A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (α,β)-complex fuzzy sets and then define α,β-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of α,β-complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.


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 (α, β)-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].
The motivation of the proposed concept is explained as follows: (1) To present a more generalized concept, i.e., (α, β)-complex fuzzy sets. (2) Note that for α = 1 and β = 2π, our proposed definition can be converted into a classical complex fuzzy set. The purpose of this paper is to present the study of (α, β)-complex fuzzy sets and (α, β)-complex fuzzy subgroups as a powerful extension of complex fuzzy sets and complex fuzzy subgroups.

Preliminaries
In this section, we describe the CFSs and CFSGs, and then we discuss the basic operations of complex fuzzy sets.

Definition 4 ([21]).
A π-fuzzy set A π of group G is called a π-fuzzy subgroup of G if

Definition 5 ([21]
). Let A = {(p, µ A (p)e iϕ A (p) ) : p ∈ G} and B = {(p, µ B (p)e iϕ B (p) ) : p ∈ G} be two CFSs of G. Then, G} be a CFSs of set P. Then, the operation of intersection and union is defined as: Then, A is called CFSG of group G if the following conditions hold.
, for all p, q ∈ G.

Algebraic Properties of (α, β)-Complex Fuzzy Subgroups
In this section, we define the hybrid models of (α, β)-CFSs and (α, β)-CFSGs. We prove that every CFSG is also (α, β)-CFSG but the converse may not be true generally, and we discuss some basic characterization of this phenomenon.

Remark 1.
It is an interesting that we obtain classical CFS A by taking the value of α = 1 and β = 2π in the above definition.
Then, A (α,β) is called (α, β)-complex fuzzy subgroupoid of group G if it satisfies the following axiom: satisfies the following axioms: Proof. Obviously.
Proof. Assume that p ∈ G. Given that G is a finite group; therefore, p has finite order n. p n = e, where e is the natural element of group G. Then, we have p −1 = p n−1 . Now, we apply the Definition 11 repeatedly. Then, we obtain Hence, we proved the claim.
. Then, from Theorem 2, we have Now, assume that Again, from Theorem 2, we have From Equations (1) and (2), we have This establishes the proof.
Proof. Let A be CFSG of group G, for any p, q ∈ G. Consider Further, we assume that This establishes the proof.

Remark 4. If A (α,β) -CFSG then it is not necessary A is CFSG.
Example 1. Let G = {e, r, s, rs} be the Klein four group.
Proof. Let x , p be any elements of G. Therefore, we have
The converse of the above result does not hold generally. In the following example, we explain this fact. Example 4. Let G = D 3 =< r, s : r 3 = s 2 = e, sr = r 2 s > 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 µ A (r 2 (rs))e iϕ A (r 2 (rs)) = 0.8e π/3 = 0.7e π/6 = µ A ((rs)r 2 )e iϕ A ((rs)r 2 ) . Now, we take αe iβ = 0.5e i π/9 , and we obtain Next, we prove that every (α, β)-CFSG of group G will be (α, β)-CFNSG of group G, for some specific values of α and β. In this direction, we prove the following results.
Theorem 8. Let A (α,β) be (α, β)-CFSG of group G such that αe iβ < re iω such that α ≤ r and This implies that µ pA α (x)e . Hence, we proved the result. Proof. Assume that A (α,β) is an (α, β)-complex fuzzy normal subgroup of group G. Then, we have Conversely, suppose that A (α,β) is constant in all conjugate classes of group G. Then, Hence, we prove the claim.
Proof. Suppose that A (α,β) is an (α, β)-complex fuzzy normal subgroup of group G. Let x, y ∈ G be element of group. Consider (p) . Let p, r ∈ G be an element.
Now, we expound two possible cases.
This implies that A (α,β) is a constant mapping and, in this case, the result holds obviously.

If min{µ
Then, from Equation (5) we have In the view of Equations (4) and (6) we have Hence, A (α,β) is constant.
Thus, is well-defined operation on G/A (α,β) . Note that the set G/A (α,β) fulfills the closure and associative axioms with respect to the well-defined binary operation . Further, is neutral element of G/A (α,β) . Clearly the inverse of every element of G/A (α,β) exist if µ pA α e iϕ pA β ∈ G/A (α,β) , and then there exists an element, called the quotient group of the G by A (α,β) .
Thus, h is homomorphism. Further, f is as well In the following result, we establish (α, β)-CFSG of the quotient group induced by the normal subgroup A e α,β .
This concludes the proof.