# On Structural Properties ofξ-Complex Fuzzy Sets and Their Applications

Authors: Aneeza Imtiaz; Umer Shuaib; Hanan Alolaiyan; Abdul Razaq; Muhammad Gulistan
Journal: Complexity
 (2020)

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## Abstract

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept ofξ-complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α,δ-cut sets of ξ-complex fuzzy sets and justify the representation of an ξ-complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ-complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ-complex fuzzy set is ξ-complex fuzzy subgroup. Furthermore, we extend the idea of ξ-complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ-complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup GAξ.


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## Body

## 1. Introduction

The competency of fuzzy logic to articulate steady adaptations from membership to nonmembership and the other way around has played an effective role to solve many physical problems. It provides us not only the powerful and meaningful representations of measuring uncertainty but also a useful approach to view the vague concepts expressed in the natural language. Despite all of the advantages of this logic, we still face immense complications to counter various physical situations based on a real-valued membership function. It is therefore quite necessary to propose an additional development of fuzzy set theory on account of set of complex numbers which is indeed an existing augmentation of real numbers. The complex fuzzy logic is linear augmentation of traditional fuzzy logic, which also allows a natural development of the difficulty based on the fuzzy logic that is impracticable to solve with superficial membership function. This particular set has a paramount part in numerous executions, in particular, advanced control systems and predicting of the periodic events, where multiple fuzzy variables are interrelated in a complex manner that cannot be effectively characterized by simple fuzzy operations. In addition, these sets are also used to solve several problems, especially the numerous periodic aspects and forecast problems. One of the far-reaching significances of studying the CFS is that they illustrate the data with uncertainty and periodicity in a much effective way.Considering the inaccuracy in decision-making, Zadeh [1] popularized the concept of fuzzy sets for the first time, in 1965. Roenfeld [2] used Zadeh’s work to invent the approach of fuzzy subgroups in 1971. Mukherjee and Bhattacharya [3] initiated the study of the fuzzy cosets along with fuzzy normal subgroups in 1984. Mashour et al. [4] studied many important features of normal fuzzy subgroups. For more details about the recent development of fuzzy subgroups, we refer to [5–7]. In addition, another important aspect of fuzzy sets is the level sets of this notion. These level sets have an incredible significance as they set up a connection between concepts of crisp and fuzzy sets. This phenomenon is utilized to generalize various ideas and techniques for crisp set hypothesis as well as fuzzy sets. In the succession of fuzzification of subgroups, the idea of level subgroup was initiated by Das [8]. Yuan and Li [9] studied the impact of these level sets in the field of intuitionistic fuzzy sets. To see more on the level sets and level subgroups, we refer to [10–12]. Buckley [13] proposed the study of complex fuzzy numbers in 1989. The author [14] applied the theory of complex fuzzy numbers to set up new techniques of differentiation. Moreover, some fundamental properties of fuzzy counter integral in the complex plane were interpreted by the same author in [15]. Zhang [16] established many important properties of complex fuzzy numbers in 1992. Ascia et al. [17] designed a competent specific fuzzy processor which effectively deals with the complex fuzzy inference system. In [18], Ramote et al. launched the study of CFS in 2002. The two novel operations, namely, reflection and rotations, of these sets were introduced in [19]. Zhang et al. [20] formulated the δ-equalities of CFS. In 2011, Chen et al. [21] developed the adaptive neuro complex fuzzy inference system. In [22], Fu and Shen utilized CFS to design a new approach of linguistic evaluation of classifier performance. Tamir and Kandel [23] provided granted bases for first order estimation, complex class theory, and complex fuzzy logic in the same year. Ma et al. [24] defined product sum accruement operator for these particular sets in 2012. Li et al. [25] presented a new self-learning complex neuro fuzzy system using the same idea. In 2014, Alkouri and Salleh [26] illustrated many important multiple distance measures defined on CFS. In 2015, Tamir et al. [27] reintroduced the concept of CFS and complex fuzzy logic in a more logical way. Al-Husban and Salleh [28] interpreted the study of complex fuzzy hyper structure over complex fuzzy space in 2016. The phenomenon of CFSG over complex fuzzy space was discussed in [29]. The operator theory has an important role in modern mathematics because it is extensively used in fuzzy theory for approximate reasoning. The techniques of t-norms and t-co-norms with respect to complex fuzzy sets were presented by Nagarajan et al. [30]. For more on fuzzy set theory, we suggest reading of [31–35]. The competency of the complex fuzzy set has played an effective role to solve many physical problems. It provides us the meaningful representations of measuring uncertainty and periodicity. Despite all of these advantages, we still face vast complications to counter various physical situations based on a complex-valued membership function. This motivates us to define the notion of ξ-complex fuzzy set (ξ-CFS) through which one can have multiple options to investigate a specific real-world situation in much efficient way by choosing appropriate value of the parameter ξ.In this article, we propose the idea ofξ-CFS as a powerful extension of classical fuzzy set and use this idea to define the notion of the cut sets of these particular sets. We explore the importance of these cut sets by proving the decomposition theorems for a ξ-CFS. Moreover, we introduce the phenomenon of ξ-complex fuzzy subgroup (ξ-CFSG) over an ξ-complex fuzzy set and investigate some of their fundamental algebraic attributes.After a brief discussion about the historical background and significance of CFS, the rest of the article is organized as follows. Section2 contains some definitions of the basic introduction of the concept of ξ-CFS and the study of α,δ-cut sets and strong α,δ-cut sets of this newly defined notion. In addition, we establish the importance of defining these ideas by viewing an ξ-CFS as a union of nested sequence of both α,δ-cut sets and strong α,δ-cut sets, respectively. Moreover, we apply ξ-CFS to a physical situation in which one may select the most suitable performance of a participant. In Section 4, we utilize the concept of ξ-CFS to propose the study of the idea of ξ-CFSG and prove that each CFSG is ξ-CFSG. Moreover, we extend the importance of these fuzzy subgroups by introducing the notions of ξ-complex fuzzy cosets and ξ-complex fuzzy normal subgroup (ξ-CFNSG) and investigate their many important algebraic aspects.


