most $n$ if and only if ${\eta }_a$ is a nilpotent subgroup of ${\mu }_a$ of nilpotent length at most $n$ for each $a\cancel{\le }inf\eta$ and $a\le \eta \left(e\right)$.

Theorem 1.76 Let $\eta \in L\left(\mu \right)$ and $L$ be a chain. Then, $\eta$ is a nilpotent $L$-subgroup of $\mu$ of nilpotent length at most $n$ if and only if ${\eta }_a^{>}$ is a nilpotent subgroup of ${\mu }_a^{>}$ of nilpotent length at most $n$, for each $a$, where $inf\eta \le a<\eta \left(e\right)$.

The following result immediately follows from the above results:

Theorem 1.77 A subgroup $H$ of a group of $G$ is nilpotent if and only if $1_H$ is a nilpotent $L$-subgroup of $1_G$.

Next, we provide the definition of a central chain and use it to characterize nilpotent $L$-subgroups.

Definition 1.16 Let $\eta \in L\left(\mu \right)$ with tip $a_0$ and tail $t_0$. Then, the chain

$$\eta ={\eta }_0\supseteq {\eta }_1\supseteq \dots \supseteq {\eta }_n\supseteq \dots$$E37

of $L$-subgroups of $\mu$ is called a central chain of $\eta$, if for each $i$, $\left[{\eta }_{i-1},\eta \right]\subseteq {\eta }_i.$ If $a_0\ne t_0$ and there exists a positive integer $m$ such that ${\eta }_m={\eta }_{t_0}^{a_0}$, where ${\eta }_{t_0}^{a_0}$ is the trivial $L$-subgroup of $\eta$ with tip $a_0$ and tail $t_0$, then

$$\eta ={\eta }_0\supseteq {\eta }_1\supseteq \dots \supseteq {\eta }_m={\eta }_{t_0}^{a_0}$$E38

is known as a central series of $\eta$. It follows that ${\eta }_i$ and $\eta$ have identical tips and also, identical tails for each $i$. Moreover, the following is easy to verify:

1. $\left[{\eta }_{i-1},\eta \right]\subseteq {\eta }_i$ if and only if ${\eta }_i\left(\left[x,y\right]\right)\ge {\eta }_{i-1}\left(x\right)\wedge \eta \left(y\right)\text{for\hspace{0.17em}each}i$,

2. ${\eta }_i\in \mathit{NL}\left(\eta \right)\text{for\hspace{0.17em}each}i.$

Now, we are in a position to study the notion of a nilpotent $L$-subgroup by making use of the concept of a central chain. The results are as follows:

Theorem 1.78 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup of $\mu$. Then, $\eta$ is nilpotent if and only if $\eta$ has a central series.

Theorem 1.79 Let $\eta \in L\left(\mu \right)$ and $\theta$ be a proper $L$-subgroup of $\eta$ such that $\eta$ and $\theta$ have the common tail $t_0$. If $\eta$ is nilpotent, then $\theta$ is also nilpotent.

The notion of set product is an extension of the notion of product of complexes in classical group theory. The following two results provides a necessary mechanism for the set product of two nilpotent $L$-subgroups of $\mu$ to be nilpotent:

Theorem 1.80 Let $\eta ,{\eta }_1,\dots ,{\eta }_{n+1}\in \mathit{NL}\left(\mu \right)$ having identical tails. If ${\eta }_i=\eta$ for k + 1 distinct values of $i$ where $0\le k\le n$, then $\left[{\eta }_1,{\eta }_2,\dots ,{\eta }_{n+1}\right]\subseteq Z_k\left(\eta \right).$

Theorem 1.81 Let $\eta$ and $\theta$ be trivial $L$-subgroups of $\mu$. Then, the set product $\eta \circ \theta$ is also a trivial $L$-subgroup of $\mu$ defined by

$$\eta \circ \theta \left(x\right)=\left\{\begin{array}{cc}\eta \left(e\right)\wedge \theta \left(e\right)& \text{if}x=e,\\ \mathit{\text{infη}}\vee \mathit{\text{infθ}}& \text{if}x\ne e,\end{array}\right.$$E39

if and only if $\mathit{\text{infη}}\vee \mathit{\text{infθ}}<\eta \left(e\right)\wedge \theta \left(e\right)$.

The final result of this subsection provides a sufficient condition for the set product of two nilpotent $L$-subgroups to be nilpotent.

Theorem 1.82 Let $\eta ,\theta \in \mathrm{NL}\left(\mu \right)$ with common tails $t_0$, such that $t_0<\eta \left(e\right)\wedge \theta \left(e\right)$ and $inf\eta \circ \theta =t_0$. If $\eta$ and $\theta$ are nilpotent of classes $c$ and $d$ respectively, then $\eta \circ \theta$ is a nilpotent $L$-subgroup of $\mu$ of class at most $c+d$.

5. Solvable and supersolvable $L$-subgroups of an $L$-group

The significance of the notion of solvability in classical group theory is beyond doubt. The studies pertaining to the notion of solvability and supersolvability frequently occur in the literature. The application of solvability in Galois Theory is established. Also, the application of nilpotency and solvability are well known in Lie groups.

5.1 Solvable $L$-subgroups of an $L$-group

In this subsection, we study the notion of derived series of an $L$-subgroup of an $L$-group in the same fashion as in classical group theory (see [29]). Here, also, $G$ would denote a group that is not perfect.

