Algebraic Properties of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 38.4212 15.2545" width="38.4212pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-246" d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"/></g><g transform="matrix(.017,0,0,-0.017,16.887,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,23.676,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g><g transform="matrix(.017,0,0,-0.017,32.224,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Complex Fuzzy Subgroups

Al-harshni, Ahad Abdullah; Alghazzawi, Dilshad

doi:https://doi.org/10.1155/2022/1426724

Mathematical Problems in Engineering

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Volume 2022 | Article ID 1426724 | https://doi.org/10.1155/2022/1426724

Algebraic Properties of -Complex Fuzzy Subgroups

Ahad Abdullah Al-harshni¹and Dilshad Alghazzawi¹

Academic Editor: Musavarah Sarwar

Received28 May 2022

Revised21 Jul 2022

Accepted17 Aug 2022

Published29 Sept 2022

Abstract

This paper defined the notion of -complex fuzzy sets, -complex fuzzy subgroupoid, and -complex fuzzy subgroups and described important examples under -complex fuzzy sets. Additionally, we discussed the conjugacy class of group with respect to -complex fuzzy normal subgroups. We define -complex fuzzy cosets and elaborate the certain operation of this analog to group theoretic operation. We prove that factor group with regard to -complex fuzzy normal subgroup forms a group and establishes an ordinary homomorphism from group to its factor group with regard to -complex fuzzy normal subgroup. Moreover, we create the -complex fuzzy subgroup of factor group.

1. Introduction

The study of group theory has a crucial orbit in mathematics owing to its utilitarian applications in different scientific domains including cryptography, mathematical biology, and chaos theory. The extensive understanding of the group allows us to evaluate the dynamic behavior of the real-world physical problems that have been effectively employed in mathematical problem modeling to tackle many complicated problems in the field of science and technology. On the other hand, the theory of fuzzy set plays a significant role for dealing uncertainty and vagueness. Zadeh [1] first proposed the concept of fuzzy sets and their operations. Subsequently, many researchers have analyzed various aspects of this theory and its applications. This theory also plays a vital role for studying differential equations. Many differential equations exhibit uncertainty and vagueness in their solutions [2]. Fuzzy set theory becomes more important to deal this type of information. Imai et al. [3] proposed the study of BCK/BCI-algebras in 1966 as a generalized conviction of set-theoretic difference. Rosenfeld [4] built up the structure of fuzzy subgroups on fuzzy sets in 1971. In 1979, fuzzy groups were redefined by Anthony and Sherwood [5]. The notions of inclusion, intersection, union, convexity, relation, etc., are extended to such sets, and different properties of these convictions in the context of fuzzy sets are established in [6]. Liu [7] introduced the concept of fuzzy invariant subgroups. In 1988, Choudhury [8] introduced the notion of fuzzy homomorphism between two groups and studied its effect on fuzzy subgroups. Mashour et. al. depicted many different key properties of fuzzy subgroups in [9]. Filep [10] extended the structure and construction of fuzzy subgroups of groups in 1992. Many researchers actively engaged the development of fuzzy set in group theory [11]. Gupta and Qi [12] presented the some typical -operators and reviewed -norm, -conorm, and negation function under -operators. Malik and Mordeson [13] defined the concept fuzzy subgroups of Abelian groups and discussed their various algebraic properties. Mishref [14] depicted the fuzzy normal series of finite groups. The idea of complex fuzzy set was introduced by Ramot et al. [15, 16] in 2002. In 2009, Zhang [17] et al. developed the various algebraic setups of complex fuzzy sets. In 2009, a new structure and construction of -fuzzy groups were described by Solairaju [18]. Al-Husban and Salleh [19]elaborated the notion of complex fuzzy hypergroups based on complex fuzzy spaces. The new applications of complex fuzzy sets in metric spaces, graph theory, group theory, and ring theory were introduced in [20–23]. Ma et al. [24] presented some new mathematical operations and laws of complex fuzzy set such as absorption law, involution law, symmetrical difference formula, simple difference, and disjunctive sum. Moreover, they developed a new algorithm using complex fuzzy sets for applications signals. Trevijano and Elorza [25] initiated the new concept of annihilator of fuzzy subgroups and discussed their various algebraic properties. The recent applications of complex fuzzy sets in ring theory and BCk/BCI algebras may be viewed in [26, 27]. Imtiaz [28] et al explored the new structure of -complex fuzzy sets and -complex fuzzy subgroups. Many authors [29–34] proposed several interesting techniques and approaches to solve complicated systems in fuzzy group theory, fuzzy ring theory, and fuzzy fractional calculus whereas the computational effects are very vague and straightforward. Gulzar [35] et al. presented the novel concept of complex fuzzy subfields. Verma [36] et al. discussed a systematic review on the advancement in study of fuzzy variational problems. Alolaiyan et al. [38] developed the structure of (α, β)-complex fuzzy subgroups.