## 2. Preliminaries

This section contains a brief review of the notion of CFS and related ideas, which are quite essential to understand the novelty of this article.Definition 1.
(see [19]). A CFS A defined on a universe of discourse U is characterized by a membership function μAm that allocates each element of Uto a unit circle C∗ in complex plane and is written as rAmeiωAm, where rAm denotes the real-valued function from U to the closed unit interval and eiωAm is a periodic function whose periodic law and principal period are 2π and 0<argAm≤2π, respectively.
Note thatωAm=argAm+2kπ,k∈Z and argAm is the principal argument.Definition 2.
(see [8]). Let 0≤α≤1 and 0≤δ≤2π. Then, the α,δ-cut set of CFS A is denoted by Aα,δ and is defined as Aα,δ=m∈U:rAm≥α,ωAm≥δ.Definition 3.
(see [31]). Let A and B be any two CFS of a universe U. Then,(1)
A is homogeneous CFS if rAm≤rAn implies ωAm≤ωAn and vice versa ∀m,n∈U(2)
A is homogeneous CFS with B if rAm≤rBn implies ωAm≤ωBn and vice versa ∀m,n∈UDefinition 4.
(see [31]) A homogeneous CFS A of G is said to be a CFSG if μAmn≥minμAm,μAn and μAm−1≥μAm∀m,n∈G.Definition 5.
(see [5]). Let A be a CFSG (G) and m∈G. Then, the complex fuzzy left coset of A in G is represented by mAand is given by mAg=μAm−1g:g∈G.
Similarly, one can define complex fuzzy right coset ofAin G.Definition 6.
(see [5]). A CFSG A of a group Gis CFNSG (G) if mA=Am,∀m∈G.


