**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 *η*∈*L*ð Þ *μ* . Then for *ax* ∈*μ*, left (right) coset of *η* in *μ* is defined as the set product *ax* ∘ *η η*ð Þ ∘ *ax* .

The following is immediate:

Let *η*∈*L*ð Þ *μ* . Then, for each *ax* ∈*μ* and *z*∈ *G*

$$a \,\, a \,\, \eta(\mathbf{z}) = a \wedge \eta(\mathbf{x}^{-1} \mathbf{z}), \quad \eta \circ a\_{\mathbf{x}}(\mathbf{z}) = a \wedge \eta(\mathbf{z} \mathbf{x}^{-1});\tag{13}$$

i.e. *ax* ∘ *η* ¼ *η* ∘ *ax* for each *ax* ∈*η:*

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 *η*∈*L<sup>μ</sup>*. Then,

$$
\forall \eta \in \mathcal{L}(\mu) \text{ if and only if } a\_{\mathbf{x}} \circ b\_{\mathbf{y}^{-1}} \in \eta \text{ for each } a\_{\mathbf{x}}, b\_{\mathbf{y}} \in \eta. \tag{14}
$$

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 *η*∈ *L*ð Þ *μ* . Then,

$$
\eta \in \text{NL}(\mu) \text{ if and only if } \ a\_{\mathbf{x}} \circ \eta = \eta \circ a\_{\mathbf{x}} \text{ for each } L \text{-point } a\_{\mathbf{x}} \in \mu. \tag{15}
$$

Clearly for *η*∈*L*ð Þ *μ* , *η*∈ *NL*ð Þ*η* . Also, we observe that for *ax*, *by* ∈ *μ*,

$$a\_{\mathfrak{x}} \circ b\_{\mathfrak{y}} = (a \wedge b)\_{\mathfrak{x}\mathfrak{y}}.\tag{16}$$

Some further characterizations of normal *L*-subgroups are given below: Theorem 1.13 Let *η*∈*L*ð Þ *μ* . Then, the following are equivalent:


iii. *ax* ∘ *η* ∘ *ax*�<sup>1</sup> ⊆ *η* for each *ax* ∈ *μ*,

iv. *ax* ∘ *by* ∘ *ax*�<sup>1</sup> ∈*η* for each *ax* ∈*μ* and *by* ∈*η*.

The following fact is well known: If *<sup>μ</sup>*∈*L<sup>X</sup>*, then *<sup>μ</sup>* <sup>¼</sup> <sup>∪</sup> *<sup>x</sup>*∈*<sup>X</sup>* ð Þ *<sup>μ</sup>*ð Þ *<sup>x</sup> <sup>x</sup>* .

This immediately yields:

Theorem 1.14 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup> *LG*. Then, *<sup>η</sup>* <sup>∘</sup> *<sup>θ</sup>* <sup>¼</sup> <sup>∪</sup> *<sup>x</sup>*<sup>∈</sup> *<sup>G</sup>* ð Þ *<sup>η</sup>*ð Þ *<sup>x</sup> <sup>x</sup>* <sup>∘</sup> *<sup>θ</sup>* <sup>¼</sup> <sup>∪</sup> *<sup>x</sup>*<sup>∈</sup> *<sup>G</sup> <sup>η</sup>* <sup>∘</sup>ð Þ *<sup>θ</sup>*ð Þ *<sup>x</sup> <sup>x</sup> :*

Hence we have that each *L*-subset of *μ* commutes with every normal *L*-subgroup of *μ*. Theorem 1.15 Let *<sup>η</sup>*<sup>∈</sup> *NL*ð Þ *<sup>μ</sup>* . Then, *<sup>η</sup>* <sup>∘</sup> *<sup>θ</sup>* <sup>¼</sup> *<sup>θ</sup>* <sup>∘</sup> *<sup>η</sup>* for each *<sup>θ</sup>* <sup>∈</sup>*L<sup>μ</sup>*.

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 *η*∈ *L*ð Þ *μ* . Define an *L*-subset *δ* of *G* as follows:

$$\delta = \bigcup\_{a\_{\mathfrak{x}} \in \mathfrak{\mu}} \{ a\_{\mathfrak{x}} : a\_{\mathfrak{x}} \bullet \eta = \eta \circ a\_{\mathfrak{x}} \}. \tag{17}$$

Then, *δ* is the largest *L*-subgroup of *μ* such that *η* is a normal *L*-subgroup of *δ*. Here *δ* is called the normalizer of *η* and is denoted by *N*ð Þ*η :* Moreover, it turns out that *N*ð Þ*η* ð Þ¼ *e μ*ð Þ*e* .

