**2. Preliminaries**

Throughout our work, the system h i *L*, ≤, ∧, ∨ denotes a complete and completely distributive lattice where'≤ ' denotes the partial ordering of *L*, the join(sup) and the meet(inf) of the elements of *L* are denoted by <sup>0</sup> ∨<sup>0</sup> and <sup>0</sup> ∧0 , 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 *Ji* f g : *i*∈ *I* be any family of subsets of a complete lattice *L* and *F* denotes the set of choice functions for *Ji*, i.e., functions *<sup>f</sup>* : *<sup>I</sup>* ! <sup>Q</sup> *<sup>i</sup>*∈*<sup>I</sup>Ji* such that *f i*ð Þ∈*Ji* for each *i* ∈*I*. Then, we say that *L* is a completely distributive lattice, if

$$\bigwedge \left\{ \bigvee\_{i \in I} J\_i \right\} = \bigvee\_{f \in F} \left\{ \bigwedge\_{i \in I} f(i) \right\}. \tag{1}$$

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 *ai* f g : *i*∈ *I* of *L* we have

$$a \bigwedge \left\{ \bigvee\_{i \in I} a\_i \right\} = \bigvee\_{i \in I} \left\{ a \bigwedge a\_i \right\}, \tag{2}$$

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 *ai* f g : *i*∈*I* of *L* we have

$$a\bigvee\left\{\bigwedge\_{i\in I} a\_i\right\} = \bigwedge\_{i\in I} \{a\bigvee a\_i\},\tag{3}$$

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

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 *LX*.

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

$$\mathbf{1}\_A(\mathbf{x}) = \begin{cases} \mathbf{1}, & \text{if } \mathbf{x} \in A, \\ \mathbf{0}, & \text{if } \mathbf{x} \notin A; \end{cases} \tag{4}$$

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

Let *<sup>μ</sup>*∈*L<sup>X</sup>*. The set f g *<sup>μ</sup>*ð Þ *<sup>x</sup>* : *<sup>x</sup>*∈*<sup>X</sup>* is called the image of *<sup>μ</sup>* and is denoted by *<sup>μ</sup>*ð Þ *<sup>X</sup>* or *Imμ*. Let sup *x*∈*X <sup>μ</sup>*ð Þ¼ *<sup>x</sup> <sup>a</sup>*<sup>0</sup> and inf *<sup>x</sup>*∈*<sup>X</sup> μ*ð Þ¼ *x t*0. We call *a*<sup>0</sup> to be the tip of *μ* and *t*<sup>0</sup> to be the

tail of *μ*. We denote the tip and tail of *μ* by sup*μ* and inf *μ* respectively. Let *a*∈*L*. Then, *<sup>μ</sup><sup>a</sup>* <sup>¼</sup> f g *<sup>x</sup>*<sup>∈</sup> *<sup>X</sup>* : *<sup>μ</sup>*ð Þ *<sup>x</sup>* <sup>≥</sup>*<sup>a</sup>* is called *<sup>a</sup>*-level set or *<sup>a</sup>*-cut of *<sup>μ</sup>*. Moreover, *<sup>μ</sup>*<sup>&</sup>gt; *<sup>a</sup>* ¼ f g *x*∈*X* : *μ*ð Þ *x* >*a* is called *a*-strong level set (or *a*-strong cut) of *μ*. Note that *μ<sup>a</sup>* ¼ *ϕ*, if *a*>*a*<sup>0</sup> and *μ*<sup>&</sup>gt; *<sup>a</sup>* <sup>¼</sup> *<sup>ϕ</sup>*, if *<sup>a</sup>*≥*a*0. Moreover, *<sup>μ</sup><sup>a</sup>* <sup>¼</sup> *<sup>X</sup>* if *<sup>a</sup>*<sup>≤</sup> *<sup>t</sup>*<sup>0</sup> and *<sup>μ</sup>*<sup>&</sup>gt; *<sup>a</sup>* ¼ *X* if *a*< *t*0. Let *Y* ⊆*X*. Then, we define *aY* ∈*LX* as follows:

$$a\_Y(\mathbf{x}) = \begin{cases} a, & \text{if } \mathbf{x} \in Y, \\ \mathbf{0}, & \text{if } \mathbf{x} \in X \backslash Y. \end{cases} \tag{5}$$