## 3. Decomposition Theorems ofξ-Complex Fuzzy Sets

In this section, we initiate the idea ofξ-CFS as a powerful extension of classical fuzzy sets. We also define the concepts of α,δ-cut sets and strong α,δ-cut sets of ξ-CFS and establish fundamental properties of these phenomena. We also prove three decomposition theorems of ξ-CFS.Definition 7.
Let A be a CFS of a universeU and ξ=αeiδ be an element of a unit circle with 0≤α≤1 and 0≤δ≤2π. Then, the CFS Aξ is called the ξ-complex fuzzy set (ξ-CFS) with respect to CFS A and is expressed as μAξm=minμAm,ξ, ∀m∈U.
The family of allξ-CFS defined on the universe U is denoted by FξU.Definition 8.
For anyΑξ and Βξ∈FξU,(1)
The union ofξ-CFS Αξ and Βξ is denoted by Αξ∪Βξ and is defined as follows:(1)μΑξ∪Βξm=rΑξ∪ΒξmeiωΑξ∪Βξm=maxrΑξm,rΒξmeimaxωAξm,ωBξm,∀m∈U.(2)
The intersection ofξ-CFS Aξ and Bξ is denoted by Aξ∩Bξ and is defined as follows:(2)μAξ∩Bξm=rAξ∩BξmeiωAξ∩Bξm=minrAξm,rBξmeiminωAξm,ωBξm,∀m∈U.(3)
The complement ofξ-CFS Aξ is denoted by Aξ' and is described as follows:(3)μAξ'm=rAξ'meiωAξ'm=1−rAξmei2π−ωAξm,∀m∈U.(4)
The product ofξ-CFS Aξ and Bξ is represented by Aξ∘Bξ and is expressed as follows:(4)μAξ∘Bξm=rAξ∘BξmeiωAξ∘Bξm=rAξm.rBξmei2πωAξm/2π.ωBξm/2π,∀m∈U.(5)
LetAnξ,n∈N be ξ-CFS of a universe U. Then, the Cartesian product of ξ-CFS Anξ is represented by A1ξ×A2ξ×⋯×Anξ and is defined in the following way:(5)μA1ξ×A2ξ×…×Anξm=rA1ξ×A2ξ×…×AnξmeiωA1ξ×A2ξ×…×Anξm=minrA1ξm1,rA2ξm2,…,rAnξmneiminωA1ξm1,ωA2ξm2,…,ωAnξmn,where m=m1,m2,⋯,mn,mi∈Ui,i∈N.Definition 9.
LetAξ and Bξ be any two ξ-CFS of a universe U. Then,(1)
Aξ is homogeneous ξ-CFS if rAξm≤rAξn implies ωAξm≤ωAξn and vice versa ∀m,n∈U(2)
Aξ is homogeneous ξ-CFS with Bξ if rAξm≤rBξn implies ωAξm≤ωBξn and vice versa ∀m,n∈U
The next result illustrates that the intersection of any twoξ-CFS is also ξ-CFS.Proposition 1.
For any twoξ-CFS Aξand Bξ, A∩Bξ=Aξ∩BξProof.
ConsiderμA∩Bξm=minμA∩Bm,ξ, m∈U.
By applying Definition8 (2), we have μA∩Bξm=minμAξm,μBξm=μAξ∩Bξm.Remark 1.
The union of any twoξ-CFS is also ξ-CFS.Definition 10.
Theα,δ-cut set of ξ-CFS Aξ is represented by Aα,δξ and is defined as follows:(6)Aα,δξ=m∈U:rAξm≥α,ωAξm≥δ,0≤α≤1,0≤δ≤2π.Definition 11.
For anyAξ∈FξU strong α,δ-cut set of Aξ is defined as A+α,δξ=m∈U:rAξm>α,ωAξm>δ,0≤α≤1,0≤δ≤2π.Definition 12.
The level setΩAξ of Aξ can be described as ΩAξ=m∈U:rAξm=α,ωAξm=δ, where α∈0,1 and δ∈0,2π.Theorem 1.
LetAξ and Bξ be any two ξ-CFS. Then, the following attributes hold for any α,α′∈0,1 and β,β′∈0,2π.(1)
A+α,δξ⊆Aα,δξ(2)
α≤α′,δ≤δ′ implies Aα',δ'ξ⊆Aα,δξ(3)
Aξ∩Bξα,δ=Aα,δξ∩Bα,δξ(4)
Aξ∪Bξα,δ=Aα,δξ∪Bα,δξ(5)
Aα,δξ'=Aξ'+1-α,2π-δProof.
(1)
In view of Definition10, for any element m∈U, rAξm>α and ωAξm>δ. It means that rAξm≥α and ωAξm≥δ. Thus, A+α,δξ⊆Aα,δξ.(2)
Letm∈U, and by applying Definition 10, we have rAξm≥α,ωAξm≥δ. Therefore, Aα′,δ′ξ⊆Aα,δξ.(3)
For anym∈Aξ∩Bξα,δ, we have rAξ∩Bξm≥α and ωAξ∩Bξm≥δ. This implies that minrAξm,rBξm≥α and minωAξm,ωBξm≥δ. Then, clearly, rAξm≥α,rBξm≥α and ωAξm≥δ,ωBξm≥δ. Therefore, m∈Aα,δξ∩Bα,δξ, and ultimately, we obtain(7)Aξ∩Bξα,δ⊆Aα,δξ∩Bα,δξ.Letm∈Aα,δξ∩Bα,δξ.By applying Definition10, in the above relations, we obtain(8)rAξm≥α,ωAξm≥δ,rBξm≥α,ωBξm≥δ.This shows thatminrAξm,rBξm≥α and minωAξm,ωBξm≥δ.Consequently,(9)Aα,δξ∩Bα,δξ⊆Aξ∩Bξα,δ.From (7) and (9), the required equality holds.(4)
For anym∈Aξ∪Bξα,δ, we obtain rAξ∪Bξm≥α and ωAξ∪Bξm≥δ. Therefore, maxrAξm,rBξm≥α and maxωAξm,ωBξm≥δ. This means that rAξm≥α,rBξm≥α and ωAξm≥δ,ωBξm≥δ.Consequently,(10)Aξ∪Bξα,δ⊆Aα,δξ∪Bα,δξ.Now, suppose thatm∈Aα,δξ∪Bα,δξ; then, m∈Aα,δξ or m∈Bα,δξ. By applying Definition 10 in the above relations, we get rAξm≥α,ωAξm≥δ or rBξm≥α,ωBξm≥δ. It further shows that maxrAξm,rBξm≥α and maxωAξm,ωBξm≥δ.Consequently,(11)Aα,δξ∪Bα,δξ⊆Aξ∪Bξα,δ.From (10) and (11), the required equality is satisfied.(5)
