[Retracted] Certain Structure of Lagrange’s Theorem with the Application of Interval-Valued Intuitionistic Fuzzy Subgroups

Abstract

This paper presents the concept of an interval-valued intuitionistic fuzzy subgroup defined on interval-valued intuitionistic fuzzy sets. We study some of the fundamental algebraic properties of interval-valued intuitionistic fuzzy cosets and interval-valued intuitionistic fuzzy normal subgroup of a given group. This idea is used to describe the interval-valued intuitionistic fuzzy order and index of interval-valued intuitionistic fuzzy subgroup. We have created numerous algebraic properties of interval-valued intuitionistic fuzzy order of an element. We also prove the interval-valued intuitionistic fuzzification of Lagrange’s theorem.

1. Introduction

The introduction of interval-valued intuitionistic fuzzy sets is based on the ideas of intuitionistic fuzzy sets and interval-valued fuzzy sets (IVIFSs). Zadeh [1] was the first to propose the concept of a fuzzy set in 1965. Rosenfeld [2] utilized this concept in 1971 to establish the concept of fuzzy groups. In the year 2000, Lee [3] described bipolar-valued fuzzy sets and their fundamental operations. In 2004, Lee [4] conducted a comparison of interval-valued fuzzy sets, IFSs, and bipolar fuzzy sets.

In 2009, Park et al. [5] investigated the IVIFS correlation coefficient and its application to multi-attribute group decision-making situations. In 2013, Chen and Li [6] used IVIFSs to evaluate students’ answer scripts. In 2013, Meng et al. [7] used an interval-valued intuitionistic fuzzy Choquet integral with respect to a generalized Shapley index to address the multi-criteria group decision-making problem. In 2013, Ye [8] used intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting to construct multi-attribute group decision-making procedures with unknown weights. In 2013, Zhang et al. [9] proposed an interval-valued intuitionistic fuzzy multi-attribute group decision-making method based on correlation coefficients. In 2014, Chen [10] presented using IVIFSs a prioritized aggregation operator-based approach to multi-criteria decision making. In 2014, Jin et al. [11] developed an interval-valued intuitionistic fuzzy continuous weighted entropy and used it to multi-criteria fuzzy group decision making. In 2014, Li [12] used interval-valued intuitionistic fuzzy information to solve decision-making difficulties in company financial performance assessment.

In 2014, Liu et al. [13] published a multi-attribute large-group decision-making method based on an interval-valued intuitionistic fuzzy principal component analysis model. In 2015, Chen and Chiou [14] published a multi-attribute decision-making method using IVIFSs. In 2015, Gupta et al. [15] developed a mixed solution technique for multi-criteria group decision making in an interval-valued intuitionistic fuzzy environment employing entropy/cross entropy. In 2015, Liu et al. [16] extended the Einstein aggregation procedures based on interval-valued intuitionistic fuzzy numbers and proved their use in group decision making.

In 2017, Chen and Huang [17] used interval-valued intuitionistic fuzzy values and linear programming to examine the multi-attribute decision-making problem. In 2017, Xian et al. [18] used IVIFSs and a weighted averaging operator to make group decisions. Shuaib et al. [19] characterized on r-interval-valued intuitionistic fuzzification of Lagrange’s theorem of r-intuitionistic fuzzy subgroups in 2017. Mu et al. [20] developed the concept of interval-valued intuitionistic fuzzy Zhengyuan aggregation operators and its application to multi-attribute decision-making problems in 2018. In 2018, Zhang [21] proposed the geometric Bonferroni means of interval-valued intuitionistic fuzzy numbers and their use in multi-attribute group decision making. In 2018, Khan and Abdullah [22] defined an interval-valued Pythagorean fuzzy grey relational analysis approach for multi-attribute decision making with incomplete weight information for multi-attribute decision making with missing weight information. In 2018, Xu [23] proposed a consensus model for interval-valued intuitionistic multi-attribute group decision making with few changes. In 2018, Gupta et al. [24] introduced the notion of multi-attribute group decision making in an interval-valued intuitionistic fuzzy environment using an extended TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) technique. In 2018, Qin et al. [25] proposed a novel technique based on ordered weighted averaging distance operators for interval-value intuitionistic fuzzy multi-criteria decision making with immediate probability. The VIKOR technique for industrial robot selection was presented by Narayanamoorthy et al. [26]. It is based on an interval-valued intuitionistic hesitant fuzzy entropy. Alolaiyan et al. [27] proposed the concept of t-intuitionistic fuzzification of Lagrange’s theorem of t-intuitionistic fuzzy subgroups in 2019. Hosinzadeh et al. [28] proposed an artificial intelligence-based prediction way to describe the flow of a Newtonian liquid/gas on a permeable flat surface in 2021. Ghasemi et al. [29] proposed a dual-phase-lag (DPL) transient non-Fourier heat transfer analysis of functional graded cylindrical material under axial heat flux.