Let $\eta \in L\left(\mu \right)$. We define inductively the following sequence of $L$-subgroups of $\mu$:

$${\eta }^{\left(0\right)}=\eta \text{and}{\eta }^{\left(i\right)}=\left[{\eta }^{\left(i-1\right)},{\eta }^{\left(i-1\right)}\right]\text{for\hspace{0.17em}each}i.$$E40

The following result is immediate:

Theorem 1.83 Let $\eta \in L\left(\mu \right)$. Then, ${\eta }^{\left(i\right)}\subseteq {\eta }^{\left(i-1\right)}$.

We study the concept of solvable $L$-subgroups of an $L$-group like its classical counterpart. For this purpose, we introduce the concept of a derived chain of an $L$-subgroup as follows:

Definition 1.17 Let $\eta \in L\left(\mu \right)$. Then, the chain

$$\eta ={\eta }^{\left(0\right)}\supseteq {\eta }^{\left(1\right)}\supseteq \cdots \supseteq {\eta }^{\left(i\right)}\supseteq \cdots$$E41

of $L$-subgroups of $\eta$ is called the derived chain of $\eta$. Clearly, the tip of ${\eta }^{\left(i\right)}$ coincides with $\eta \left(e\right)$. Also, ${\eta }^{\left(i\right)}\in \mathrm{NL}\left(\eta \right)$. Moreover, if $\eta \in$ NL$\left(\mu \right)$, then ${\eta }^{\left(i\right)}\in$ NL$\left(\mu \right)$.

Now, we are in a position to formulate the definition of a solvable $L$-subgroup of an $L$-group.

Definition 1.18 Let $\eta \in L\left(\mu \right)$ with tip $a_0$ and tail $t_0$ and $a_0\ne t_0$. If the derived chain

$$\eta ={\eta }^{\left(0\right)}\supseteq {\eta }^{\left(1\right)}\supseteq \cdots \supseteq {\eta }^{\left(i\right)}\supseteq \cdots$$E42

terminates finitely to the trivial $L$-subgroup ${\eta }_{t_0}^{a_0}$, then $\eta$ is known as a solvable $L$-subgroup of $\mu$. If $m$ is the least non negative integer such that ${\eta }^{\left(m\right)}={\eta }_{t_0}^{a_0}$, then the series

$$\eta ={\eta }^{\left(0\right)}\supseteq {\eta }^{\left(1\right)}\supseteq \cdots \supseteq {\eta }^{\left(m\right)}={\eta }_{t_0}^{a_0}$$E43

is called the derived series of $\eta$ and $m$ is said to be the solvable length of $\eta$. If $\eta$ is a solvable $L$-subgroup of $\mu$, then we simply write $\eta$ is solvable. Clearly, the tip and tail of the members ${\eta }^{\left(i\right)}$ of derived series coincide with the tip and the tail of the trivial $L$-subgroup ${\eta }_{t_0}^{a_0}$.

Next, we provide some results pertaining to the member’${\eta }^{\left(i\right)}$’ of the derived chain which are peculiarities of $L$-setting.

Theorem 1.84 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then,

$$\mathit{Im}{\eta }^{\left(i\right)}\subseteq \mathit{Im}\eta \cup \left\{\mathit{inf}\eta \right\}.$$E44

Theorem 1.85 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then,

1. ${\eta }^{\left(i\right)}$ possesses sup-property,

2. ${\eta }^{\left(i\right)}$ and $\eta$ are jointly supstar.

The following result justifies the naturality of the extension of the notion of derived chain:

Theorem 1.86 Let $H$ be a subgroup of $G$. Then for each $i$, ${\left(1_H\right)}^{\left(i\right)}=1_{H^{\left(i\right)}}$.

For the justification of these notions, we also provide the level subset and strong level subset characterizations of the members of derived series.

Theorem 1.87 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then, for each $a\nleqq \mathit{\text{infη}}$ and $a\le \eta \left(e\right)$, ${\left({\eta }^{\left(i\right)}\right)}_a={\left({\eta }_a\right)}^{\left(i\right)}\text{for\hspace{0.17em}each}i$.

Theorem 1.88 Let $L$ be a chain and $\eta \in L\left(\mu \right)$. Then for each $a$, where $inf\eta \le a<\eta \left(e\right)$, ${\left({\eta }^i\right)}_a^{>}={\left({\eta }_a^{>}\right)}^i$.

Finally, we obtain the level subset and strong level subset characterizations for solvable $L$-subgroups.

Theorem 1.89 Let $\eta \in L\left(\mu \right)$ and possesses sup-property. Then, $\eta$ is a solvable $L$-subgroup of $\mu$ of solvable length at most n if and only if ${\eta }_a$ is a solvable subgroup of ${\mu }_a$ of solvable length at most n for each $a\nleqq \mathit{inf}\eta$ and $a\le \eta \left(e\right)$.

Theorem 1.90 Let $\eta \in L\left(\mu \right)$ and $L$ be a chain. Then, $\eta$ is a solvable $L$-subgroup of $\mu$ of solvable length at most $n$ if and only if ${\eta }_a^{>}$ is a solvable subgroup of ${\mu }_a^{>}$ of solvable length at most $n$ for each $a$, where $inf\eta \le a<\eta \left(e\right)$.

In view of the above results, the following is immediate:

Theorem 1.91 A subgroup $H$ of a group of $G$ is solvable if and only if $1_H$ is a solvable $L$-subgroup of $1_G$.

Now, we provide the definition of a solvable series and use it to characterize solvable $L$-subgroups.