Letm∈Aα,δξ′, then μAξ′m=1−rAξmei2π−ωAξm implying that 1−rAξm≥α, 2π−ωAξm≥δ.
It follows thatrAξm≤1−α, ωAξm≤2π−δ which shows that m∉Aξ+1−α,2π−δ.
Therefore,m∈Aξ′+1−α,2π−δ and hence(12)Aα,δξ′⊆Aξ′+1−α,2π−δ.
Now, supposem∈Aξ′+1−α,2π−δ; then, m∉Aξ+1−α,2π−δ.
This implies that1−α≥rAξm, ωAξm≤2π−δ, α≤1−rAξm, and δ≤2π−ωAξm.
It means thatm∈Aα,δξ′; therefore,(13)Aξ′+1−α,2π−δ⊆Aα,δξ′.
From (12) and (13), the required result is satisfied.Theorem 2.
LetAξ and Bξ be any two ξ-CFS. Then, the following attributes hold for all α,α′∈0,1 and δ,δ′∈0,2π(1)
α≤α′,δ≤δ′ implies A+α′,δ′ξ⊆A+α,δξ(2)
Aξ∩Bξ+α,δ=A+α,δξ∩B+α,δξ(3)
Aξ∪Bξα,δ=Aα,δξ∪Bα,δξTheorem 3.
The following properties hold for any family ofξ-CFS Aiξ, i∈I.(1)
∪i∈IAiξα,δ⊆∪i∈IAiξα,δ(2)
∩i∈IAiξα,δ=∩i∈IAiξα,δProof.
(1)
Letm∈∪i∈IAiξα,δ; in view of Definition 10, we get rAioξm≥α and ωAioξm≥δ.The above relation holds only ifSuprAiξm≥α and SupωAiξm≥δ, that is, ∪i∈Iriξm≥α and ∪i∈Iωiξm≥δ.It follows thatm∈∪i∈IAiξα,δ.(2)
Letm∈∩i∈IAiξα,δ; by using Definition 10, we obtain rAioξm≥α, ωAioξm≥δ.
The above inequality holds only ifinfrAiξm≥α, infωAiξm≥δ, that is, ∩i∈Iriξm≥α and ∩i∈Iωiξm≥δ. This implies that m∈∩i∈IAiξα,δ. Hence, ∩i∈IAiξα,δ⊆∩i∈IAiξα,δ. Now, suppose that m∈∩i∈IAiξα,δ. Again, by applying Definition 10, we get rAiξm≥α, ωAiξm≥δ. Then, obviously m∈∩i∈IAiξα,δ. Consequently, ∩i∈IAiξα,δ⊆∩i∈IAiξα,δ.Theorem 4.
Any family ofξ-CFS Aiξ: i∈I admits the following properties:(1)
∪i∈IAiξ+α,δ=∪i∈IAiξ+α,δ(2)
∪i∈IAiξ+α,δ⊆∪i∈IAiξ+α,δTheorem 5.
Any twoξ-CFS Aξ and Bξ satisfy the following relations:(1)
Aξ⊆Bξ if and only if Aα,δξ⊆Bα,δξ(2)
Aξ=Bξ if and only if Aα,δξ=Bα,δξProof.
(1)
SupposeAξ⊆Bξ. Assume that there exist αo∈0,1 and δo∈0,2π such that Aαo,δoξ⊆Bαo,δoξ.It means thatmo∈U such that mo∈Aαo,δoξ, but mo∉Bαo,δoξ. Then, rAξmo≥αo, ωAξmo≥δo, rBξmo<αo, and ωBξmo<δo. Hence, rBξmo<rAξmo and ωBξmo<ωAξmo, which is contradiction to our supposition.Conversely, letAα,δξ⊆Bα,δξ. Consider Aξ⊆Bξ, implying that mo∈U, such thatrAξmo>rBξmo, ωAξmo>ωBξmo. This further shows that mo∈Aα,δξ and mo∉Bα,δξ, which contradicts our assumption.(2)
In a similar way, we can obtain the required equality.Theorem 6.
Any twoξ-CFS Aξ and Bξ satisfy the following characteristics:(1)
Aξ⊆Bξ if and only if A+α,δξ⊆B+α,δξ(2)
Aξ=Bξ if and only if A+α,δξ=B+α,δξTheorem 7.
Everyξ-CFS Aξ satisfies the following relations:(1)
Aα,δξ=∩α′<αδ′<δAα′,δ′ξ(2)
A+α,δξ=∪α<α′δ<δ′Aα′,δ′ξ(3)
Aα,δξ=∩α′<αδ′<δA+α′,δ′ξ(4)
A+α,δξ=∪α<α′δ<δ′A+α′,δ′ξProof.
(1)
In view of Theorem1 (2),(14)Aα,δξ⊆∩α′<αδ′<δAα′,δ′ξ,for all α′<α and δ′<δ.
Next, supposem∈∩α′<αδ′<δAα′,δ′ξ. Again, by applying Theorem 1 (2), we have rAξm≥α' and ωAξm≥δ'.
The application of the given condition in the above relation yields thatrAξm≥α and ωAξm≥δ. This implies that m∈Aα,δξ.
Hence,(15)∩α′<αδ′<δAα′,δ′ξ⊆Aα,δξ.
From (14) and (15), the required equality holds. The remaining parts can be proved in a similar manner. In the following definitions, we present a new approach to define ξ-CFS which is quite necessary to establish the proofs of decomposition theorems.Definition 13.
LetAα,δξ be a α,δ-cut set of Aξ∈FξU. Then, the ξ-CFS Aα,δξ∗ with respect to Aα,δξ is defined as follows:(16)Aα,δξ∗m=αeiδ,0,ifm∈Aα,δξ,otherwise.Definition 14.
LetA+α,δξ be a strong α,δ cut set of Aξ∈FξU. Then, the ξ-CFS A+α,δξ∗ with respect to A+α,δξ can be described as follows:(17)A+α,δξ∗m=αeiδ,0,ifm∈A+α,δξ,otherwise.
The subsequent result illustrates the decomposition of anξ-CFS as a union of ξ-CFS Aα,δξ∗.Theorem 10.