This immediately leads us to the following result:

Let *η*∈*L*ð Þ *μ* . Then, *η*∈ *NL*ð Þ *μ* if and only if *N*ð Þ¼ *η μ*.

On the other hand, if we replace the parent *L*-group *μ* by 1*G*, then we have:

Theorem 1.18 Let *η*∈*L G*ð Þ. Then, fð Þ*η* is an normal *L*-subgroup of *G* if and only if *N*ð Þ¼ *η* 1*G*.

As a consequence, we recover the classical result:

Theorem 1.19 Let *H* be a subgroup of *G*. Then, *x*∈ *N H*ð Þ if and only if 1*<sup>H</sup>* ∘ 1*<sup>x</sup>* ¼ 1*<sup>x</sup>* ∘ 1*H*.

Below we provide some more properties related to the normalizer so defined: Theorem 1.20 Let *η*, *θ* ∈*L*ð Þ *μ* . Then

$$\text{i. } N(\eta) \cap N(\theta) \subseteq N(\eta \cap \theta),$$

$$\text{iii.}\ N(\eta)\cap N(\theta)\subseteq N(\eta\circ\theta)\text{ provided }\eta\circ\theta\in L(\mu).$$

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* ! *K* be a group homomorphism and *x*∈ *G*. If *y*∈ *K*, then the set of all preimages of '*yf x*ð Þ' is precisely the set of all elements of the form '*ux*' where *f u*ð Þ¼ *y*.

Theorem 1.22 Let *<sup>f</sup>*ð*v*<sup>Þ</sup> be a group homomorphism and *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*Lμ*. Then, *f*ð Þ¼ *η* ∘ *θ f*ð Þ*η* ∘*f*ð Þ*θ* .

Moreover,

Theorem 1.23 Let *<sup>f</sup>* : *<sup>G</sup>* ! *<sup>K</sup>* be a group homomorphism and *<sup>ν</sup>*∈*L K*ð Þ.If *<sup>θ</sup>* <sup>∈</sup> *<sup>L</sup><sup>ν</sup>* , then *f* �<sup>1</sup> *<sup>θ</sup>* <sup>∘</sup> *bf x*ð Þ <sup>¼</sup> *<sup>f</sup>* �1 ð Þ*θ* ∘ *bx*.

The above results are helpful in establishing the following:

Theorem 1.24 Let *f* : *G* ! *K* be a group homomorphism. Then, for *μ*∈ *L G*ð Þ and *ν*∈*L K*ð Þ

i. *f N*ð Þ ð Þ*η* ⊆ *N f*ð Þ ð Þ*η* for each *η*∈*L*ð Þ *μ* ,

$$\text{iii}.f^{-1}(N(\theta)) \subseteq N(f^{-1}(\theta))) \text{ for each } \theta \in L(\nu).$$

### **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 *<sup>η</sup>*<sup>∈</sup> *<sup>L</sup><sup>μ</sup>*. Let *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>∨</sup> *x*∈ *G* f g *η*ð Þ *x* and define an *L*-subset ^*η* of *G* by

$$\hat{\eta}(\mathfrak{x}) = \underset{a \le a\_0}{\text{y}} \{ \mathfrak{a} : \mathfrak{x} \in \langle \eta\_a \rangle \}. \tag{18}$$

Then, ^*η*∈*L*ð Þ *μ* and ^*η* ¼ h i*η* . Moreover, h i*η* ð Þ¼ *e* ∨*<sup>x</sup>*<sup>∈</sup> *<sup>G</sup>*f g *η*ð Þ *x* .

The above theorem is used to establish the following:

Theorem 1.26 Let *<sup>η</sup>*<sup>∈</sup> *NL<sup>μ</sup>*. Then, h i*<sup>η</sup>* <sup>∈</sup> *NL*ð Þ *<sup>μ</sup>* .