Definition 1.19 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. If ${\eta }_{t_0}^{a_0}$ is the trivial $L$-subgroup of $\eta$, then a series

$$\eta ={\eta }_0\supseteq {\eta }_1\supseteq \cdots \supseteq {\eta }_n={\eta }_{t_0}^{a_0}$$E45

of $L$-subgroups of $\eta$ is said to be a solvable series for $\eta$, if for each $i$

$$\left[{\eta }_{i-1},{\eta }_{i-1}\right]\subseteq {\eta }_i.$$E46

It follows that for each i, ${\eta }_i$ and $\eta$ have identical tips as well as identical tails. Moreover, the following is easy to verify:

$$\left(i\right)\left[{\eta }_{i-1},{\eta }_{i-1}\right]\subseteq {\eta }_i\text{if\hspace{0.17em}and\hspace{0.17em}only\hspace{0.17em}if}{\eta }_i\left(\left[x,y\right]\right)\ge {\eta }_{i-1}\left(x\right)\wedge {\eta }_{i-1}\left(y\right)\text{for\hspace{0.17em}each}i,$$

$$\left(\mathrm{ii}\right){\eta }_i\in \mathit{NL}\left({\eta }_{i-1}\right)\text{for\hspace{0.17em}each}i.$$

The following is characterization relates the concept of solvability with that of solvable series:

Theorem 1.92 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. Then, $\eta$ is solvable if and only if $\eta$ has a solvable series.

The above characterization helps us in obtaining the following:

Theorem 1.93 Let $\eta ,\theta \in L\left(\mu \right)$ be proper $L$-subgroups having identical tails. If $\eta$ is a solvable $L$-subgroup and $\theta \subseteq \eta$, then $\theta$ is also solvable.

Our next result establishes the fact that every nilpotent $L$-subgroup is solvable.

Theorem 1.94 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup. Then, every central series of $\eta$ is a solvable series.

In classical group theory, every nilpotent group is solvable. We obtain the same result in $L$-setting:

Theorem 1.95 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup. If $\eta$ is nilpotent, then $\eta$ is also solvable.

The nature of image and pre-image of a solvable $L$-subgroup under a group homomorphism is ascertained in the following results:

Theorem 1.96 Let $f:G\to H$ be a group homomorphism and $\eta \in L\left(\mu \right)$. Let $\mathit{inf}\eta =\mathit{inf}f\left(\eta \right)$. If $\eta$ is solvable, then $f\left(\eta \right)$ is also solvable.

Theorem 1.97 Let $f:G\to H$ be a group homomorphism having solvable kernel and $\nu \in L\left(H\right)$. Let $\rho \in L\left(\nu \right)$ and $\mathit{inf}\rho =\mathit{inf}{f}^{-1}\left(\rho \right)$. If $\rho$ is solvable, then $f^{-1}\left(\rho \right)$ is also solvable.

So far in our studies, we have not dealt with normal series and subinvariant series in $L$-setting. Here we introduce these concepts and utilize them to characterize solvable $L$-subgroups (see [32]). We start with:

Definition 1.20 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. Then, a sequence ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n$ of $L$-subgroups of $\mu$ is said to be a normal (subinvariant) series of $\eta$ if

$$\eta ={\lambda }_0\supseteq {\lambda }_1\supseteq \dots \supseteq {\lambda }_n={\eta }_{t_0}^{a_0}$$E47

and ${\lambda }_i\in \mathit{NL}\left(\mu \right)\left({\lambda }_i\in \mathit{NL}\left({\lambda }_{i-1}\right)\right)$ for each $i$.

Remark: If ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n$ is a normal (subinvariant) series of $\eta$, then ${\lambda }_i\left(e\right)=\eta \left(e\right)=a_0$ and $inf{\lambda }_i=inf\eta =t_0$ for each $i$.

Definition 1.21 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. Then, a normal (subinvariant) series ${\lambda }_0=\eta \supseteq {\lambda }_1\supseteq \dots \supseteq {\lambda }_n={\eta }_{t_0}^{a_0}$ of $\eta$ is said to be a normal (subinvariant) series with Abelian factors if for each $i$, the factor group $\frac{{\left({\lambda }_{i-1}\right)}_a}{{\left({\lambda }_i\right)}_a}$ is Abelian for each $t_0<a\le a_0$.

The following theorem extends a famous result of classical group theory pertaining to the notion of solvability to $L$-setting:

Theorem 1.98 Let $L$ be a dense chain and $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. Then, the following are equivalent:

1. $\eta$ is solvable,

2. $\eta$ has a normal series with Abelian factors,

3. $\eta$ has a subinvariant series with Abelian factors.

Below we propose a definition of supersolvable subgroups in $L$-setting:

Definition 1.22 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. Then, $\eta$ is said to be a supersolvable $L$-subgroup if $\eta$ has a normal series ${\lambda }_0=\eta \supseteq {\lambda }_1\supseteq \dots \supseteq {\lambda }_n={\eta }_{t_0}^{a_0}$ such that each factor group $\frac{{\left({\lambda }_{i-1}\right)}_a}{{\left({\lambda }_i\right)}_a}$ for each $a$, where $t_0<a\le a_0$, is cyclic.

It has been established earlier, that $L$-subgroups of nilpotent $L$-subgroups and solvable $L$-subgroups are nilpotent and solvable respectively. It is well known that the class of supersolvable groups is also closed under the formation of subgroups in classical group theory. Here, we extend this property of supersovability to $L$-group theory.

Theorem 1.99 Let $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. Let $\theta \in L\left(\eta \right)$ be such that $\eta$ and $\theta$ have identical tips and also identical tails. If $\eta$ is supersolvable, then $\theta$ is also supersolvable.

