On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups

In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (<inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG) and prove that every IVFSG is <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG. We use this ideology to define the concepts of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF cosets, <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG and investigate the condition under which a <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFS is <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> relative to the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFNSG and acquire a correspondence between each <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of its quotient group. Furthermore, we define the index of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG and establish the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-interval valued fuzzification of Lagrange’s theorem of any <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG of a finite group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>.


I. INTRODUCTION
In the late eighteenth century, Lagrange's Theorem appeared in the literature. It was basically discovered to resolve the problem of finding roots of the equation of degree greater than 4 and its association with symmetric volumes. In 1870, Lagrange's expressed a modification of this theorem. Pietro Abbati gave the first complete proof of this theorem about thirty years after the Lagrange's modification. This theorem has a significant role in the development of modern group theory. It is an incredible asset to investigate finite groups; as it gives an exact review about subgroups of a finite group. This theorem produces a very effective sign of Fermat's Little Theorem, which is very valuable in cryptography and numerous different fields.
The theory of fuzzy logic offers a mathematical method to apprehend the uncertainty related to human cerebral process like thoughtful and intellectual. It also handles issues of uncertainty and lexical imprecision. In this theory the The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau . elements of a universe are permitted to be partially accommodated by the set. It is determined by indeterminate boundaries. Consequently, fuzzy set follows infinite-valued logic. Practically, the accomplishment of usage of the fuzzy set hypothesis depends upon a decision of membership function that we make. In spite of this, there are many physical problems in which scientist don't have careful learning of the limit that should be taken. The limitation of this theory is the case when which we do not have exact information of the membership function. In these cases, it is reasonable to declare each component of the fuzzy set of membership grades by methods of interval. These perceptions rise the development of fuzzy sets called the theory of IVFS. IVFSs are basically used in medical diagnosis, approximate reasoning and image processing.
In 1965, the idea of fuzzy sets was firstly presented by Zadeh [1]. In 1967, fuzzy sets were refined in terms of L-fuzzy set by Gougen [2]. The author [3] utilized this thought to innovate the hypothesis of fuzzy subgroups in 1971. Later on, Anthony and Sherwood [4] reviewed fuzzy subgroups on the basis of the study of t-norm. Das [5] derived level subgroups of a fuzzy group. The notions of fuzzy normal VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ subgroups and fuzzy cosets were presented by Mukherjee and Bhattacharya [6]. Bhattacharya [7] developed numerous fundamental characterizations of fuzzy subgroups. Liu [8] commenced the study of fuzzy invariant subgroups and fuzzy ideals. For more detail about the development of fuzzy subgroups, we refer to [9]- [11] and [12]. Gupta and Qi [13] reviewed the theory of t-norm and t co-norm. Zadeh [14] proposed the concept of IVFS in 1975. An important interpretation of IVFS was given Turksen [15] in 1986. Gorzalczany [16] applied this particular concept to formulate an estimate technique for verbal choice into mathematical approximations. Atanassov and Gargov [17] initiated the study of IV intuitionistic fuzzy set in 1989. Roy and Biswas [18] described IVF relations and obtained their various important results in 1992. In 1995, the study of IVF-SGs was initiated by Liu [8]. Mondal and Samanta [19] characterized the topology of IVFS and talked about its properties in 1999. Later on, Gehrke et al. [20] investigated the theory of IVFS defined over t-norms in 2001. Lee [21] gave a detailed comparison of IVFS with bipolar VFS and intuitionistic fuzzy set in 2004. Mondal and Samanta [22] contemplated the connectedness in topology of IVFS in 2005. Jun et al. [23] utilized this phenomenon to characterize the thought of IVF pre-open sets and IVF α-open sets and examine their interrelationship in 2006. For more recent development in IVFSs, we refer to [24]- [26]. In 2012, a fuzzy diagnostic method based on IV intuitionistic fuzzy sets was recommended by Choi et al. [27]. Ahn et al. [28] presented an important application of this phenomenon in the medical diagnosis of headache. The author [29] introduced the concept of IVF coset in 2011. In 2013, Lee et al. [30] proved that an IVFSG cannot be visualized as the union of two suitable IVFSGs. Liu and Luo [31] gave useful methods to construct entropy of IVFS in 2015.
The rest of the paper is designed as: In the section 2 the ideas of IVFS and IVFSG are discussed. We present the concepts of -IVFS and -IVFSG and some of their basic algebraic properties in section 3. Section 4 deals with views of -IVF coset, -IVF quotient group and the index of -IVFSG. Furthermore, we conclude this section by establishing the -interval valued fuzzification of Lagrange's theorem of -IVFSG.

