1. Introduction
In 1965, Zadeh [
1] presented the theory of fuzzy sets and discussed their initiatory results. The action of fuzzy set theory is a decisive structure to deal vagueness and uncertainty in real life problems. Thus, crisp sets commonly do not have suitable response and feedback for actual worldly conditions of happening issues. In addition, this particular set plays a remarkable role in various scientific fields with wide applications in topological spaces, medical diagnosis determination, coding theory, computer sciences, and module theory.
The idea of fuzzy subgroups was introduced by Rosenfeld [
2] in 1971. The abstraction of fuzzy subrings was proposed by Liu [
3]. Later, these notions were discussed in [
4,
5,
6]. Atanassov [
7] initiated the theory of intuitionistic fuzzy sets and established the basic algebraic properties of intuitionistic fuzzy sets. The complex numbers with fuzzy sets were combined by Buckley [
8]. Kim [
9] presented the idea of the fuzzy order of elements of a group. Ajmal [
10] established the fuzzy homomorphism theorems of fuzzy subgroups. He also discussed the fuzzy quotient group and correspondence theorem.
Ray [
11] initiated the notion of Cartesian product of fuzzy subgroup. Moreover, the volume and intricacy which exist in the collected information of our daily life are developing rapidly with the phase shift of data. Then, there is regularly existing various sorts of uncertainty in that information is represented with complicated problems in different disciplines, such as biology, economics, social science, computer science, mathematics, and environmental science. With the development of science and technology, the decision making problems are becoming increasingly difficult.
To overcome this drawback, Ramot et al. [
12,
13] presented the generalized form of fuzzy set by combining a phase term, called a complex fuzzy set. The efficiency of complex fuzzy logic in the respect of membership has a powerful role to deal with concrete problems. It is highly valuable for calculating unevenness, and also it is very useful way to address ambiguous ideas. Despite its efficacy, we have serious problems about physical features on complex membership related function.
Thus, it is highly essential to formulate extra theories of complicated fuzzy set relating intricate set members. This reasoning is a direct version of traditional fuzzy logic, which results in problem related fuzzy reasoning. Thus, it is not favorable for superficial membership function. This set has a very specific role in wide variety of applications in modern commanding systems especially those that forecast periodic events in which a number of variables are interconnected in complex ways and fuzzy operations cannot run it effectively.
The fundamental set theoretic operations of complex fuzzy sets were presented by Zhang et al. [
14]. Recently, the possible applications, which explain the novel ideas, including complex fuzzy sets in forecasting issues, solar activity, and time series were investigated by Thirunavukarasu et al. [
15]. The complex fuzzy sets have wide applications in decision making, image restoration, and reasoning schemes. Ameri et al. [
16] invented the of Engel fuzzy subgroups in 2015. The notion of complex vague soft sets were defined by Selvachandran [
17]. Al-Husban and Salleh [
18] developed a connection between complex fuzzy sets and group theory in 2016.
Singh et al. [
19] discussed the link between complex fuzzy set and metric spaces. In 2016, Thirunavukarasu et al. [
20] depicted the abstraction of complex fuzzy graph and find energy of this newly defined graph. In 2017, Alsarahead and Ahmed [
21,
22,
23] presented a new abstraction of complex fuzzy subgroup. They also introduced novel conception complex fuzzy subring and complex fuzzy soft subgroups. These abstractions are completely different from Rosenfeld fuzzy subgroups [
2] and Liu fuzzy subring [
3]. The parabolic fuzzy subgroups were introduced by Makamba and Murali [
24]. Then, certain algebraic properties of Engel fuzzy subgroups were discussed in [
25].
Moreover, the Mohamadzahed et al. [
26] established the novel concept of nilpotent fuzzy subgroup. The fuzzy homomorphism structures on fuzzy subgroups was discussed by Addis [
27]. The algebraic structure between fuzzy sets and normed rings were proposed in [
28]. Gulistan et al. [
29] presented the notion of
-complex fuzzy hyper-ideal and investigated many algebraic properties of this phenomena. Liu and Shi [
30] presented a novel framework to fuzzification of lattice, which is known as an
M-hazy lattice. The complex fuzzy sub-algebra commenced in [
31].
Currently, the novel environment of complex fuzzy set in decision making problems has been used frequently [
32,
33,
34,
35,
36,
37,
38], owing to the existence of complex fuzzy information in several practical situations. Gulzar et al. [
39] published a inventive theory of complex fuzzy subfields. The new development about
Q-complex fuzzy subring was introduced in [
40]. For other useful results of fuzzy subgroups not mentioned in the article, readers are referred to [
41,
42,
43,
44,
45,
46,
47].
The motivation of the proposed concept is explained as follows: (1) To present a more generalized concept, i.e., -complex fuzzy sets. (2) Note that for and , our proposed definition can be converted into a classical complex fuzzy set. The purpose of this paper is to present the study of -complex fuzzy sets and -complex fuzzy subgroups as a powerful extension of complex fuzzy sets and complex fuzzy subgroups.