The nilpotency of the commutator of a supersolvable subgroup is also retained in $L$-setting. The result is as follows:

Theorem 1.100 Let $L$ be an upper well ordered chain and $\eta \in L\left(\mu \right)$ be a proper $L$-subgroup with tip $a_0$ and tail $t_0$. If $\eta$ is supersolvable, then the commutator $L$-subgroup $\left[\eta ,\eta \right]$ is nilpotent.

5.2 Zassenhaus theorem and Schreier refinement theorem

In the earlier subsection, we have already introduced the concept of a normal (subinvariant) series of an $L$-subgroup of an $L$-group. In classical group theory, two normal series of a group are said to be equivalent if it is possible to set up a one to one correspondence between the factors of two series such that the paired factors are isomorphic. This is obtained by a certain type of factorization of a group into factor groups. Then, a generalization of second isomorphism theorem which is called Zassenhaus Theorem is used to establish this fact. Here in $L$-setting for the equivalence of two normal (subinvariant) series, we consider the factorization of each level of the $L$-subgroups in the spirit of classical group theory. Then, we extend Zassenhaus Theorem to $L$-setting and utilize it to establish a version of Schreier Refinement Theorem.

Below we extend the definitions of a refinement of a normal (subinvariant) series and equivalent normal (subinvariant) series (see [32]):

Definition 1.23 Let $\eta \in L\left(\mu \right)$ with tip $a_0$ and tail $t_0$ such that $a_0\ne t_0$. A normal (subinvariant) series of $\eta$

$$\eta ={\theta }_0\supseteq {\theta }_1\supseteq {\theta }_2\supseteq \dots \supseteq {\theta }_m={\eta }_{t_0}^{a_0}$$E48

is said to be a refinement of a normal (subinvariant) series of $\eta$

$$\eta ={\eta }_0\supseteq {\eta }_1\supseteq {\eta }_2\supseteq \dots \supseteq {\eta }_n={\eta }_{t_0}^{a_0}$$E49

if ${\eta }_0,{\eta }_1,{\eta }_2,\dots ,{\eta }_n$ is a subsequence of ${\theta }_0,{\theta }_1,{\theta }_2,\dots ,{\theta }_m$.

Definition 1.24 Let $\eta \in L\left(\mu \right)$ with tip $a_0$ and tail $t_0$ such that $a_0\ne t_0$. Then, two normal (subinvariant) series $\eta ={\eta }_0\supseteq {\eta }_1\supseteq {\eta }_2\supseteq \dots \supseteq {\eta }_n={\eta }_{t_0}^{a_0}$ and $\eta ={\theta }_0\supseteq {\theta }_1\supseteq {\theta }_2\supseteq \dots \supseteq {\theta }_m={\eta }_{t_0}^{a_0}$ of an $L$-subgroup $\eta$ are said to be equivalent if for each fixed $a\le a_0$, the factors of the series

$${\eta }_a={\left({\eta }_0\right)}_a\supseteq {\left({\eta }_1\right)}_a\supseteq {\left({\eta }_2\right)}_a\supseteq \dots \supseteq {\left({\eta }_m\right)}_e=\left\{e\right\}$$E50

can be put in one to one correspondence with the factors of the series

$${\eta }_a={\left({\theta }_0\right)}_a\supseteq {\left({\theta }_1\right)}_a\supseteq {\left({\theta }_2\right)}_a\supseteq \dots \supseteq {\left({\theta }_m\right)}_e=\left\{e\right\}$$E51

in such a way that the paired factors are isomorphic.

Now, we extend Zassenhaus Theorem to $L$-setting:

Theorem 1.101 (Zassenhaus Theorem) Let $\eta ,\theta \in L\left(\mu \right).$ Let ${\eta }_1\in \mathit{NL}\left(\eta \right)$ and ${\theta }_1\in \mathit{NL}\left(\theta \right)$. Then,

$$\left(\eta \cap {\theta }_1\right)\circ {\eta }_1⊲\left(\eta \cap \theta \right)\circ {\eta }_1\text{and}\left({\eta }_1\cap \theta \right)\circ {\theta }_1⊲\left(\eta \cap \theta \right)\circ {\theta }_1.$$E52

Also, there is an isomorphism such that

$$\frac{{\left(\eta \cap \theta \right)}_a{\left({\eta }_1\right)}_a}{{\left(\eta \cap {\theta }_1\right)}_a{\left({\eta }_1\right)}_a}\cong \frac{{\left(\eta \cap \theta \right)}_a{\left({\theta }_1\right)}_a}{{\left({\eta }_1\cap \theta \right)}_a{\left({\theta }_1\right)}_a}\text{for\hspace{0.17em}each}a\le a_0;$$E53

where $a_0={\eta }_1\left(e\right)\wedge {\theta }_1\left(e\right).$

The following theorem provides an extension of Schreier Refinement Theorem to $L$-setting:

Theorem 1.102 (Schreier Refinement Theorem) Let $L$ be an upper well ordered chain. Then, every two normal series of an $L$-subgroup have refinements that are equivalent.

6. Normal closure and subnormal $L$-subgroups of an $L$-group

The study of a normal closure is important in classical group theory. The concept arises due to the fact that certain subgroups of a group are away from being normal. It is the the smallest subgroup containing a given subgroup which is normal in the group. This notion leads to some refined concepts in classical group theory such as normal closure series and subnormality.