In particular, if *Y* is singleton say f g*y* , then *a*f g*<sup>y</sup>* is called *L*-point or *L*-singleton and is denoted by *ay*. We say that the *L*-point *ay* ∈*μ* if *μ*ð Þ *x* ≥*a*. The union ∪ *i*∈*I μ<sup>i</sup>* and the intersection ∩ *μ<sup>i</sup>* of any family *μ<sup>i</sup>* f g : *i* ∈*I* of *L*-subsets of *X* are, respectively, defined by:

$$\left(\bigcup\_{i\in I} \mu\_i\right)(\mathbf{x}) = \bigvee\_{i\in I} \mu\_i(\mathbf{x}) \text{ and} \left(\bigcap\_{i\in I} \mu\_i\right)(\mathbf{x}) = \bigwedge\_{i\in I} \mu\_i(\mathbf{x}),\tag{6}$$

for each *x*∈ *X*. Let *η*, *μ*∈*L<sup>X</sup>*. Then, *η* is said to be contained in *μ*, if we have *η*ð Þ *x* ≤ *μ*ð Þ *x* for each *x*∈*X* and is written as *η*⊆*μ* or *μ* ⊇ *η*.

Theorem 1.1 Let *η*, *θ* ∈*LX*. Then

*i*∈*I*

i. *η*⊆ *θ* if and only if *η<sup>a</sup>* ⊆ *θ<sup>a</sup>* for each *a*∈*L*,

ii. *η*⊆ *θ* if and only if *η*<sup>&</sup>gt; *<sup>a</sup>* ⊆*θ* <sup>&</sup>gt; *<sup>a</sup>* for each∈*L* � f g1 , provided *L* is a chain.

Theorem 1.2 Let f g *<sup>μ</sup><sup>i</sup> <sup>i</sup>*∈*<sup>I</sup>* <sup>⊆</sup>*L<sup>μ</sup>*. Then

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

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

$$\begin{aligned} \text{iii.} \left(\bigcup\_{i\in I} \mu\_i\right)\_d^> &= \bigcup\_{i\in I} (\mu\_i)\_a^> \text{ for all } a \in L \sim \{1\}, \text{ provided } L \text{ is a chain;}\\ \text{iv.} \left(\bigcap\_{i\in I} \mu\_i\right)\_a^> &\subseteq \bigcap\_{i\in I} (\mu\_i)\_a^> \text{ for all } a \in L \sim \{1\}, \text{ provided } L \text{ is a chain.} \end{aligned}$$

Let f be a function from *<sup>X</sup>* into *<sup>Y</sup>*, and let *<sup>μ</sup>*∈*LX* and *<sup>ν</sup>*<sup>∈</sup> *LY*. Then, the image fð Þ *<sup>μ</sup>* of *μ* under f and the pre-image f�<sup>1</sup> ð Þ*η* of *η* under f are *L*-subsets of *Y* and *X* respectively defined by:

$$\mathbf{f}(\mu)(y) = \bigvee\_{\mathbf{x} \in \mathbf{f}^{-1}(y)} \{ \mu(\mathbf{x}) : \mathbf{x} \in X \} \text{ and } \mathbf{f}^{-1}(\eta)(\mathbf{x}) = \eta(\mathbf{f}(\mathbf{x})).\tag{7}$$

If f�<sup>1</sup> ð Þ¼ *y ϕ*, then fð Þ *μ* ð Þ¼ *y* 0 since the least upper bound of the empty set in *L* is 0. The set product *μ* ∘ *η* of *μ*, *η*∈*L<sup>S</sup>* , where *S* is a groupoid, is an *L*-subset of *S* defined by

$$\mu \circ \eta(\mathbf{x}) = \sup\_{\mathbf{x} = \mathbf{y}\mathbf{x}} \{\mu(\mathbf{y}) \land \eta(\mathbf{z})\}. \tag{8}$$

If *x* cannot be factored as *x* ¼ *yz* in *S*, then *μ* ∘ *η*ð Þ *x* being the least upper bound of the empty set in *L* is 0.