In this article, we present a new concept -complex fuzzy set -CFS. First, we introduce some basic notions and some important properties which are needed in our paper. Second, in section three, we define -complex fuzzy subgroups -CFSGs and -complex fuzzy normal subgroups and study their properties. Finally, we define the -complex fuzzy cosets, and we obtain the quotient group regarding to -complex normal fuzzy group -CFNSG.

2. Preliminaries

In this section, we recall important concepts of complex fuzzy set and complex fuzzy subgroup.

Definition 1. (see [1]). A fuzzy subset is just like a function from a universal set to a unit interval [0, 1].

Definition 2. (see[15]). A CFS of the universe of discourse is mapping from nonempty set to unit disk and is described by the rule , where is set of complex numbers. The is complex membership function of CFS , where .

Definition 3. (see [22]). Let be a underlying universe and be fuzzy subset. Then, -fuzzy subset of is described as

Definition 4. (see [22]). Let be a group. A -fuzzy set of is a -fuzzy subgroup of if(1) for all (2) for all

Definition 5. (see [22]). Let be a underlying universe where and be CFSs of . Then,(1)A CFS is homogeneous CFS if , if and only if (2)A CFS is homogeneous CFS with if, such that if and only if

Definition 6. (see [17]). Let and be a CFSs of set . Then, the following set-theoretic operation is defined as(1)(2)

Definition 7. (see [22]). Let be a group and be a homogeneous CFS of . Then, is said to be CFSG of if the given below axioms are satisfied.(1)(2) , for all

Definition 8. (see [22]). Let be group. A CFS of is called a CFNSG of if: for all .

3. -Complex Fuzzy Subgroups

In this section, we define -CFSG and study their properties. We investigate the properties of -CFNSGs and erect the quotient group with respect to -CFNSGs.

Definition 9. Let be complex fuzzy set of group . For any and , then the complex fuzzy set is an -complex fuzzy set of with regard to complex fuzzy set and is defined as .

Example 1. Consider a universal set . The complex fuzzy set of is described as . In the view Definition 9, we get -complex fuzzy set of with respect to complex fuzzy set , for the value of , and , as follows: .

Definition 10. Let and be two -CFSs of . Then, A CFS is homogeneous -CFS if for all , we have if and only if . A -CFS is homogeneous -CFS with if for all , we have if and only if .

Remark 1. If and in the above definition, we obtain a classical complex fuzzy set.

Remark 2. Let and be two -CFSs of . Then, .

Definition 11. Let be an -CFS of group , for and . A -complex fuzzy subgroupoid of is a fuzzy algebraic structure which satisfies the following condition:

Example 2. Consider the group of fourth roots of unity under usual multiplication. DefineTake with and . Then, for each . Thus, is -complex fuzzy subgroupoid of .

Definition 12. Let be an CFS of group for and . A -CFSG of the group is a fuzzy algebraic structure that satisfies the following axioms:(1) (2) for all

Example 3. Consider the dihedral group . Let be CFS of such thatDefine for the value and as follows:Take , , and , we haveMoreover,Similarly, both conditions of Definition 12 are satisfied for all elements of . Hence, is CFSG of .

Remark 3. If be an CFS of the group for . Then,

Theorem 1. Let be an CFSG of the group . Then,(1)(2) which implies that

Proof. The proof of this theorem is obvious.

Theorem 2. Let be finite group. If is an complex fuzzy subgroupoid of , then is -CFSG of .

Proof. Assume that is a member of , thus, the order of is a finite number such that , where is the identity element of the group . We know that by the application of Definition 11. Then, we have

Theorem 3. Let be a group. If be an -CFSG of and for some , , then for all .

Proof. Given that . Then, from Theorem 2, we haveConsiderNow, assume thatAgain, from Theorem 2, we haveTherefore, we obtainFrom (12) and (15), we have

Theorem 4. .If is CFSG of the group , then is an -CFSG of .