6.1 Normal closure series and subnormality

In order to define a suitable notion of normal closure in $L$-setting which can be used to introduce the notion of subnormality, we start with the following (see [30]):

Definition 1.25 Let $\eta \in L\left(\mu \right)$. Define an $L$-subset ${\mathit{\mu \eta \mu }}^{-1}$ of $G$ as follows:

$${\mathit{\mu \eta \mu }}^{-1}\left(x\right)=\underset{\begin{array}{c}x={\mathit{zyz}}^{-1}\\ y,z\in G\end{array}}{\bigvee }\left\{\eta \left(y\right)\wedge \mu \left(z\right)\right\}\text{for\hspace{0.17em}each}x\in G.$$E54

We call the $L$-subset ${\mathit{\mu \eta \mu }}^{-1}$, conjugate of $\eta$ in $\mu$. Clearly, $\eta \subseteq {\mathit{\mu \eta \mu }}^{-1}\subseteq \mu$. Moreover, $\underset{x\in G}{\vee }\left\{{\mathit{\mu \eta \mu }}^{-1}\left(x\right)\right\}=\eta \left(e\right)$ and ${\mathit{\mu \eta \mu }}^{-1}\left(x\right)={\mathit{\mu \eta \mu }}^{-1}\left(x^{-1}\right)$ for each $x\in G$. The normal closure of $\eta$ in $\mu$ is defined as the $L$-subgroup of $\mu$ generated by the conjugate ${\mathit{\mu \eta \mu }}^{-1}$. It is denoted by ${\eta }^{\mu }$. Thus ${\eta }^{\mu }=⟨{\mathit{\mu \eta \mu }}^{-1}⟩$.

The above defined notion satisfies the characteristic properties of a normal closure and retains the usual group theoretic relationship with the concepts of commutator subgroups and set product in $L$-setting. The result are as follows:

Theorem 1.103 Let $\eta \in L\left(\mu \right)$. Then, ${\eta }^{\mu }$ is the least normal L-subgroup of $\mu$ containing $\eta$.

Theorem 1.104 Let $\eta \in L\left(\mu \right)$. Then, $\left[\mu ,\eta \right]\circ \eta \in L\left(\mu \right)$.

Theorem 1.105 Let $\eta ,\theta \in L\left(\mu \right)$. Then,

1. $\left[\eta ,\theta \right]\subseteq {\eta }^{\mu }\cap {\theta }^{\mu },$

2. ${\eta }^{\mu }=\eta \circ \left[\eta ,\mu \right].$

Before we embark on the study of normal closure series and subnormality in $L$-setting, we generalize notion of conjugacy and provide the definition of a conjugate of an $L$-subset by an $L$-subset (see [10, 33]).

Definition 1.26 Let $\eta ,\theta \in L^{\mu }$. Define an $L$-subset ${\mathit{\theta \eta \theta }}^{-1}$ of $G$ as follows:

$${\mathit{\theta \eta \theta }}^{-1}\left(x\right)=\underset{x={\mathit{zyz}}^{-1}}{\bigvee }\left\{\eta \left(y\right)\wedge \theta \left(z\right)\right\}\text{for\hspace{0.17em}each}x\in G.$$E55

We call the $L$-subset ${\mathit{\theta \eta \theta }}^{-1}$ the conjugate of $\eta$ by $\theta$. Clearly, ${\mathit{\theta \eta \theta }}^{-1}\subseteq \mu .$ Hence the $L$-subgroup $⟨{\mathit{\theta \eta \theta }}^{-1}⟩\in L\left(\mu \right)$ and is denoted by ${\eta }^{\theta }$.

Following theorem is instrumental in the development of this subsection:

Theorem 1.106 Let $\eta ,\theta \in L\left(\mu \right)$. Then,

1. $\eta \subseteq {\mathit{\theta \eta \theta }}^{-1}$ provided $\eta \left(e\right)⩽\theta \left(e\right),$

2. $\underset{x\in G}{\bigvee }\left\{{\mathit{\theta \eta \theta }}^{-1}\left(x\right)\right\}={\mathit{\theta \eta \theta }}^{-1}\left(e\right)=\eta \left(e\right)\wedge \theta \left(e\right),$

3. ${\mathit{\theta \eta \theta }}^{-1}\left(x\right)={\mathit{\theta \eta \theta }}^{-1}\left(x^{-1}\right)\text{for\hspace{0.17em}each}x\in G,$

4. ${\mathit{\theta \eta \theta }}^{-1}\left({\mathit{gxg}}^{-1}\right)\ge {\mathit{\theta \eta \theta }}^{-1}\left(x\right)\wedge \theta \left(g\right)\text{for\hspace{0.17em}each}x,g\in G.$

Firstly, we discuss here some properties of conjugate of $L$-subsets where the $L$-subsets in question are $L$-subgroups. The significance of such properties have already been shown in classical group theory for establishing certain properties of $i$th normal closure of a subgroup of a group.

Theorem 1.107 Let $\theta \subseteq \gamma$ and $\eta \left(e\right)=\theta \left(e\right)$. Then,

Let $\eta ,\theta ,\gamma \in L\left(\mu \right)$.

$${\left({\eta }^{\gamma }\right)}^{\theta }={\eta }^{\gamma },$$E56

$${\left({\eta }^{\theta }\right)}^{\gamma }={\eta }^{\gamma },$$E57

$${\eta }^{\theta \circ \eta }={\eta }^{\theta }.$$E58

Theorem 1.08 Let $\eta ,\theta ,\gamma \in L\left(\mu \right)$ be such that $\gamma \left(e\right)=\eta \left(e\right)=\theta \left(e\right).$ Then, ${\left({\eta }^{\theta }\right)}^{\gamma }={\eta }^{\gamma \circ \theta }$.

Theorem 1.09 Let $\eta ,\theta \in L\left(\mu \right)$ and ${\eta }^{\theta }=\eta .$ Then, $\eta \circ \theta \in L\left(\mu \right)$.