We organized this article as follows:
Section 2 contains the basic notions of complex fuzzy sets (CFSs), complex fuzzy subgroups (CFSGs), and associated results, which are important for our paper. In
Section 3, we expound the abstraction of
-complex fuzzy set (CFS) and
-complex fuzzy subgroup (
-CFSG). We prove that every complex fuzzy subgroup (CFSG) is also an
-complex fuzzy subgroup (
-CFSG) and discuss fundamental properties of this newly defined CFSG. In
Section 4, we explicate the
-complex fuzzy cosets and
-complex fuzzy normal subgroups (
-CFNSGs) and investigate many algebraic characteristics of these specific groups. Moreover, we find the quotient group with respect to
-complex fuzzy cosets and prove the
-complex fuzzy quotient group. We initiate the definition of the index of
-CFSG and develop the
-complex fuzzification of Lagrange’s Theorem.
3. Algebraic Properties of -Complex Fuzzy Subgroups
In this section, we define the hybrid models of -CFSs and -CFSGs. We prove that every CFSG is also -CFSG but the converse may not be true generally, and we discuss some basic characterization of this phenomenon.
Definition 9. Let be CFS of group G, for any and such that and or Then, the set is called an -CFS and is defined as:where and . In this paper, we shall use and , as a membership function of -CFSs and , respectively.
Definition 10. Let and be a two -CFSs of Then,
- 1.
A -CFS is homogeneous -CFS if, for all , we have if and only if
- 2.
A -CFS is homogeneous -CFS with if, for all , we have if and only if
In this article, we take -CFS as homogeneous -CFS.
Remark 1. It is an interesting that we obtain classical CFS A by taking the value of and in the above definition.
Remark 2. Let and be two -CFSs of group G. Then,
Definition 11. Let be an -CFS of group G for and . Then, is called -complex fuzzy subgroupoid of group G if it satisfies the following axiom: Definition 12. Let be an -CFS of group G for and . Then, is called -CFSG of group G if it satisfies the following axioms:
- 1.
- 2.
for all .
Remark 3. If is -CFSG of group G for . Then, Theorem 1. If is an -CFSG of group G, for all . Then,
- 1.
- 2.
which implies that
Theorem 2. Let be an -complex fuzzy subgroupoid of a finite group, and then is -CFSG of finite group.
Proof. Assume that
. Given that
G is a finite group; therefore,
p has finite order
n.
, where
e is the natural element of group
G. Then, we have
Now, we apply the Definition 11 repeatedly. Then, we obtain
Hence, we proved the claim. □
Theorem 3. If is an -CFSG of a group G, let and , and then , for all .
Proof. Given that
. Then, from Theorem 2, we have
Consider
This establishes the proof. □
Theorem 4. Every CFSG of group G is also -CFSG of G.
Proof. Let
A be CFSG of group
G, for any
Consider
Further, we assume that
This establishes the proof. □
Remark 4. If -CFSG then it is not necessary A is CFSG.
Example 1. Let be the Klein four group. One can see that is not CFSG of G. Take and then easily we can see that , for all Then, we obtain Therefore, , for all Moreover, .
Thus, . Hence, is -CFSG.
Theorem 5. Let A be a CFS of group G such that Let such that and , where and and Then, is an -CFSG of G.
Proof. Note that
implies that
, which implies that
, for all
which implies that
Hence,
is
-CFSG of
G. □
Theorem 6. If and are two -CFSGs of G, then is also -CFSG of G.
Proof. Given that
and
are two
-CFSGs of
G, for any
Consequently,
is
-CFSG of
G. □
Remark 5. The union of two -CFSGs may not be -CFSG.
Example 2. Consider a symmetric group of all permutation of four elements. Define two -CFSGs and of for the value given as:This implies thatTake , and . Therefore, and Clearly, we can see thatHence, this proves the claim. 4. -Complex Fuzzification of Lagrange’s Theorem
In this section, we investigate the algebraic attributions of -CFNSGs. We start the study of the concept of -complex fuzzy cosets of -CFSG and develop a quotient group induced by this particular CFNSGs. We also establish -CFSG of classical quotient group and prove some important properties of these CFNSGs. Moreover, we discuss -complex fuzzification of Lagrange’s heorem.
Definition 13. Let be an -CFSG of group G, where and . Then, the -CFS of G is called a -complex fuzzy left coset of G determined by and p and is described as: Similarly we can define -complex fuzzy right coset of of G determined by and p and is described as: The following example illustrates the notion of -complex fuzzy cosets of .
Example 3. Take a permutation group of order Define -CFSG of G for the value of and as follows:From Definition 13, we have Hence, the -complex fuzzy left coset of in G for is as follows:Similarly, one can find -complex fuzzy left coset of , for each Definition 14. Let be an -CFSG of group G, where and . Then, is called a -CFNSG if Equivalently -CFSG is -CFNSG of group G if: , for all .
Note that -CFNSG is classical CFNSG of G.
Remark 6. Let be an -CFNSG of group Then, for all .