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

Theorem 1.27 Let *η*∈ *L<sup>μ</sup> <sup>s</sup>* . Then, define an *L*-subset ^*η* of *G* by

$$\hat{\eta}(\mathfrak{x}) = \underset{a \in \operatorname{Im} \eta}{\operatorname{\mathbf{y}}} \{ \mathfrak{a} : \mathfrak{x} \in \langle \eta\_a \rangle \}. \tag{19}$$

Then, ^*η*∈*L*ð Þ *μ* and ^*η* ¼ h i*η* . Moreover, ^*η* possesses sup-property and Im ^*η*⊆Im *η:* The following result is an immediate consequence of the above theorem: Theorem 1.28 Let *η*∈ *L<sup>μ</sup> <sup>s</sup>* . If *a*<sup>0</sup> ¼ ∨ *x*∈ *G* f g *η*ð Þ *x* , then for each *b*≤*a*0, *η<sup>b</sup>* h i ¼ h i*η <sup>b</sup>*.

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 *<sup>Z</sup>* be the group of integers under addition, and let 2*<sup>n</sup>* h i be the subgroup of *Z* generated by 2*<sup>n</sup>*, where *n* is a fixed positive integer. Then the direct product *Z* � *Z* contains subgroups

$$\langle \langle \mathcal{2}' \rangle \times \langle \mathcal{2}' \rangle \text{ for each } r, s = 0, 1, 2, \dots \tag{20}$$

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

$$\mu(\mathbf{x}) = \begin{cases} 0 & \text{if } \mathbf{x} \in \mathbb{Z} \times \mathbb{Z} \sim \langle 2 \rangle \times \mathbb{Z}, \\ \frac{3}{4} & \text{if } \mathbf{x} \in \langle 2 \rangle \times \mathbb{Z}. \end{cases} \tag{21}$$

$$\begin{cases} 0 & \text{if } \mathbf{x} \in \mathbf{x} \in \mathbb{Z} \times \mathbb{Z} \sim \langle 2 \rangle \times \mathbb{Z}, \\ \frac{1}{2} \left( 1 - \frac{1}{2^n} \right) & \text{if } \mathbf{x} \in \langle 2^n \rangle \times \mathbb{Z} \sim \langle 2^{n+1} \rangle \times \mathbb{Z}, \text{ where } n = 1, 2, 3, \dots \\ 0 & \text{if } \mathbf{x} \in \langle 0 \rangle \times \mathbb{Z} \sim \langle 0 \rangle \times \langle 2 \rangle, \\ \frac{3}{4} \left( 1 - \frac{1}{4^n} \right) & \text{if } \mathbf{x} \in \langle 0 \rangle \times \langle 2^n \rangle \sim \langle 0 \rangle \times \langle 2^{n+1} \rangle, \text{ where } n = 1, 2, 3, \dots \\ \frac{3}{4} & \text{if } \mathbf{x} = (0, 0). \end{cases} \tag{22}$$

Here *A* � *B* means usual set difference.Clearly, *η*⊆*μ*, *η* 6¼ *μ* and *μ*∈*L G*ð Þ. Observe that *<sup>η</sup>* does not possess sup-property and for *<sup>t</sup>* <sup>¼</sup> <sup>1</sup> 2 ,

$$
\langle \mathbf{0} \rangle \times \langle \mathbf{2} \rangle = \left\langle \eta\_{\frac{1}{2}} \right\rangle \subset \langle \eta \rangle\_{\frac{1}{2}} = \langle \mathbf{0} \rangle \times Z. \tag{23}
$$

Moreover,

Theorem 1.29 Let *<sup>η</sup>*<sup>∈</sup> *<sup>L</sup><sup>μ</sup>* and *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>∨</sup> *x*∈ *G <sup>η</sup>*ð Þ *<sup>x</sup>* . If *<sup>L</sup>* is a chain, then *<sup>η</sup>*<sup>&</sup>gt; *a* � � <sup>¼</sup> h i*<sup>η</sup>* <sup>&</sup>gt; *<sup>a</sup>* 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 *η*∈*L<sup>μ</sup>* with tip *a*0. Then, *η* is said to be a characteristic *L*-subset of *μ* if

$$
\eta(T\mathbf{x}) \ge \eta(\mathbf{x}) \text{ for each } T \in A(\mu\_a) \text{ and for each } a \le a\_0; \tag{24}
$$

where *A μ<sup>a</sup>* ð Þ is the group of automorphisms of *μa*. We denote the set of all characteristic *L*-subsets of *μ* by *CL<sup>μ</sup>* .