In order to introduce the notion of subnormality of an $L$-subgroup of an $L$-group, we define a descending series. For $\eta \in L\left(\mu \right)$, define a series of $L$-subgroups of $\mu$ inductively as follows:

$${\eta }_0=\mu ,{\eta }_1={\eta }^{\mu },{\eta }_2={\eta }^{{\eta }_1},\dots ,{\eta }_i={\eta }^{{\eta }_{i-1}}\dots$$E59

By Theorem 1.103, ${\eta }_1$ is the smallest normal $L$-subgroup of $\mu$ containing $\eta$ and ${\eta }_2$ is the smallest normal $L$-subgroup of ${\eta }_1$ containing $\eta$ and so on.Thus, we have

$$\eta \subseteq \cdots ⊲{\eta }_{i+1}⊲{\eta }_i⊲\cdots ⊲{\eta }_1⊲{\eta }_0=\mu .$$E60

This inductively defined series is known as the normal closure series of $\eta$ in $\mu$ and we call ${\eta }_i$ the ith normal closure of $\eta$ in $\mu$. It is easy to verify that ${\eta }_i\left(e\right)=\eta \left(e\right)$ for each $i$.

Theorem 1.110 Let $\eta ,\theta \in L\left(\mu \right)$ such that $\eta \left(e\right)=\theta \left(e\right).$ Let ${\eta }_i$ be the ith normal closure of $\eta$ in $\mu$ and ${\eta }_i^{\theta }={\eta }_i.$ Then, ${\eta }_{i+1}\in \mathit{NL}\left({\eta }_i\circ \theta \right).$

Now, define the notion of a subnormal $L$-subgroup of an $L$-group as follows:

Definition 1.27 Let $\eta \in L\left(\mu \right)$ and ${\eta }_i$ be the $\mathit{ith}$ normal closure of $\eta$ in $\mu$. If there exists a non negative integer $m$ such that

$${\eta }_m=\eta ⊲{\eta }_{m-1}⊲\cdots ⊲{\eta }_0=\mu ,$$E61

then $\eta$ is known as a subnormal $L$-subgroup of $\mu$ with defect $m$. We shall denote a subnormal $L$-subgroup $\eta$ of $\mu$ with defect $m$ by $\eta \stackrel{m}{⊲}\mu$. If $\eta$ is a subnormal $L$-subgroup of $\mu ,$ then we shall write $\eta$ is subnormal in $\mu$.

Remark: Obviously $m$ equals 0 if $\eta =\mu$ and $m=1$ if $\eta \in \mathit{NL}\left(\mu \right)$ and $\eta \ne \mu$.

The following theorem shows that the normal closure series is the fastest descending normal series [33]:

Theorem 1.111 Let $\eta \in L\left(\mu \right)$ and

$$\eta \subseteq \cdots ⊲{\eta }_{i+1}⊲{\eta }_i⊲\cdots ⊲{\eta }_1⊲{\eta }_0=\mu$$E62

be the normal closure series of $\eta$. If there exists a descending series ${\gamma }_0=\mu ,{\gamma }_1,\dots ,{\gamma }_i\dots$ of $L$-subgroups of $\mu$ such that

$$\eta \subseteq \dots ⊲{\gamma }_{i+1}⊲\dots {\gamma }_1⊲{\gamma }_0=\mu ,$$E63

then ${\eta }_i\subseteq {\gamma }_i.$

Following is the definition of a subnormal series of an $L$-group:

Definition 1.28 Let $\eta \in L\left(\mu \right)$. A finite series ${\theta }_0=\mu ,{\theta }_1,{\theta }_2,\dots ,{\theta }_m=\eta$ of $L$-subgroups of $\mu$ such that

$$\eta ={\theta }_m⊲{\theta }_{m-1}⊲\dots ⊲{\theta }_0=\mu$$E64

is said to be a subnormal series of $\eta .$.

We shall describe the notion of a subnormal $L$-subgroup through the notion of above defined subnormal series. The following result inter-connects these two concepts:

Theorem 1.112 Let $\eta \in L\left(\mu \right).$ Then, $\eta$ is a subnormal $L$-subgroup of $\mu$ having defect $m$ if and only if $\eta$ has a subnormal series

$$\eta ={\gamma }_m⊲\dots ⊲{\gamma }_{i+1}⊲\dots {\gamma }_1⊲{\gamma }_0=\mu ,$$E65

of length $m$ and $m$ is the smallest length of such a subnormal series.

The results given below are established with the help of the above theorem:

Theorem 1.113 Let $\eta$ be a subnormal $L$-subgroup of $\mu$ with defect $m$.

1. Let $\theta \in L\left(\mu \right).$ Then, $\eta \cap \theta$ is a subnormal $L$-subgroup of $\theta$ with defect $c$ where $c\le m.$ In particular, $\eta$ is a subnormal $L$-subgroup of $\lambda$ where $\lambda \in L\left(\mu \right)$ such that $\eta \subseteq \lambda$ with defect $c$ where $c\le m,$

2. Let $\theta \in \mathit{NL}\left(\mu \right).$ Then, $\eta \circ \theta$ is a subnormal $L$-subgroup of $\mu$ with defect $c$ where $c\le m.$

It can be seen easily that the intersection of any finite set of subnormal $L$-subgroups is again subnormal. More generally:

Theorem 1.114 Let $\left\{{\theta }_i:i\in I\right\}$ be a family of subnormal $L$-subgroups such that defect of ${\theta }_i$ is $m_i$ where $m_i\le m.$ Then, $\underset{i\in I}{\cap }$${\theta }_i$ is a subnormal $L$-subgroup of $\mu$ with defect $c$ where $c\le m$.