(first decomposition theorem). For everyAξ∈FξU, Aξ=∪α∈0,1δ∈0,2πAα,δξ∗.Proof.
SupposerAξm=uandωAξm=v for a particular m∈U; then,(18)∪α∈0,1δ∈0,2πAα,δξ∗m=Supα∈0,1δ∈0,2πAα,δξ∗m=maxSupα∈0,uδ∈0,vAα,δξ∗m,Supα∈u,1δ∈v,2πAα,δξ∗m.
Choose anyα∈u,1 and δ∈v,2π, then rAξm=u<αandωAξm=v<δ. Therefore, Aα,δξ∗m=0ei0. On the contrary, for any choice of α∈0,u and β∈0,v, we have rAξm=u≥αandωAξm=v≥δ.
Therefore,Aα,δξ∗m=αeiδ, which implies that ∪α∈0,1δ∈0,2πAα,δξ∗m=Supα∈0,uδ∈0,vαeiδ=ueiv=μAξm. The following result describes the decomposition of Aξ as a union of A+α,δξ∗.Theorem 11.
(second decomposition theorem). For everyAξ∈FξU, Aξ=∪α∈0,1δ∈0,2πA+α,δξ∗.Proof.
SupposerAξm=uandωAξm=v for a particular m∈U, then(19)∪α∈0,1δ∈0,2πA+α,δξ∗m=Supα∈0,1δ∈0,2πA+α,δξ∗m=maxSupα∈0,uδ∈0,vA+α,δξ∗m,Supα∈u,1δ∈v,2πA+α,δξ∗m=Supα∈0,uδ∈0,vαeiδ=ueiv=μAξm.
The decomposition ofξ CFS Aξ as a union of level sets can be established by the following result.Theorem 12.
(third decomposition theorem). For everyAξ∈FξU, Aξ=∪α∈ΩAξδ∈ΩAξAα,δξ∗.Proof.
The proof is analogous to that of Theorem10.
The following example illustrates the algebraic fact stated in first decomposition theorem.Example 1.
Consider theξ-CFS:(20)Aξ=0.2ei0.5πm1+0.4eiπm2+0.6ei1.2πm3+0.8ei1.8πm4.
Forα=0.2 and δ=0.5π, ξ-CFS Aα,δξ∗ with respect to ξ-CFS Aξ is given by(21)A0.2,0.5πξ∗=0.2ei0.5πm1+0.2ei0.5πm2+0.2ei0.5πm3+0.2ei0.5πm4.
Corresponding toα=0.4 and δ=π, we have(22)A0.4,πξ∗=0ei0m1+0.4eiπm2+0.4eiπm3+0.4eiπm4.
Corresponding toα=0.6 and δ=1.2π,(23)A0.6,1.2πξ∗=0ei0m1+0ei0m2+0.6ei1.2πm3+0.6ei1.2πm4.
Also, forα=0.8 and δ=1.8π,(24)A0.8,1.8πξ∗=0ei0m1+0ei0m2+0ei0m3+0.8ei1.8πm4.
Consequently,Aξ=∪Aα,δξ∗.
In the following example, we apply the concept ofξ-CFS to judge the performance of an artist in an art competition.Example 2.
LetX = {a, b, c, d, e, f} be the list of 10 artists competing in an art competition. After the initial screening based on sketch designing, the performance of each artist is given in Table 1.
The graphical interpretation of the above performance of the artists is displayed in Figure1.
Letξ=αeiβ be a parameter to select a candidate based on the performance to draw a sketch of a healthy environment. The judgment procedure has two phases α and β, where α denotes the level of roughness of the drawing and β denotes the use of inappropriate color in the drawing. Table 2 indicates the performance of the artists after the first phase for α=0.7.
The graphical interpretation of the qualified artists of stage one is presented in Figure2.
Table3 indicates the performances of the qualified artists for phase two.
The graphical interpretation of the artists of stage two is presented in Figure3.
The following outcomes indicate the performance of the qualified artists after phase two forβ=0.5π are ωAξa=0.5πa dm ωAξh=0.4π and final score is μAξa=0.2ei0.5π and μAξh=0.3ei0.4π. At this stage, we will use the score function to compare the performance of the artists. For this, we may take a=0.2276 and h=0.2998. The above information shows that artist “h” may be considered the best of all the artists.Table 1
Performance of all artists after initial screening.
ArtistsPerformance of the artistsrAa0.2rAb0.91rAc0.5rAd0.7rAe0.4rAf0.61rAg0.72rAh0.3rAi0.8rAj0.9Figure 1
A graphical overview of Table1.Table 2
Performance of the artists after the first phase.
ArtistsRoughness of the sketchrAξa0.2rAξc0.5rAξd0.7rAξe0.4rAξf0.61rAξh0.3Figure 2
A graphical interpretation of Table2.Table 3
Performance of the artists after the second phase.
ArtistsPerformance of the artistsωAa0.5πωAc0.6πωAd1.5πωAe0.6πωAf0.7πωAh0.4πFigure 3
A graphical interpretation of Table3.