This paper is organized as follows. Section 2 contains basic definitions of interval-valued intuitionistic fuzzy order of an element of interval-valued intuitionistic fuzzy subgroup and the related result which are very useful to build up the consequent investigation of this paper. We construct the algebraic properties of interval-valued intuitionistic fuzzy order of an element of interval-valued intuitionistic fuzzy subgroup of a finite cyclic group in Section 3. In Section 4, we extend the study of this notion to introduce interval-valued intuitionistic left cosets and index of interval-valued intuitionistic fuzzy subgroups. Moreover, we develop Lagrange’s theorem by using the notion of interval-valued fuzzy information and establish some key fundamental algebraic aspects.

2. Interval-Valued Intuitionistic Fuzzy Order of an Element of Interval-Valued Intuitionistic Fuzzy Subgroup

This section reviews some fundamental concept of IVIFSs and interval-valued intuitionistic fuzzy subgroup along with the relevant results.

Definition 1. Let be non-empty set. An interval-valued fuzzy set defined on is given by , where and are two fuzzy sets of such that , for all .
On the other hand, an interval-valued fuzzy set of is specified as , where is the set of all intervals within , and is expressed as such that .

Definition 2. Let be an ordinary set. Then, is designed by , where is designed by , where and , where and .

Definition 3. An of group is known as an of group if it satisfies the following axioms: and .

Theorem 1. Let an of a group and ; then, and for all if and only if and .

Proof. Assume that and for all . By replacing by , we have the required result.
Conversely, if . Since is , and for all . Now , .
We haveBut , , and this shows thatFrom (1) and (2), we have , .
Similarly, we can show that .

Definition 4. Let be an of a group and be an element of the group. The interval-valued intuitionistic fuzzy right coset of of is defined as

Definition 5. An is known as an of group , if , and , for all .

Definition 6. Consider an of a group , which is finite and . Then, the interval-valued intuitionistic fuzzy order (IVIFO) of is named as and is defined as
, whereThe algebraic information can be observed in the following example.

Example 1. Let be a symmetric group of order 6. Then, an of is defined asClearly, .

Theorem 2. forms a subgroup of .

Proof. As , is a non-empty set. By Definition 6, for arbitrary two elements , we have
Since is an , , and which implies that . Thus, . Consequently, is a subgroup of .

Corollary 1. Assume that there exists an of a group ; then, the of any element of divides ’s order.

Proof. By Theorem 2 and Lagrange’s theorem, anyone can show that the of any element of always divides group ’s order.

Theorem 3. Let be an of a group and . Then, .

Proof. Let ; then, . This means that , and , for all . Thus, . Consequently, and .
The next result produces a relation between the of any element of and the order of that element in .

Theorem 4. Let be an of a group and ; then, divides .

Proof. Assume that and consider a subgroup of . In view of Definition 6, we have , and similarly, we can have . This indicates that Consequently, forms subgroup of and divides This means that divides , and therefore divides .

Definition 7. The of of is denoted by and can be obtained by computing the greatest common divisor of the of all elements of

Example 2. Let be a symmetric group of order 6. An of is defined asClearly, .
The of in is 1.
In the following result, we prove the condition that .

Theorem 5. Let an of a group and ; then, where is an integer.

Proof. This result is clear for and . For ,Assume the statement is true for
Now,which completes the induction.
If , thenSimilarly, .
Therefore, we can easily prove , for any integer .

Remark 1. If , then , for any integer .

Theorem 6. Let and and . Then, .

Proof. We are aware that if , then , for . So,But .
Consequently, .
Similarly, we can easily prove for the lower case.
Therefore, we can prove .

Theorem 7. Let such that , , for all . Then, both divide .

Proof. Let be a non-identity element and . Suppose does not divide ; then, .
By Theorem 6, we have . But , , so .
As such, it is a contradiction, and thus divides .
Similarly, we can easily prove divides .

Theorem 8. If , then for some integer

Proof. Suppose that .
ConsiderSimilarly, we can easily prove for the lower case.
Therefore, . We can also prove for the lower limit.
By Theorem 7, we have that divides
Moreover, since , , for some . Now,We know that and hence
Similarly, we can easily prove for the lower case.
Therefore, . By applying Theorem 7, we get .
Consequently, .

Theorem 9. Let be an of a group and ; then, .

Proof. Since is of , and , for all . This means that ; as such, . In addition, we know that , for all Therefore, .
In the following theorem, we illustrate another form of of elements of .

Theorem 10. Let be an of a group and be any fixed element; then, for all .

Proof. By Definition 5, we have , and . So, .
Consequently, .

Theorem 11. Let be an of a group ; then, for all .

Proof. Since , by Theorem 10, .
So, we have .

Theorem 12. Let for all . If , where , then .

Proof. Assume that and . Since for some As such, . Similarly, we can prove and . Hence,

Theorem 13. Assume that for all . Then, .

Proof. Proof. Suppose that and . By Theorem 5, we have . By Theorem 7, we have .
Similarly, we can easily prove for the non-membership function.

Theorem 14. If and for all , then .