Our next result determines the transitivity of the notion of subnormality.

Theorem 1.115 Let $\eta ,\theta \in L\left(\mu \right)$ such that $\eta$ is a subnormal $L$-subgroup of $\theta$ with defect $m$ and $\theta$ is a subnormal $L$-subgroup of $\mu$ with defect $n$. Then, $\eta$ is a subnormal $L$-subgroup of $\mu$ with defect $m+n$.

The following theorem establishes that the subnormality in $L$-setting is also preserved under the action of a homomorphism and its inverse image:

Theorem 1.116 Let $\eta \in L\left(\mu \right)$ and $f:G\to K$ be a group homomorphism. Then,

1. if $\eta$ is a subnormal $L$-subgroup of $\mu$ with defect $n$, then $f\left(\eta \right)$ is a subnormal $L$-subgroup of $f\left(\mu \right)$ with defect $m$ where $m\le n$,

2. if $\eta$ is a subnormal $L$-subgroup of $\mu$ with defect $n$, then $f^{-1}\left(\eta \right)$ is a subnormal $L$-subgroup of $f^{-1}\left(\mu \right)$ with defect $m$ where $m\le n$, provided that the group homomoirphism $f$ is onto.

6.2 Subnormal $L$-subgroups and nilpotency

In this subsection, we characterize subnormal $L$-subgroups by the usual group theoretic subnormality of the level subsets of the given $L$-subgroups. We shall refer this as a level subset characterization of subnormality. Then, this characterization is used to establish that when the lattice $L$ is an upper well ordered chain, then every $L$-subgroup of a nilpotent $L$-group is subnormal. For this purpose, we need to develop a necessary mechanism (see [10, 33]). Here, we present:

Theorem 1.117 Let $\eta \in L\left(\mu \right)$ be such that $\mu \text{and}\eta$ are jointly supstar. Then,

$$\mathit{Im}{\eta }^{\mu }\subseteq \mathit{Im}\left({\mathit{\mu \eta \mu }}^{-1}\right)\subseteq \mathit{Im\mu }\cup \mathit{Im\eta }.$$E66

More generally we have:

Theorem 1.118 Let $\eta \in L\left(\mu \right)$ be such that $\mu \text{and}\eta$ are jointly supstar. Then, for each i

$$\mathit{Im}{\eta }_{i+1}\subseteq \mathit{Im}\left({\eta }_i{\mathit{\eta \eta }}_i^{-1}\right)\subseteq \mathit{Im\mu }\cup \mathit{Im\eta },$$E67

where ${\eta }_i$ is the $\mathit{ith}$ normal closure of $\eta$ in $\mu$.

Theorem 1.119 Let $\eta \in L\left(\mu \right)$ be such that $\mu \text{and}\eta$ are jointly supstar. Then for each $i$,

1. $\eta$ and ${\eta }_i$ are jointly supstar.

2. if $\eta \in L\left(\mu \right)$ and $\mu \text{and}\eta$ be jointly supstar, then the $L$-subset ${\eta }_i{\mathit{\eta \eta }}_i^{-1}$ possesses sup-property

Below, we discuss the level subsets and strong level subsets of the normal closure of $\eta$ in $\mu$.

Theorem 1.120 Let $\eta \in L\left(\mu \right).$ Then,

1. ${\left({\eta }^{\mu }\right)}_a={\left({\eta }_a\right)}^{{\mu }_a}\text{for\hspace{0.17em}each}a\le \eta \left(e\right),\text{provided}\mu \text{and}\eta \mathrm{are}\text{supstar},$

2. ${\left({\eta }^{\mu }\right)}_a^{>}={\left({\eta }_a^{>}\right)}^{{\mu }_a^{>}}\text{for\hspace{0.17em}each}a<\eta \left(e\right),\text{provided}$L is a chain.

Theorem 1.121 Let $\eta \in L\left(\mu \right)$ be such that $\eta$ and $\mu$ are jointly supstar. Then, $\eta$ is subnormal having defect at most $n$ if and only if each level subset ${\eta }_a$ is subnormal having defect atmost $n$ where $a\le \eta \left(e\right)$.

Theorem 1.112 Let $G$ be a group and $H$ be its subgroup. Then, $H$ is a subnormal subgroup of $G$ if and only if $1_H$ is a subnormal $L$-subgroup of $1_G$.

The strong level subset characterization of subnormal $L$-subgroup can be obtained easily.

Theorem 1.123 Let $L$ be chain and $\eta \in L\left(\mu \right)$. Then, $\eta$ is subnormal having defect at most $n$ if and only if each strong level subset ${\eta }_a^{>}$ is subnormal having defect atmost $n$ where $a<\eta \left(e\right)$.

Theorem 1.124 Let $L$ be an upper well ordered chain. Let $\eta \in L\left(\mu \right)$ and $\eta$ be nilpotent having tip $a_0$ and tail $t_0$ and $a_0\ne t_0$. If $\theta \in L\left(\eta \right)$ and has the tail $t_0$, then $\theta$ is a subnormal $L$-subgroup of $\eta$.

In a forthcoming paper [10], we develop a mechanism in order to tackle the join problem for subnormal $L$-subgroup and we prove that:

Theorem 1.125 Let $\eta$ and $\theta$ be subnormal $L$-subgroups of $\mu$. Let $\eta \left(e\right)=\theta \left(e\right)$ and $\eta \circ \theta \in L\left(\mu \right)$. Then, the following are equivalent:

1. $\eta \circ \theta$ is subnormal in $\mu$,

2. ${\eta }^{\theta }$ is subnormal in $\mu$,

3. $\left[\eta ,\theta \right]$ is subnormal in $\mu$.