II. PRELIMINARIES
This section contains a brief review of the basic definitions of IVFS and IVFSG which are quite essential to understand the novelty of this article.
Definition 1 [12]: A fuzzy set is a function from a universe Z to a closed unit interval.
Definition 6 [12]: A fuzzy set M of group Z is called a fuzzy subgroup, if M admits the following conditions: Definition 7 [29]: An IVFS M of group Z is an interval valued fuzzy subgroup (IVFSG) of Z if M satisfies the following conditions. i Definition 8 [29]: Let M be an IVFSG and x 1 be an element of a group. The interval valued fuzzy right coset of an IVFSG M of Z is defined as: in shading pictures by choosing a suitable value of . Our methodology depends on a solitary picture, where the amplified picture is gained by joining various developed rectangles. To make each rectangle we utilize the -IVFS that we have recently connected with the picture, keeping up the power of the first pixel in the focal point of the rectangle and filling in the rest utilizing the connection between that pixel and its neighbors.
In the following theorem, we show that the intersection of any two -IVFS is a -IVFS.
Theorem 13: For any two -IVFS M and N of Z , Proof: In the view of definition (10), we have Theorem 15: Let M be a -IVFSG of Z and x 1 , In the view of definition (14), we have By the combination of (3.1) and (3.2), we obtain Similarly, the upper case can be proved.
ii. Note that, in the view of definition (14), we have This shows that M L (e) ≥ M L (x 1 ) , x 1 ∈ Z . The upper case can be proved in the same way.
The application of given condition in the above relation yields that Moreover, By using theorem (15)(i), in the above relation, we get The application of given condition in the above relation yields that The comparison of the relations (3.3) and (3.4) gives the required equality. Similarly, one can prove the above two inequalities for the upper case as well.
The following result illustrates the relationship between an IVFSG and -IVFSG of a group.
Theorem 16 : In the view of definition (14), for any x 1 , x 2 ∈ Z , we have Moreover, .
Similarly, one can prove the upper case for M , that is, and The following example shows that the converse of the previous theorem is not true.
In the following result, we show that any two -IVFSG obey the intersection property.
Theorem 18: Intersection of two -IVFSG is a -IVFSG of Z .
Proof: Let M and N be any two -IVFSG of Z . By definition (14), for each which implies that Moreover, Consequently, Similarly, one can prove the above two inequalities for the upper case, that is, . Thus, the intersection of any two -IVFSG is -IVFSG of Z . Remark 19: Union of two -IVFSG may not be a -IVFSG of a group Z .
Example 20: Consider IVFS M and N of group of integers Z under addition as follows: The -IVFSG M and N of Z corresponding to = [0.3, 0.7] are given by: The values of M (9) and N (2) The values of M ∪ N at x 1 = 9 and x 2 = 2 are given by It is quite evident from the above discussion that In the following theorem, we show that a group cannot be written as a union of two proper -IVFSG. Thus, in either case, we find contradiction.
can be established in the same way, that is, Moreover, for any Conversely, for any and M (x 2 ) = [ν 1 , ψ 1 ]. Then clearly, x 1 ∈ M [ν,ψ] and Then Moreover, By using the definition (14), in the above relation we obtain . Applying the given condition in the above relation yields that The comparison of relations (3.5) and (3.6), gives the required equality.
The upper case can be proved in the same way.