The set *LX* 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 ⊆. Its maximal and minimal elements are 1*<sup>X</sup>* and 0*X*, respectively. Here 1*<sup>X</sup>* and 0*<sup>X</sup>* are *L*-subsets of *X* which map each element of *X* to 1 and 0, respectively. Moreover, the lattice *P X*ð Þ of all subsets of *X* can be isomorphically embedded into the lattice *LX*.

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 *μ*∈*LG*. Then, *μ* is called an *L*-subgroup of *G* if.

$$\text{i. } \mu(\kappa y) \ge \mu(\kappa) \land \mu(y) \text{ for each } \kappa, y \in G,$$

$$\text{iii. } \mu(\mathfrak{x}^{-1}) = \mu(\mathfrak{x}) \text{ for each } \mathfrak{x} \in G.$$

The set of all *L*-subgroups of *G* is denoted by *L G*ð Þ. From the definition, it is clear that the tip of *μ* is attained at the identity element of *G*.

Definition 1.2 Let *μ*∈ *L G*ð Þ. Then, *μ* is said to to be normal *L*-subgroup of *G* if *μ*ð Þ¼ *xy μ*ð Þ *yx* for each *x*, *y*∈ *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 *μ*∈ *LG*. Then, the *L*-subgroup of *G* generated by *μ*, denoted by h i *μ* , is defined as the smallest *L*-subgroup of *G* which contains *μ*, i.e.,

$$\langle \mu \rangle = \cap \{ \nu : \mu \subseteq \nu, \nu \in L(\mathcal{G}) \}. \tag{9}$$

The set *L G*ð Þ is a complete lattice under the ordering of *L*-set inclusion where the meet'∧' and join'∨' of an arbitrary family f g *η<sup>i</sup> <sup>i</sup>* <sup>∈</sup>*<sup>I</sup>* in *L G*ð Þ are defined, respectively, by: *Development of* L*-Group Theory DOI: http://dx.doi.org/10.5772/intechopen.110387*

$$\square\_{i \in I} \eta\_i = \left( \bigcap\_{i \in I} \eta\_i \right) \text{ and } \bigvee\_{i \in I} \eta\_i = \left\langle \bigcup\_{i \in I} \eta\_i \right\rangle. \tag{10}$$

Let *η*, *μ*∈*L<sup>G</sup>* such that *η*⊆*μ*. Then, *η* is said to be an *L*-subset of *μ*. The set of all *<sup>L</sup>*-subsets of *<sup>μ</sup>* is denoted by *<sup>L</sup><sup>μ</sup>:* Moreover, if *<sup>η</sup>*, *<sup>μ</sup>*∈*L G*ð Þsuch that *<sup>η</sup>*⊆*μ*, then *<sup>η</sup>* is said to be an *L*-subgroup of *μ*. The set of all *L*-subgroups of *μ* is denoted by *L*ð Þ *μ* . Here we mention that the set *L*ð Þ *μ* of all *L*-subgroups of *μ* is a complete sublattice of the lattice *L*ð Þ 1*<sup>G</sup>* .

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

Theorem 1.3 Let *η*∈*L<sup>μ</sup>*. Then,


The following results discuss homomorphic image and pre-image of an *L*-subgroup: Theorem 1.4 Let *η*∈ *L*ð Þ *μ* and f : *G* ! *H* be a group homomorphism. Then, fð Þ*η* is an *L*-subgroup of fð Þ *μ* .

Theorem 1.5 Let *η*, *μ*∈*L H*ð Þ with *η*∈ *L*ð Þ *μ* and f : *G* ! *H* be a group homomorphism. Then, f�<sup>1</sup> ð Þ*<sup>η</sup>* is an *<sup>L</sup>*-subgroup of f�<sup>1</sup> ð Þ *μ* .