Proof. Proof. Suppose , and . Now considerWe know thatFrom (14) and (15), we have .
Similarly, .
Likewise, .
By Theorem 7, we have the relationSince , or .
Assume that ; then,By using Theorem 7,By equations (17) and (18), we have .
From Theorem 13 and equations (17) and (18), we have . This means thatFrom (16) and (19), we get the result.

Remark 2. Let and be any two of group . If and , then for all .

Theorem 15. If and are any two of a group such that and , then .

Proof. As and are finite, the of every element of and is finite. Let and be the sets consisting of s of elements in and , respectively. Remark 2 gives that divides for all . Then, gcd of every elements of divides the gcd of every elements of As a result, .

3. Properties of of Elements in in a Finite Cyclic Group

This section examines the of elements of in cyclic groups and their elementary properties.

Lemma 1. Assume that there exists an of a cyclic group and are any two generators of ; then, .

Proof. Assume that . Since are two generators of , .
Since for some we have , . Next, by Theorem 6, .

Theorem 16. Let be an on a finite cyclic group . The following results hold for all .(1)If , then .(2)If divides , then divides .

Proof. Let be a generator of ; then, and where . We have and . In view of Theorem 8, we have and . From Theorem 3, we have .(1)Since , then . This shows that . From the above relation, we have . Consequently, .(2)Since divides , . This implies that In addition, as , divides .

Corollary 2. Let be an of a finite cyclic group of order If , then for all .

Corollary 3. Let an of a group of order If divides , then .

Theorem 17. Let be an of a group and be a cyclic subgroup of . For all , if divides , then .

Proof. Suppose and for some . Let for some . It follows that . Thus, . As such, . Similarly, we can prove for the lower limit of a non-membership function.
Likewise, .
In the following example, we show that Theorem 17 is not valid for all .

Example 3. Let be a group of order 6. Then, and of is defined asWe know that in .
Clearly, divides , but .

4. Interval-Valued Intuitionistic Fuzzification of Lagrange’s Theorem

This part recapitulates the concept pertaining to the index of . In addition, interval-valued intuitionistic fuzzification of Lagrange’s theorem of is studied.

Theorem 18. Assume that there exists an of a group and is the set of all interval-valued intuitionistic fuzzy left cosets of by Then, forms a group withDefine a mapping byThen, is an of

Proof. Let such thatThen, we must show thatBy Definition 4,Now,Using Definition 4 in (23) givesandNow, replace by in (27), and we haveSubstitute with in (28), and we haveBut . Since is , and for all . Thus, (26) now yieldsSimilarly, .
This shows that .
Consequently, for all .
The lower case can be proved in the same way.
Similarly, we can show thatThis shows that this is a well-defined composition.
We can view that the inverse of is for .
Hence, is a group.
Now, let where .
ConsiderSimilarly, the lower case can be established.
As such,Moreover,The lower case can be proved in the same way.
Similarly, .
This shows that is a of .

Definition 8. Assume that there exists an of a finite group ; define a mapping by
for all , which is called an interval-valued intuitionistic fuzzy quotient group.

Theorem 19. Let be an . Then, establish a homomorphism from to defined by for all with kernel .

Proof. Let . Then,which indicates that is a natural homomorphism.
Moreover,In view of Definition 4, we haveUsing Theorem 2 in the above relation yields .
Consequently,

Remark 3. Note that .

Definition 9. Let be an of finite group . Then, is called the index of and is denoted by .

Theorem 20 (interval-valued intuitionistic fuzzification of Lagrange’s theorem). If is an of a finite group , then the index of of divides the order of .

Proof. By Theorem 19, we have a homomorphism from to where .
As is finite, it is trivial that is also finite.
DefineBy Theorem 19, we have .
Now we partition group into disjoint union of cosets.
Considerwhere . Now we prove that for each coset in relation (41), there exists an coset in ; also, its corresponding counterpart is injective. Take a coset . Let ; then,Thus, maps every element of into the interval-valued intuitionistic fuzzy coset
Now, we give a relation ƫ between set and set byThe correspondence ƫ is injective.
For this, let ; then, .
By using (40), we have ; this means that , and hence ƫ is injective.
It is clear from the above discussion that are equal, since divides .

Corollary 4. Assume that there exists an of a finite group ; then, divides the order of .

The index of of a finite group can be obtained from the following relation.

Remark 4. .
The algebraic information can be observed in the following examples.

Example 4. Let be a group of order 6. The of is defined asThe set of all interval-valued intuitionistic fuzzy left cosets of is given byThis means that .

Example 5. Let be a cyclic group of order 4. The of is defined asThe set of all interval-valued intuitionistic fuzzy left cosets of is given byThis means that .

5. Conclusion

In this article, we have fostered the idea of of an element and have demonstrated the basic algebraic characteristic of these phenomena. Besides, we have created numerous algebraic properties of interval-valued intuitionistic fuzzy order of an element and have presented the interval-valued intuitionistic fuzzification of Lagrange theorems.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This study was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge with thanks the DSR for technical and financial support.