IV. -INTRTVAL VALUED FUZZIFICATION OF LAGRANGE'S THEOREM OF -INTERVAL VALUED FUZZY SUBGROUP
The idea of the index of -IVFSG is presented in this section. Furthermore, -interval valued fuzzification of Lagrange's theorem of -IVFSG is performed. Definition 25: Let M be a -IVFSG of a group Z and ∈ D (I).
For any x 1 ∈ Z , the -IVF right coset of M in Z is represented by M x 1 and is defined as: Likewise, the notion of -IVF left coset x 1 M can be established.
Theorem 26: Let M be a -IVFSG of Z and let g 1 , g 2 be any elements of Z , then Then g 1 M L (g 1 ) = g 2 M L (g 1 ) and g 1 M L (g 2 ) = g 2 M L (g 2 ).
In view of definition (25), we have M L g −1

Conversely, suppose that
Let x 1 ∈ Z , then in view of definition (25), we have Using the given condition in the above relation yields that The upper case can be proved in the same way. Consequently, g 1 M = g 2 M Theorem 27: Let M be a -IVFSG of Z . If M g 1 = M g 2 for any g 1 , g 2 ∈ Z , then M (g 1 ) = M (g 2 ).
Proof: Let g 1 , g 2 ∈ Z and M g 1 = M g 2 . Then M g 1 (g 2 ) = M g 2 (g 2 ) implying that M g 2 g −1 VOLUME 8,2020 This means that In view of (4.1) and (4.2), we obtain Similarly, the above equality can be obtained for the upper case. Definition 28: The -interval valued fuzzy normal subgroup ( -IVFNSG) of a given -IVFSG M of Z is defined as: Theorem 29: Let M be a -IVFNSG of a group Z , then for all elements in Z, M x 1 In the light of the definition (25), we have Proof: Let M ∈ -IVFNSG of Z and for any x 1 , Therefore, Similarly, the upper case can be established.
Consider z 1 ∈ Z , then which implies that By using same arguments, we have M U x −1 Moreover,

Using (4.3) and (4.4), we get
. Similarly, according to same argument, we have In the following theorem, we see that every IVFNSG is a -IVFNSG of Z . Theorem 31: Every IVFNSG is a -IVFNSG of Z .
Proof: Let M be an IVFNSG of Z and x 1 be a fixed element in Z , then The application of definition (25) yields that .
Theorem 32: Suppose that M ∈ -IVFNSG of Z and Z M is the set of all -IVF cosets of M in Z . Define a binary operation on Z M in the following way: forms a group.