## 4. Algebraic Attributes ofξ-Complex Fuzzy Subgroups

In this section, we innovate the notion ofξ-CFSG defined on ξ-CFS and establish fundamental algebraic characteristics of this phenomenon.Definition 15.
A homogeneousξ-CFS Αξ of a group G is called ξ-complex fuzzy subgroup (ξ-CFSG) if Αξ admits the following conditions:(1)
μΑξmn≥minμΑξm,μΑξn(2)
μΑξm−1≥μΑξm, ∀m,n∈G
The family of allξ-CFSG defined on the group G is denoted by FξG.Proposition 2.
Eachξ-CFSG (G) Αξ satisfies the following properties:(1)
μAξm≤μAξe(2)
μAξmn−1=μAξe⟹μAξm=μAξn, ∀m,n∈GProof.
(1)
Letm∈G; then,(25)μΑξe=μΑξmm−1≥minμΑξm,μΑξm−1=μAξm.(2)
Letm,n∈G; then,(26)μAξm=μAξmn−1n≥minμAξmn−1,μAξn=minμAξe,μAξn=μAξn.
In the following result, we investigate the condition under which a given CFS isξ-CFSG.Proposition 3.
LetA be a CFS (G), such that μAm−1=μAm∀m∈G.
Moreover,ξ≤q, where q=infμAm:m∈G. Then, Aξ is ξ-CFSG (G).Proof.
By using the given conditions for anym∈G, we obtain μAm≥ξ. The application of Definition 7 in the above inequality yields that μAξm=ξ. Therefore, μAξmn=minμAmn,ξ and μAξmn≥minμAξm,μAξn, for all m,n∈G. Moreover, by using the given condition μAm−1=μAm, we get μAξm−1=μAξm. The following result shows that every CFSG is always ξ-CFSG.Proposition 4.
Every CFSGA is ξ-CFSG of a group G.Proof.
By using Definition7, for all m,n∈G, we have μAξmn=minμAmn,ξ. The application of Definition 13 in the above relation gives us μAξmn≥minμAξm,μAξn.
Moreover,(27)μAξm−1=minμAm−1,ξ≥minμAm,ξ=μAξm.
Hence,A is a ξ-CFSG (G).Remark 2.
The converse of Proposition4 does not hold in general. This algebraic fact may be viewed in the following example.Example 3.
The CFSA defined on a G=1,−1,i−i is given as(28)Am=0.2eiπ1+0.4eiπ−1+0.4ei1.2π−i+0.3ei0.9πi.
Theξ-CFSG (G) corresponding to the value ξ=0.1ei0.5π is given by(29)Aξm=0.1ei0.5π1+0.1ei0.5π−1+0.1ei0.5π−i+0.1ei0.5πi.
Moreover,A is not CFSG (G) as A does not satisfy Definition 4.
The following result indicates that intersection of any twoξ-CFSG is also ξ-CFSG.Proposition 5.
For any twoAξ, Bξ∈FξG, A∩Bξ=Aξ∩Bξ.Proof.
By using Proposition1, for any two elements m,n∈G,(30)μA∩Bξmn=μAξ∩Bξmn=minμAξmn,μBξmn.
The application of Definition13 in the above relation gives that(31)μΑ∩Βξmn=minμΑ∩Βξm,μΑ∩Βξn.
Moreover,(32)μΑ∩Βξm−1=μΑξ∩Βξm−1≥minμΑξm,μΒξm=μΑ∩Βξm.
This concludes the proof.Theorem 13.
Aξ is a ξ-CFSG (G) if and only if Aξ′ is a ξ-CFSG (G).Proof.
LetAξ be a ξ-CFSG. Then,(33)μAξ′mn=1−rAξmnei2π−ωAξmn.
By using Definition13, we obtain(34)μAξ′mn≥1−minrAξm,rAξnei2π−minωAξm,ωAξn=max1−rAξm,1−rAξneimax2π−ωAξm,2π−ωAξn≥min1−rAξm,1−rAξneimin2π−ωAξm,2π−ωAξn.
By using Definition8 (3) in the above relation, we get μAξ′mn=minμAξ′m,μAξ′n.
Moreover,(35)μAξ′m−1=1−rAξm−1ei2π−ωAξm−1=1−rAξmei2π−ωAξm=μAξ′m.
Conversely, letAξ′ be a ξ-CFSG. Assume that(36)μAξmn=rAξmneiωAξmn=1−1−rAξmnei2π−2π−ωAξmn≥1−min1−rAξm,1−rAξnei2π−min2π−ωAξm,2π−ωAξn=maxrAξm,rAξneimaxωAξm,ωAξn≥minrAξm,rAξneiminωAξm,ωAξnμAξmn=minμAξm,μAξn.
Moreover,(37)μAξm−1=1−1−rAξm−1ei2π−2π−ωAξm−1=rAξmeiωAξm=μAξm.Definition 16.
LetΑξ∈FξG, α∈0,1, and δ∈0,2π. Then, the subgroup Aα,δξ with rAξe≥α and ωeAξ≥δ is called the level subgroup of ξ-CFSG Αξ.
In the subsequent result, we establish necessary and sufficient condition for anξ-CFS to be ξ-CFSG.Theorem 14.
Aξ-CFS Aξ of G is a ξ-CFSG (G) if and only if each of its level set ΩAξ with rAξe≥α and ωeAξ≥δ is a subgroup of G.Proof.