Theorem 7. If A is CFNSG of group G, then is an -CFNSG of G.
Proof. Let
be any elements of
G. Therefore, we have
Consequently,
is
-CFNSG of
□
The converse of the above result does not hold generally. In the following example, we explain this fact.
Example 4. Let be the dihedral group.
Let A be a CFS of G and described as:Note that A is not a CFNSG of group G. For . Now, we take , and we obtain Next, we prove that every -CFSG of group G will be -CFNSG of group G, for some specific values of and . In this direction, we prove the following results.
Theorem 8. Let be -CFSG of group G such that such that and , where and and Then, is a -complex fuzzy normal subgroup of group G.
Proof. Given that implies that which implies that , .
Thus, for any .
Similarly,
This implies that Hence, we proved the result. □
Theorem 9. Let be an -CFSG of a group G, then is an -complex fuzzy normal subgroup if and only if is constant in the in the conjugacy class of group G.
Proof. Assume that
is an
-complex fuzzy normal subgroup of group
G. Then, we have
Conversely, suppose that
is constant in all conjugate classes of group
G. Then,
Hence, we prove the claim. □
Theorem 10. If is an -CFSG of a group G, then is an -complex fuzzy normal subgroup if and only if
Proof. Suppose that
is an
-complex fuzzy normal subgroup of group
G. Let
be element of group. Consider
Conversely, suppose that
Let
be an element.
Now, we expound two possible cases.
This implies that is a constant mapping and, in this case, the result holds obviously.
Hence, is constant. □
Theorem 11. Let be -CFNSG of group G. Then, the set is a normal subgroup of group G.
Proof. We know that because . Let be any elements.
Consider
This implies that
. However,
Therefore,
This implies that
which implies that
. Further,
However,
. Thus,
is subgroup of group
G. Moreover, let
and
. We have
. This implies that
. Hence,
is a normal subgroup. □
Theorem 12. Let be an -CFNSG of group G. Then,
- 1.
- 2.
Proof. For any , we have .
Consider,
Therefore,
.
Conversely, let
implies that
Interchange the role of
p and
q, and we obtain
. Therefore,
.
(ii) Similarly one can prove that as part (i). □
Theorem 13. Let be an -CFNSG of group G and and b be any elements in G. If and , then
Proof. Given that and . This implies that
Consider, . As is normal subgroup of G. Thus, . Consequently, .
In the following result, we establish -complex fuzzy quotient group analog to the classical quotient group. □
Theorem 14. Let be the collection of all -complex fuzzy cosets of -CFNSG of G. Then, the binary operator 🟉 is a well-defined operation of set and is defined as .
Proof. We have
, for any
Let
be any element, then
Thus, 🟉 is well-defined operation on
. Note that the set
fulfills the closure and associative axioms with respect to the well-defined binary operation 🟉. Further,
=
is neutral element of
. Clearly the inverse of every element of
exist if
, and then there exists an element,
such that
=
As a result,
is a group. The group
is called the quotient group of the
G by
. □
Lemma 1. Let be natural homomorphism from group G onto and defined by the rule, with the kernel h =.
Proof. Let
be any elements of group
G, and then
Thus,
h is homomorphism. Further,
f is as well
□
In the following result, we establish -CFSG of the quotient group induced by the normal subgroup .
Theorem 15. Let be a normal subgroup of G. If is -CFSG, then, the -CFS of is also a -CFSG of , where
Proof. First, we shall prove that
is well-defined. Let
then
for some
.
Therefore,
is well-defined.
This concludes the proof. □
Remark 7. If is an -CFSG of a group G, let and , for all then .
Definition 15. Let be a -complex fuzzy subgroup of finite group Then, the cardinality of the set of all -complex fuzzy left cosets of G by is called the index of -complex fuzzy subgroup and is denoted by
Theorem 16. (-Complex Fuzzification of Lagrange’s Theorem): Let us assume that there exists a -complex fuzzy subgroup of finite group Then, the index of -complex fuzzy subgroup of G divides the order of
Proof. By Lemma 1, we have a natural homomorphism h from G to .
Define a subgroup . By applying the Definition 13 and , we have . This implies that , by Remark 7, which shows that Therefore, is contained in Now, we take any element and using the fact is subgroup of G, we have From Theorem 13, the elements which means that which implies that Hence, is contained in . From this discussion, we can say that
Now, we define the partition of the group
G into the disjoint union of right cosets, and this is defined as
where
Now, we prove that, to each coset
in relation
there exists an
-complex fuzzy coset
in
, and this corresponding is injective.
Consider any coset
Let
then
Thus,
h maps each element of
into the
-complex fuzzy coset
Now, we establish a natural correspondence
between the set
and the set
defined by
The correspondence
h is injective.
For this, let then . By using , we have which means that , and hence is injective. It is quite clear from the above discussion that are equal. Hence, divides □
Example 5. Consider as a finite permutation group of order 6. The -complex fuzzy subgroup of G corresponding to the value and is defined asThe set of all -complex fuzzy left cosets of G by is given by:This means that