Proof. Assume that is a CFSG of , for all . ConsiderFurther, we assume that

Theorem 5. Let and be two -CFSGs of , then is also -CFSGs of .

Proof. Given that and be two -CFSGs of , for any . ConsiderFurthermore,Consequently, is of .

Remark 4. The converse of Theorem 10 may not be true.

Example 4. Consider a permutation group . Consider two -CFSGs and of , we take are described asThis implies thatTake and . Therefore,
and .
Hence, this proves the claim.

Definition 13. Let be an -CFSG of the group . Then, -CFS of is said to be a -complex fuzzy left coset of established by and . It is defined asOn the same way, we describe the -complex fuzzy right coset is described as

Example 5. Take a group of order 8. Describe -CFSG of , and , such thatFrom Definition 13, we have . Hence, the -complex fuzzy left coset of in for is defined asIn the same way, we can define -complex fuzzy right coset of , for all .

Definition 14. Let be group and be an -CFSG of . Then, is said to be a complex fuzzy normal subgroup (invariant) -CFNSG if . Equivalently, -CFSG is -CFNSG of the group if , for all .

Remark 5. Let be an -CFNSG of the group . Then, for all .

Theorem 6. Every CFNSG of the group is an -CFNSG of .

Proof. Suppose that are elements of . Then,This implies thatwhich implies thatThis implies that . Consequently, is -CFNSG of .

Theorem 7. Let be an -CFSG of the group . Then, remains the same in the conjugacy class of if and only if is an -CFNSG.

Proof. Suppose that is an -CFNSG of the group . Then,Conversely, assume that remains the same in every conjugate class of . Then,

Theorem 8. Let be a group and be an -CFSG of . Then, is an -CFNSG if and only if .

Proof. Take is an -CFNSG of . Let be two members of . Assume thatConversely, suppose that . Let Now, we present two potential cases:

Case 1. If
Then, we obtain , which means that remains a constant, and the result is valid trivially.

Case 2. If .
Then, from (36), we haveFrom Equations (35) and (37), we haveHence, is a constant

Theorem 9. If is an -complex fuzzy normal subgroup of , Then, the set is an invariant subgroup of .

Proof. We have because . Let be any members. Consider which means that . However, . Therefore, . It means that ; hence, . Furthermore, . However, . Thus, is subgroup of group . Let and . We know . This implies that . Therefore, is invariant subgroup.

Theorem 10. Let is an -CFNSG of . Then,(i) if and only if (ii) if and only if

Proof. (i)For any , we have . Consider Therefore, . Conversely, let implies that . Consider Exchange the function of and , and we get . Therefore, .(ii)In the same way, we can show that as part (i).

Theorem 11. Let be a group and be an -CFNSG of and let and be any members in . If and . Then, .

Proof. We have and . This means that . Consider . As is normal subgroup of . Thus, . As result, .

Theorem 12. Assume that is the assemblage of all -complex fuzzy cosets of is an -CFNSG of the group . Therefore, we can define a well-defined binary operation of and is described as .

Proof. We have and for any . Let by any element. Then, = = . ConsiderTherefore, is well-defined operation and satisfied the associative property on . Furthermore, , and this means that is an identity element of . Obviously, the inverse of each member of exists if , and then, there exists an element such that . Consequently, is a group.

Lemma 1. Let be group then we can define to be a classical homomorphism from onto and is described by with the kernel .

Proof. Let be an arbitrary member of , and then,Therefore, is a homomorphism. In the same way, is also a homomorphism

Theorem 13. Let be a normal subgroup of . If is -CFSG, then the -CFS of is also -CFSG of , where .

Remark 6. If is an -CFSG of , , and , for all , then .

4. Conclusion

We have presented a new theory of -CFSs and -CFSGs as a powerful extension of complex fuzzy sets and complex fuzzy subgroups. Moreover, we have created the -CFSG of the factor group. Each concept introduced in this paper is explained with clear examples. This algebraic structure will play an important role to handle many other group theory problems under a new complex fuzzy environment [37, 38].

Data Availability

No real data were used to support this study. The data used in this study are hypothetical and anyone can use them by citing this article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The authors would like to thank the Deanship of Scientic Research of King Abdulaziz University, Jeddah, Saudi Arabia, for technical and financial support.

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Copyright © 2022 Ahad Abdullah Al-harshni and Dilshad Alghazzawi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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