SupposeΩAξ with rAξe≥α and ωeAξ≥δ is a subgroup of G. Assume that rAξm=α, rAξn=α1, ωAξm=δ, and ωAξn=δ1, for any elements m,n∈G. By using Definition 10 in the above relations, we have m∈ΩAξ and n∈ΩA1ξ. By applying Theorem 1 (2) for α<α1 and δ<δ1, we have n∈ΩAξ, since ΩAξ is a subgroup of G. Therefore, mn∈ΩAξ. It shows that rAξmn≥minrAξm,rAξn=α and ωAξmn≥minωAξm,ωAξn=δ.
In view of Definition10, we have(38)rAξm−1≥α=rAξm,ωAξm−1≥δ=ωAξm.
Consequently,Aξ is a ξ CFSG (G). Conversely, suppose Αξ∈FξG. Let ΩAξ be an arbitrary level subgroup of Αξ. Obviously, ΩAξ is nonempty as e∈ΩAξ, where e is the identity element of G. For any elements m,n∈ΩAξ and using the fact that Αξ∈FξG, we have rAξmn≥minrAξm,rAξn=α and ωAξmn≥minωAξm,ωAξn=δ. It follows that mn∈ΩAξ. Moreover, for any element m∈Aξ and using the fact that Αξ∈FξG, we have rAξm−1≥rAξm=α andωAξm−1≥ωAξm=δ. Therefore, m−1∈ΩAξ implying that ΩAξ is a subgroup of G.Definition 17.
LetAξ be a ξ-CFSG (G) and m∈G. Then, the ξ-complex fuzzy left coset of Aξ in G is represented by mAξand is given by mAξg=minμAm−1g,ξ:g∈G=μAξm−1g.
Similarly, one can define theξ-complex fuzzy right coset of Aξ in G.Definition 18.
Aξ-CFSG Aξ of a group G is ξ-complex fuzzy normal subgroup (ξ-CFNSG) of G if mAξ=Aξm, ∀m∈G.
The following result illustrates another characteristic ofξ-CFNSG.Proposition 6.
Everyξ-CFNSG Aξ admits the following property:(39)μAξmn=μΑξnmfor allm,n∈G.Proof.
By using Definition16, we have mAξ=Αξm,∀m∈G
By using Definition15, the above equation gives that(40)mAξn−1=Aξmn−1∀n∈G,μAξnm−1=μAξmn−1.
This shows thatμAξnm=μAξmn. In the following consequence, we explore the condition under which an ξ-CFSG is ξ-CFNSG (G).Proposition 7.
For anyAξ∈FξG with ξ≤q, where q=InfμAm:m∈G, then Aξ is a ξ-CFNSG (G).Proof.
By using the given condition for anym∈G, we have μAm≥ξ. The application of Definition 7 in the above inequality yields that μAξm=ξ.
Therefore,(41)Aξmg=mAξgfor allg∈G.
Hence,(42)Aξm=mAξfor allm∈G.
The following result shows that every CFNSG (G) isξ-CFNSG (G).Proposition 8.
Every CFNSG (G)Α is ξ-CFNSG (G).Proof.
By using Definition6 for element m∈G, we have mAg=Amg. By applying Definition 5, the above relation gives that μAm−1g=μAgm−1. So, minμAm−1g,ξ=minμAgm−1,ξ implying that mAξg=Aξmg. Consequently, mAξ=Aξm.Remark 3.
The converse of Proposition8 does not hold in general. This algebraic fact may be viewed in the following example.Example 4.
The CFNSGAdefined on a group G=a,b:a2=b2=ab2=1,ba=a2b is given by(43)Am=1ei2π1+0.9ei1.8πa+0.7ei1.7πa2+0.5ei1.6πb+0.4eiπab+0.3ei0.7πa2b.
Theξ-CFNSG (G) corresponding to the value ξ=0.3ei0.7π is given by(44)Aξm=0.3ei0.7π1+0.3ei0.7πa+0.3ei0.7πa2+0.3ei0.7πb+0.3ei0.7πab+0.3ei0.7πa2b.
Moreover,A is not CFNSG (G) because(45)μAab=0.4eiπ≠0.3ei0.7π=μAba.Theorem 15.
For any twoξ-CFNSG Aξ and Bξ, A∩Bξ=Aξ∩Bξ.Proof.
By using Proposition1 for any element m∈G, we have(46)μA∩Bξgm−1=μAξ∩Bξgm−1=minμAξgm−1,μBξgm−1.
The application of Definition16 in the above relation is given as(47)μA∩Bξgm−1=minμAξm−1g,μBξm−1g=μAξ∩Bξm−1g.
Thus,μA∩Bξm−1g=μA∩Bξgm−1.Theorem 16.
Aξ is a ξ-CFNSG if and only if Aξ′ is a ξ-CFNSG.Proof.
LetAξbe a ξ-CFNSG. Then,(48)mAξ′=μAξ′m−1g=1−rAξm−1gei2π−ωAξm−1g.
By using Definition16, we obtain(49)mAξ′=1−rAξgm−1ei2π−ωAξgm−1=μAξ′gm−1.
Hence,mAξ′=Aξ′m.
Conversely, letAξ′ be a ξ-CFNSG. Assume that(50)mAξ=μAξm−1g=rAξm−1geiωAξm−1g=1−1−rAξm−1gei2π−2π−ωAξm−1g=1−μAξ′m−1g=1−μAξ′gm−1=μAξgm−1.
Thus,mAξ=Aξm.Proposition 9.
LetAξbe a ξ-CFSG (G). Then, the set GΑξ=m∈G:μAξm=μAξe is a normal subgroup of G.Proof.