This means that
Also, we know that, M L (e) ≥ M L gy −1 0 x −1 0 implying that Similarly, we have By comparing (4.5) and (4.6), we have In the light of same arguments, we have, Hence, Z M , is a group.
Note that Z M is known as the − IVF quotient group induced by M .
Theorem 33:Let M be a -IVFNSG of Z , and M : Then M is a -IVFSG of Z M .
Proof: In view of given mapping and the application of definition (14), we have Moreover, The upper case can be proved in the same way. This completes the proof. Note that, M is known as -IVF sub-quotient group determined by M . (4.7) We also know that (4.8) By the comparison of (4.7) and (4.8), we get Hence Z M is a subgroup of Z . Moreover, by applying theorem (30) and using the normality of M for any element x 1 ∈ Z M and x 2 ∈ Z , we have Similarly, one can established the upper case as well. Consequently, Theorem 35: Every -IVFNSG M of Z , admits the following properties: In view of definition (25), we have which shows that x −1 1 x 2 ∈ Z M . VOLUME 8, 2020 Conversely, let x −1 1 x 2 ∈ Z M . By applying definition (25) on x 1 M L for any element z 1 ∈ Z , we have which implies that The application of given condition in the above relation yields that The comparison of (4.9) and (4.10) gives the required equality for M L . Likewise, we can prove the upper case. Consequently, In view of definition (25), we have which means that x 1 x −1 2 ∈ Z M . Conversely, let x 1 x −1 2 ∈ Z M . By applying definition (25) on M L x 1 for any element z 1 ∈ Z , we have The application of given condition in the above relation yields that The comparison of relations (4.11) and (4.12) gives the required equality for M L . Likewise, the upper case can be proved in the similar way.
Proof: The application of theorem (35) and using the given condition on M , we have x −1 1 m and x −1 2 n ∈ Z M . Consider In the following result, we establish a natural homomorphism between groups Z and its quotient group by -IVFNSG M .
Theorem 37: Let M ∈ -IVFNSG and x 1 ∈ Z . The map which shows that ∅ is a natural homomorphism. Moreover, We obtain a correspondence between each -IVF(N)SG of Z M and -IVF(N)SG of Z in the following theorem. Similarly, the above relation can be obtained for the upper case of S .
which shows that S ∈ -IVFSG(Z). Moreover, it is quite easy to view that if S ∈ −IVFNSG Z M , then S is -IVFNSG of Z . Proof: A natural homomorphism π : Z → Z M is obtained by using theorem (37). Consider a subgroup H 1 of Z of the form By applying definition (25) on h 1 ∈ H 1 and g ∈ Z , we have Using the theorem (15)(i) in the above relation gives that M (h 1 ) = M (e), which shows that h 1 ∈ Z M and therefore (4.13) Now, for any element h 1 ∈ Z M and using the fact that Z M ≤ Z , we have In the light of theorem (24) for the elements h −1 1 and g belong to Z M , we obtain the following relation M h 1 = M e which means that h 1 ∈ H 1 and hence Therefore, in view of (4.13) and (4.14), we get Now we partition the group Z into disjoint union of cosets. Consider The correspondence π is one-to-one. Thus, Hence, π is one-to-one. It is quite clear form the above discussion that Z M = k which implies that Z M divides the |Z | as k divides |Z |.
The above algebraic fact is illustrated in the following example.
Example 41: Consider the finite group of order 6 as follows: This shows that S 3 M = 3.

V. CONCLUSION
In this paper, we initiated the study of -IVFSG and proved many algebraic characteristics of this notion. We defined -IVFNSG, a quotient group of Z by -IVFNSG and established numerous fundamental algebraic aspects of these concepts. Moreover, we obtained a correspondence between each -IVF(N)SG of a group Z and -IVF(N)SG of its quotient group. Furthermore, we performed the -interval valued fuzzification of Lagrange's theorem of -IVFSG of a finite group Z .
UMER SHUAIB was born in Faisalabad, in June 1980. He received the Ph.D. degree in the field of group theory from Quaid-e-Azam University, Islamabad, Pakistan, in 2016. He is currently working as an Assistant Professor of mathematics with Government College University, Faisalabad, Pakistan. His research interests include group theory and its generalization, fuzzy logic, and cryptography. He has several publications to his credit. He has supervised 22 M.S. theses so far.
ABDUL RAZAQ received the Ph.D. degree from the Department of Mathematics, Quaid-i-Azam University, Islamabad, in 2015. He is currently working as an Assistant Professor with the University of Education Lahore, Jauharabad Campus. His main research interests include algebra, analysis, and cryptography.
HANAN ALOLAIYAN received the Ph.D. degree in mathematics from King Saud University, Saudi Arabia. She is currently working as an Assistant Professor with the Department of Mathematics, King Saud University. She has published a number of research articles in reputable journals. Her research interests include algebra, analysis, and cryptography.
MUHAMMAD SHAHRAM SAIF was born in July 1995. He received the B.S. degree in mathematics from the University of Education Township Campus Lahore and the M.Phil. degree from Government College University Faisalabad. His area of specialization is interval valued fuzzy subgroups and its applications.
AYESHA RAFIQ is currently a Researcher in the field of group theory and generalizations. She is currently working as an Assistant Professor with the Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, Pakistan. Her interests include in the domain of the graph theory, cryptography, and the modular group.