Let *η*∈ *L*ð Þ *μ* be such that *η* is non-constant and *η* 6¼ *μ*. Then, *η* is said to be a proper *L*-subgroup of *μ*. Clearly, *η* is a proper *L*-subgroup of *μ* if and only if *η* has distinct tip and tail and *η* 6¼ *μ*.

Definition 1.4 Let *η*∈*L*ð Þ *μ* . Then, *η* is said to be a trivial *L*-subgroup of *μ* if its chain of level subgroups contains only f g*e* and *G:*.

Definition 1.5 Let *<sup>η</sup>*∈*L*ð Þ *<sup>μ</sup>* . Then, define an *<sup>L</sup>*-subset *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> of *μ* as follows:

$$\eta\_{t\_0}^{a\_0}(\boldsymbol{y}) = \begin{cases} a\_0, \text{if } \boldsymbol{y} = \boldsymbol{e}, \\\ t\_0, \text{if } \boldsymbol{y} \neq \boldsymbol{e}; \end{cases} \tag{11}$$

where *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>*ð Þ*<sup>e</sup>* and *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> inf *<sup>η</sup>*. Clearly, *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> is a trivial *L*-subgroup of *μ* and is called the trivial *L*-subgroup of *η*.

From now onwards, we denote *μ* 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 *μ* 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 *η*∈ *L<sup>μ</sup>*. Then, *η* is said to be a normal *L*-subset of *μ* if

$$
\eta\left(y^{-1}\mathbf{x}y\right) \ge \eta(\mathbf{x}) \land \mu(y) \text{ for each } \mathbf{x}, y \in G. \tag{12}
$$

The set of all normal *<sup>L</sup>*-subsets of *<sup>μ</sup>* is denoted by NL*<sup>μ</sup>*. Moreover, if *<sup>η</sup>*<sup>∈</sup> *<sup>L</sup>*ð Þ *<sup>μ</sup>* , then *<sup>η</sup>* is said to be a normal *L*-subgroup of *μ*. The set of all normal *L*-subgroups of *μ* is denoted by NL(*μ*). The following result follows immediately by the definition of normal *L*-subsets:

Theorem 1.6 Let f g *<sup>η</sup><sup>i</sup> <sup>i</sup>*∈*<sup>I</sup>* <sup>⊆</sup> *NL<sup>μ</sup>* be any family. Then,

$$\begin{aligned} \text{i. } \bigcap\_{i \in I} \eta\_i \in \mathcal{NL}^\mu, \\\\ \text{ii. } \bigcup\_{i \in I} \eta\_i \in \mathcal{NL}^\mu. \end{aligned}$$

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 *η*∈*L<sup>μ</sup>*. Then,


Theorem 1.8 Let *η*∈*L*ð Þ *μ* . Then,


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

Theorem 1.9 Let *η*, *μ*∈*L G*ð Þ such that *η*∈ NLð Þ *μ* and f : *G* ! *H* be an onto group homomorphism. Then, fð Þ*η* is a normal *L*-subgroup of fð Þ *μ* .

Theorem 1.10 Let *H* be a group and *μ*, *η*∈*L H*ð Þ be such that *η*∈ NLð Þ *μ* . Let f : *G* ! *H* be a group homomorphism. Then, f�<sup>1</sup> ð Þ*<sup>η</sup>* is a normal *<sup>L</sup>*-subgroup of f�<sup>1</sup> ð Þ *μ* .

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 *η*∈*L<sup>μ</sup>* . Then, *η* is said to have sup-property if for each non-empty subset *A* of *G*, there exists *a*<sup>0</sup> ∈ *A* such that ∨ *a*∈ *A η*ð Þ¼ *a η*ð Þ *a*<sup>0</sup> . The set of all *L*-subsets of *μ* possessing sup-property is denoted by *L<sup>μ</sup> s* .