Obviously,GAξ≠∅ as e∈GAξ. By applying Definition 13 for any two elements m,n∈G, we have(51)μAξmn−1≥minμAξm,μAξn−1=μAξe.
This shows thatμAξmn−1≥μAξe, but μAξmn−1≤μAξe. Consequently, mn−1∈GAξ. Furthermore, in view of Definition 16 for any element m∈GΑξ and g∈G, we obtain(52)μAξg−1mg=μAξm=μAξe.
This implies thatg−1mg∈GΑξ. Hence, GΑξ is a normal subgroup of G.Proposition 10.
Everyξ-CFNSG satisfies the following relation;
IfmAξ=uAξ and nAξ=vAξ, then mnAξ=uvAξ.Proof.
SincemAξ=uAξ and nAξ=vAξ, therefore m−1u,n−1v∈GAξ.
Consider(53)mn−1uv=n−1m−1uv=n−1m−1unn−1v=n−1m−1unn−1v∈GAξ.
It follows thatmn−1uv∈GAξ. Consequently, mnAξ=uvAξ.Proposition 11.
Everyξ CFNSG Aξ admits the following characteristics:(1)
mAξ=nAξ if and only if m−1n∈GAξ(2)
Aξm=Aξn if and only if mn−1∈GAξProof.
(1)
LetmAξ=nAξand m,n∈GAξ. Then, by applying Definition 7, we have(54)μAξm−1n=minμAm−1n,ξ=mAξn.
By using the given condition in the above equations, we obtain(55)μAξm−1n=nAξn=minμAn−1n,ξ=minμAe,ξ.
So,μAξm−1n=μAξe, implying that m−1n∈GAξ. Conversely, suppose that m−1n∈GAξ. This implies that μAξm−1n=μAξe. By applying Definition 16 for any element z∈G, we have(56)mAξz=minμAm−1z,ξ=μAξm−1z=μAξm−1nn−1z.
By using Definition13 in the above equation, we obtain(57)mAξz=minμAξe,μAξn−1z=μAξn−1z.
Consequently,mAξz=nAξz. The remaining part can be proved as the first part.Definition 19.
For anyξ-CFNSG Aξ of G, we define the set of all ξ-complex fuzzy left cosets of G by Aξ as G/Aξ=mAξ:m∈G. This set forms a group under the following binary operation mAξnAξ=mnAξ. This particular quotient group is called quotient group of G by ξ-CFNSG Aξ.
In the following result, we establish a natural epimorphism between group and its quotient group defined in Definition17.Theorem 17.
For anyξ-CFNSG Aξ of G, there exist a natural epimorphism φ:G⟶G/Aξ, defined by m⟶mAξ,m∈G with kerφ=GAξ.Proof.
The surjectivity of the functionφ is quite obvious. Moreover, for any elements m,n∈G, we have φmn=mAξnAξ=φmφn. Therefore, φ is an epimorphism. Moreover, obvious φ is surjective. Now,(58)kerφ=m∈G:φm=eAξ=m∈G:mAξ=eAξ=m∈G:me−1∈GAξ=GAξ.
In the following result, we establish an isomorphic correspondence between quotient group ofG by ξ-CFNSG Aξ and quotient group G by GAξ.Theorem 18.
LetAξ be ξ-CFNSG and GAξ be normal subgroup of G. Then, there exist an isomorphism between G/Aξ and G/GAξ.Proof.
Define a mappingφ:G/Aξ⟶G/GAξ as φmAξ=mGAξ. For any xAξ,yAξ∈G/Aξ, we have(59)φmAξnAξ=φmnAξ=mnGAξ=mGAξnGAξ=φmAξφnAξ.
This shows thatφ is homomorphism.
Moreover, for anymAξ,nAξ∈G/Aξ, we have(60)φmAξ=φnAξ,mGAξ=nGAξ,n−1mGAξ=GAξ,which shows that n−1mGAξ∈GAξ.
By applying Theorem16 in the above relation yields that mAξ=nAξ. Moreover, the surjective case is quite obvious. Consequently, φ is an isomorphism between G/Aξ and G/GAξ.


## 5. Conclusion

In this paper, we first present theξ-CFS which is completely a new notion. We have utilized this phenomena to define the α,δ-cut sets and strong α,δ-cut sets and have proved the representation of an ξ-CFS in the framework of these sets. Moreover, the notions of ξ-CFSG and level subgroups of these groups have also been defined in this article. In addition, a necessary and sufficient condition for an ξ-CFS to be a ξ-CFSG has also been investigated. Moreover, an isomorphism has been established between the quotient groups of a group G by its ξ-CFNSG and a normal subgroup GAξ.



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