7. Discussion and analysis

The subject matter discussed in this work presents a systematic and compatible theory of $L$-subgroups(fuzzy subgroups). In this work, firstly we replace ordinary fuzzy subsets by lattice valued fuzzy subsets. This puts our work beyond the purview of Tom Head’s metathoerem [35]. This is contrary to the situation of fuzzy group theory, where the results which are obtained, become simple instances of application of metatheorem. Secondly, throughout our work, we have replaced the parent structure of an ordinary group by an $L$-subgroup which is called an $L$-group. In order to carry out our studies successfully, we need to use the notion of normality of a fuzzy subgroup in a fuzzy group defined by Wu [22] rather than the notion of normality of a fuzzy subgroup in a group introduced by Liu [21]. Without following this approach, we can not construct various types of series of $L$-subgroups, such as the subinvariant series, normal closure series, subnormal series or derived series: even the results of classical group theory such as a characteristic subgroup of a normal subgroup of a group is a normal subgroup of the given group or the property of transitivity of characteristic subgroup can not be formulated and extended to the $L$-setting. By following Wu’s normality, we have successfully extended the above results to the $L$-setting. Moreover, we have proved that the commutator $L$-subgroup of an $L$-group is a characteristic $L$-subgroup. Furthermore, we proved that a commutator $L$-subgroup of a supersolvable $L$-subgroup is nilpotent. Thus an application of Wu’s normality establishes a very high degree of compatibility among various notions studied in $L$-group theory.

Another deviation in our work from the work of earlier researchers in fuzzy group theory is the construction of the trivial $L$-fuzzy subgroup. This is a proper fuzzification of the notion of trivial subgroup of an ordinary group and it makes possible a successful study of various types of series, arising while dealing with the notions of nilpotency, solvability, supersolvability, subnormality etc.. While defining a trivial $L$-subgroup, the notion of infimum of an $L$-set comes into play. So is the case at various other places in our investigations where the infimum of an $L$-subgroup plays a significant role. This is due to the fact that we are carrying out our investigations in an $L$-group rather than an ordinary group and we use Wu’s normality in place of Liu’s normality. There are several peculiarities of $L$-setting which involves the notion of infimum of $L$-subgroups. For example Theorem 1.60 states that the infimum of the set product of two $L$-subsets lies in between the meet and the join of the infimums of given $L$-subsets. This result has been used in Theorem 1.81 wherein we have obtained a necessary and sufficient condition for the set product of two trivial $L$-subgroups of an $L$-group to be a trivial $L$-subgroup. This result is instrumental in establishing a sufficient condition for the set product of two nilpotent $L$-subgroups to be nilpotent. Moreover, the role of infimum (tail) of $L$-subgroups is displayed while dealing with homomorphic and inverse homomorphic images of commutator and solvable $L$-subgroups. To show that the the concepts of nilpotency, solvability and supersolvability are closed under the formation of subgroups, the notion of infimum is again helpful. The infimums (tails) of all the members of almost all the series discussed in our work are identical with the tails of their respective trivial $L$-subgroups.

Last but not the least, another pleasing feature of our study is the formation of lattices of normal $L$-subgroups and characteristic $L$-subgroups of an $L$-group. In the process, the notion of sup-property has played a very significant role. We obtain a characterization of sup-property by using the notion of image set of the given $L$-subset. This characterization of sup-property forms the basis for our generalization. This gave rise to the notions of supstar family of $L$-subsets and jointly supstar $L$-subsets. The notion of image of an $L$-subset is intimately related with the above mentioned concepts. Theorem 1.36 shows that each member of supstar family possesses sup-property. Further, in Theorem 1.70, a relationship has been established between the image of ith members of descending central series of an $L$-subgroup and the image of the given $L$-subgroup. A similar relationship has been obtained in Theorem 1.84 and Theorem 1.120.

These concepts are subsequently used in the development of $L$-group theory. A co-ordinated approach between all these concepts paved a way for a successful development of $L$-group theory.

8. Conclusion

It has been said earlier that $L$-group theory developed in [10, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] is a very rich generalization of classical group theory. For if we replace the lattice $L$ by the closed unit interval $L$, then we retrieve fuzzy group theory. Moreover, if we replace $L$ by the two elements set {0,1}, we obtain results of classical group theory as simple corollaries of corresponding results of $L$-group theory.

In view of this development we suggest the researchers, working in the areas of other fuzzy algebraic structures to shift their studies to lattice valued fuzzy sets ($L$-subsets). Also, in order to obtain consistency, the parent structure of classical algebra should be replaced by the corresponding fuzzy algebraic structures.

For those who are involved in active research in these areas, we propose here few research problems: The formation of quotient structure in fuzzy algebraic structures has been a problem child since its very inception. Its proper formulation is still awaited. We invite the researchers to construct a quotient of $L$-group $\mu$ by a normal $L$-subgroup in $\mu$ in the sense of Wu [22]. The second problem which is likely to be tackled more easily is related to nilpotent $L$-subgroups of an $L$-group which is discussed in the present work, that is, the investigation of nilpotency by upper central series. The upper central series is not yet formulated in the theory of $L$-subgroups.

Finally, to mention the further richness of this generalization, we emphasize that here we study the group theoretic properties of posets of subgroups of a group or in particular chains of subgroups of a group rather than properties of a single subgroup. This way, $L$-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. Therefore the approach adopted in the studies of L-group theory can be looked upon as a modernization of the approach of classical group theory.

Dedication

I dedicate this work to Prof. Naseem Ajmal, one of the greatest fuzzy algebraists and my Ph. D. supervisor.