It is easy to see that *μ* is a characteristic *L*-subset of itself.

Theorem 1.30 Let *η*∈*L<sup>μ</sup>* with tip *a*0. Then, *η*∈*CL<sup>μ</sup>* if and only if *η<sup>a</sup>* is a characteristic subset of *μ<sup>a</sup>* for each *a*≤*a*0.

The set of all normal *L*-subsets of the *L*-group *μ* is denoted by *NL<sup>μ</sup>*. The following results are immediate:

Theorem 1.31 Let *η*∈*CL<sup>μ</sup>* . Then, *η*∈ *NL<sup>μ</sup>*. Theorem 1.32 Let *η*∈*CL<sup>μ</sup>* with tip *a*0. Then,

$$T(\eta|\mu\_a) = \eta|\mu\_a \quad \text{for all } T \in A(\mu\_a); \quad \text{where} \quad a \le a\_0. \tag{25}$$

The set of all *L*-subsets of *μ* possessing sup-property is denoted by *L<sup>μ</sup> <sup>s</sup>* . The following result has been discussed in the fuzzy setting in [27]:

Theorem 1.33 Let *η*, *θ* ∈*L<sup>μ</sup> <sup>s</sup>* . Then, *η* ∪ *θ* and *η* ∩ *θ* ∈*L<sup>μ</sup> <sup>s</sup>* provided *L* is a chain.

Now, since the meet and the join operations in *L<sup>μ</sup>* are defined to be the intersection and the union of *L*-subsets respectively, the set *L<sup>μ</sup> <sup>s</sup>* constitutes a sublattice of *L<sup>μ</sup>* 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 *η*∈ *L<sup>μ</sup>*. Then, *η* possesses sup-property if and only if each nonempty subset of *Imη* 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 sup*A* ¼ *a*0, then *a*<sup>0</sup> ∈ *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 f g *<sup>η</sup><sup>i</sup> <sup>i</sup>* <sup>∈</sup>*<sup>I</sup>* <sup>⊆</sup> *<sup>L</sup><sup>μ</sup>* be an arbitrary family. Then, f g *<sup>η</sup><sup>i</sup> <sup>i</sup>* <sup>∈</sup>*<sup>I</sup>* is said to be a supstar family if ∪ *i*∈*I Imη<sup>i</sup>* is a supstar subset of *L*. In particular, a pair of *L*-subsets *η* and

*θ* is said to be jointly supstar if the set f g *η*, *θ* is a supstar family.

*Development of* L*-Group Theory DOI: http://dx.doi.org/10.5772/intechopen.110387*

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

Theorem 1.35 Let *η*∈*L<sup>μ</sup>*. Then, *η* has sup-property if and only if *Imη* is a supstar subset of *L*.

Theorem 1.36 Let f g *<sup>η</sup><sup>i</sup> <sup>i</sup>*∈*<sup>I</sup>* <sup>⊆</sup>*L<sup>μ</sup>* be a supstar family. Then,

i. *η<sup>i</sup>* possesses sup-property for each *i*∈ *I*,

ii. ⋃ *i*∈ Ω *η<sup>i</sup>* possess sup-property, where Ω ⊆*I*.

Moreover, we have:

*i*∈*I*

Theorem 1.37 If f g *<sup>η</sup><sup>i</sup> <sup>i</sup>* <sup>∈</sup>*<sup>I</sup>* <sup>⊆</sup> *<sup>L</sup><sup>μ</sup>* is a maximal supstar family, then f g *<sup>η</sup><sup>i</sup> <sup>i</sup>* <sup>∈</sup>*<sup>I</sup>* 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 f g *<sup>η</sup><sup>i</sup> <sup>i</sup>* <sup>∈</sup>*<sup>I</sup>* <sup>⊆</sup>*CL<sup>μ</sup>*. Then,

$$\begin{aligned} \text{i. } \bigcap\_{i \in I} \eta\_i \in CL^{\mu}, \\\\ \text{ii. } \bigcup\_{i} \eta\_i \in CL^{\mu} \text{ provided } \{\eta\_i\}\_{i \in I} \text{ is a support family.} \end{aligned}$$

In view of Theorem 1.31, *CL<sup>μ</sup>* ⊆ *NL<sup>μ</sup>* ⊆ *L<sup>μ</sup>*. By Theorem 1.6, *NL<sup>μ</sup>* is closed under arbitrary unions and intersections. Hence *NL<sup>μ</sup>* is a complete sublattice of *L<sup>μ</sup>*. Further by Theorem 1.38, *CL<sup>μ</sup>* is closed under arbitrary intersections with the greatest element *μ*. Thus *CL<sup>μ</sup>* is a lower complete sublattice of *L<sup>μ</sup>* and is a complete lattice in its own right. Moreover, if both *CL<sup>μ</sup>* and *NL<sup>μ</sup>* are supstar families and the lattice *L* is a chain, then by Theorem 1.31 and Theorem 1.36, *CL<sup>μ</sup>* ⊆ *NL<sup>μ</sup>* ⊆ *L<sup>μ</sup> <sup>s</sup>* . By Theorem 1.38, *CL<sup>μ</sup>* is closed under arbitrary unions with the least element identically zero function. So *CL<sup>μ</sup>* is an upper complete sublattices of *NL<sup>μ</sup>*. Similarly by Theorem 1.6 and Theorem 1.36, *NL<sup>μ</sup>* is an upper complete sublattices of *L<sup>μ</sup> s* .

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 *η*, *ν*∈*L<sup>μ</sup>*. Then,

i. ð Þ *<sup>η</sup>* <sup>∘</sup> *<sup>ν</sup>* <sup>&</sup>gt; *<sup>a</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>&</sup>gt; *<sup>a</sup> ν*<sup>&</sup>gt; *<sup>a</sup>* for each *a*∈*L* � f g1 , provided *L* is a chain,

ii. ð Þ *η* ∘ *ν <sup>a</sup>* ¼ *ηaν<sup>a</sup>* for each *a*∈ *L*, provided *η* and *ν* are jointly supstar.

This helps us in establishing the following:

Theorem 1.40 Let *η*, *θ* ∈*CL<sup>μ</sup>* be jointly supstar. Then, *η* ∘ *θ* ∈*CL<sup>μ</sup>* .

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

Definition 1.12 Let *η*∈*L*ð Þ *μ* . Then, *η* is said to be a characteristic *L*-subgroup of *μ* if *η*∈*CL<sup>μ</sup>* . We denote by *CL*ð Þ *μ* the set of all characteristic *L*-subgroups of *μ*.

Clearly, an *L*-group *μ* 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 *η*∈*L*ð Þ *μ* . Then, *η*∈*CL*ð Þ *μ* if and only if *η<sup>a</sup>* is a characteristic subgroup of *μ<sup>a</sup>* foreach *a*≤*η*ð Þ*e* .

Moreover, we have the following results: Theorem 1.42 Let *η*∈*CL*ð Þ *μ* . Then, *η*∈ *NL*ð Þ *μ* . Theorem 1.43 Let *η*∈*CL*ð Þ *μ* . Then,

$$|T(\eta|\mu\_a)| = \eta|\mu\_a \text{ for all } T \in A(\mu\_a); \quad \text{where} \quad a \le \eta(e). \tag{26}$$

Theorem 1.44 Let *η*, *θ* ∈*CL*ð Þ *μ* be jointly supstar. Then, *η* ∘ *θ* ∈*CL*ð Þ *μ* . Theorem 1.45 Let *θ* ∈ *NL*ð Þ *μ* and *η*∈*CL*ð Þ*θ* . Then, *η*∈ *NL*ð Þ *μ* .

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 *θ* ∈*CL*ð Þ *μ* and *η*∈*CL*ð Þ*θ* . Then, *η*∈*CL*ð Þ *μ* .

Let us denote by *CLs*ð Þ *μ* the set of all characteristic *L*-subgroups of *μ*, 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 *<sup>η</sup>*∈*CL<sup>μ</sup>* and possesses sup-property. Then, h i*<sup>η</sup>* <sup>∈</sup>*CLs*ð Þ *<sup>μ</sup>* .

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 *NL*ð Þ *μ* is a complete sublattice of *L*ð Þ *μ :*.

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

Theorem 1.49 The set *Ls*ð Þ *μ* of all *L*-subgroups of *μ* each member of which possesses sup-property, is a sublattice of *L*ð Þ *μ* .

Theorem 1.50 The set *NLs*ð Þ *μ* of all normal *L*-subgroup of *μ* each member of which possesses sup-property, is a sublattice of *Ls*ð Þ *μ* and hence of *L*ð Þ *μ* .

Theorem 1.51 The set *CLs*ð Þ *μ* of all characteristic *L*-subgroups of *μ* each member of which possesses sup-property, is a sublattice of *NLs*ð Þ *μ* and hence of *L*ð Þ *μ* .

The following diagram provides the lattice structure of the sublattices of the lattice *L G*ð Þ provided the lattice *L* is a chain (**Figure 1**):

If *Lt*ð Þ *μ* denotes the set of all *L*-subgroups of *μ* each member of which has the same tip *t*, where *t* ∈*Im μ*, then the following result is easy to verify:

Theorem 1.52 The set *Lt*ð Þ *μ* is a sublattice of *L*ð Þ *μ* .

Moreover, we have:

Theorem 1.53 The lattice *L*ð Þ *μ* is a disjoint union of its sublattices *Lt*ð Þ *μ* . That is *L*ð Þ¼ *μ* ⋃ *Lt*ð Þ *μ* .

*t*∈*Im μ*

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

Theorem 1.54 The set *Lnt*ð Þ *μ* of all normal *L*-subgroups of *μ* with the same tip *t*, is a sublattice of *Ln*ð Þ *μ* and hence of *L*ð Þ *μ* .

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

Theorem 1.55 The set *Lst*ð Þ *μ* of all *L*-subgroups of *μ* with the same tip *t* and each member of which possesses sup-property, is a sublattice of *Ls*ð Þ *μ* and hence of *L*ð Þ *μ* .

Theorem 1.56 The set *Lnst*ð Þ *μ* of all normal *L*-subgroups of *μ* with the same tip *t* and each member of which possesses sup-property, is a sublattice of *Lnt*ð Þ *μ* and *Lst*ð Þ *μ* and hence of *L*ð Þ *μ* .

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

Theorem 1.57 The set *Lcst*ð Þ *μ* of all characteristic *L*-subgroups of *μ* with the same tip *t* and each member of which possesses sup-property, is a sublattice of *NLst*ð Þ *μ* and hence of *L*ð Þ *μ* provided the lattice *L* is a chain.

Now, let us consider an arbitrary lattice *<sup>L</sup>*. Let *<sup>L</sup>*<sup>∗</sup> ð Þ *<sup>μ</sup>* denote a subclass of the lattice *<sup>L</sup>*ð Þ *<sup>μ</sup>* consisting of all the supstar families. Then, *<sup>L</sup>*<sup>∗</sup> ð Þ *<sup>μ</sup>* is a complete sublattice of *<sup>L</sup>*ð Þ *<sup>μ</sup>* . Now if *CL*<sup>∗</sup> *st* , *CL*<sup>∗</sup> *<sup>s</sup>* ð Þ *<sup>μ</sup>* , *NL*<sup>∗</sup> *<sup>s</sup>* ð Þ *<sup>μ</sup>* , *<sup>L</sup>*<sup>∗</sup> *<sup>s</sup>* ð Þ *<sup>μ</sup>* , *NL*<sup>∗</sup> *st* ð Þ *<sup>μ</sup>* and *<sup>L</sup>*<sup>∗</sup> *st* ð Þ *μ* denote the set of all supstar families of *L*-subsets in *CLst*, *CLs*ð Þ *μ* , *NLs*ð Þ *μ* , *Ls*ð Þ *μ* , *NLst*ð Þ *μ* and *Lst*ð Þ *μ* respectively, then it follows that

$$\text{CL}\_{st}^{\*} \subseteq \text{NL}\_{st}^{\*} \subseteq \text{L}\_{st}^{\*} \subseteq \text{L}\_{s}^{\*} \left(\mu\right) \text{ and } \text{CL}\_{st}^{\*} \subseteq \text{CL}\_{s}^{\*} \subseteq \text{NL}\_{s}^{\*} \subseteq \text{L}\_{s}^{\*} \left(\mu\right); \tag{27}$$

are sublattices of *<sup>L</sup>*ð Þ *<sup>μ</sup>* . Note that *<sup>L</sup>*<sup>∗</sup> *<sup>s</sup>* ð Þ *μ* ⊆*Ls*ð Þ *μ* . Moreover, we mention that it is not known whether *CL*ð Þ *μ* is a sublattice of *NL*ð Þ *μ* . 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**).
