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

Authors: Doha A. Kattan; Maria Amin; Abdul Bariq
Journal: Journal of Function Spaces
 (2022)

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## 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.


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## Body

## 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. Section2 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.
LetX be non-empty set. An interval-valued fuzzy set M defined on X is given by =m,ML,MU,∀m∈X, where ML and MU are two fuzzy sets of X such that ML≤MU, for all m∈X.
On the other hand, an interval-valued fuzzy setIVFS of X is specified as MX:X⟶Int0,1, where Int0,1 is the set of all intervals within 0,1, and is expressed as M=ML,MU such that ML≤MU.Definition 2.
LetK be an ordinary set. Then, P:K⟶0,1×0,1 is designed by P=m,MPm,NPm|m∈K, where MP:K⟶Int0,1 is designed by MP=MPL,MPU, where MPL≤MPL and NP:K⟶Int0,1is designed byNP=NPL,NPL, where NPL≤NPL and SupMPm+SupNPm≤1.Definition 3.
AnIVIFSP of group G is known as an IVIFSG of group G if it satisfies the following axioms: μPUmn−1≥minμPUm,μPUn,μPLmn−1≥minμPLm,μPLn and νPUmn−1≤maxνPUm,νPUn,νPLmn−1≤maxνPLm,νPLn,∀m,n∈G.Theorem 1.
LetP an IVIFSG of a group G and m∈G; then, μPUmn=μPUm,μPLmn=μPLm and νPUmn=νPUm,νPLmn=νPLm for all n∈G if and only if μPUn=μPUe,μPLn=μPLe and νPUn=νPUe,νPLn=νPLe.Proof.
Assume thatμPUmn=μPUm,μPLmn=μPLm and νPUmn=νPUm,νPLmn=νPLm for all n∈G. By replacing n by e, we have the required result.
Conversely, ifμPUn=μPUe,μPLn=μPLe. Since P is IVIFG, μPUn≤μPUe,μPLn≤μPLe and νPUn≥νPUe,νPLn≥νPLe for all n∈G. Now μPUmn−1≥minμPUm,μPUn, μPLmn−1≥minμPLm,μPLn.
We have(1)μPUmn≥μPUn,μPLmn≥μPLn.
ButμPUn=μPUnmm−1≥minμPUm,μPUnm, μPLn=μPLnmm−1≥minμPLm,μPLnm, and this shows that(2)μPUn≥μPUnm,μPLn≥μPLnm,for allm∈G.
From (1) and (2), we have μPUmn=μPUn, μPLmn=μPLn.
Similarly, we can show thatνPUnm=νPUn,νPLnm=νPLn.Definition 4.
LetP be an IVIFSG of a group G and m1 be an element of the group. The interval-valued intuitionistic fuzzy right coset of IVIFSGP of G is defined as(3)μm1Pg=μPUgm1−1,μPLgm1−1=μPgm1−1for allg∈G,νm1Pg=νPUgm1−1,νPLgm1−1=νPgm1−1.Definition 5.
AnIVIFSGP is known as an IVIFNSG of group G, if μPUmm−1n=μPUn, μPLmm−1n=μPLn and νPUn=νPUnmm−1,νPLn=νPLnmm−1, for all m,n∈G.Definition 6.
Consider anIVIFSGP of a group G, which is finite and m∈G. Then, the interval-valued intuitionistic fuzzy order (IVIFO) of m is named as IVIFOPm and is defined as
IVIFOPm=Qm, where(4)Qm=r∈G:μPUr≥μPUm,μPLr≥μPLmandνPUr≤νPUm,νPLr≤νPLm.
The algebraic information can be observed in the following example.Example 1.
LetG=e,a,a2,b,ab,a2b be a symmetric group of order 6. Then, an IVIFSGP of G is defined as(5)P=e,0.6,1,0,0.4>,<a,0.6,1,0,0.4>,<a2,0.6,1,0,0.4>,<b,0.3,0.35,0.5,0.55>,<ab,0.3,0.35,0.5,0.55>,<a2b,0.3,0.35,0.5,0.55.
Clearly,IVIFOPe=IVIFOPa=3,IVIFOPa2=3,IVIFOPb=IVIFOPab=IVIFOPa2b=6.Theorem 2.
Qm forms a subgroup of G.Proof.
Asm∈Qm, Qm is a non-empty set. By Definition 6, for arbitrary two elements r,q∈Qm, we have μPUr≥μPUm,μPLr≥μPLmandνPUr≤νPUm,νPLr≤νPLm.Also,μPUq≥μPUm,μPLq≥μPLmandνPUq≤νPUm,νPLq≤νPLm
SinceP is an IVIFSG, μPUrq−1≥minμPUr,μPUq, μPLrq−1≥minμPLr,μPLq and νPUrq−1≤maxνPUr,νPUq,νPLrq−1≤maxνPLr,νPLq which implies that μPUrq−1≥μPUm,μPLrq−1≥μPLmandνPUrq−1≤νPUm,νPLrq−1≤νPLm. Thus, rq−1∈Qm. Consequently, Qm is a subgroup of G.Corollary 1.
Assume that there exists anIVIFSGP of a group G; then, the IVIFO of any element of P divides G’s order.Proof.
By Theorem2 and Lagrange’s theorem, anyone can show that the IVIFO of any element of IVIFSG always divides group G’s order.Theorem 3.
LetP be an IVIFSG of a group G and e≠m∈G. Then, IVIFOPe≤IVIFOPm.Proof.
Letr∈Qm; then, μPUr=μPUm,μPLr=μPLmandνPUr=νPUm,νPLr=νPLm. This means that μPUr≥minμPUm, μPLr≥minμPLm and νPUr≤maxνPUm,νPLr≤maxνPLm, for all m∈G. Thus, r∈Qm. Consequently, Qe⊆Qm and IVIFOPe≤IVIFOPm.
The next result produces a relation between theIVIFO of any element of P and the order of that element in G.Theorem 4.
LetP be an IVIFSG of a group G and m∈G; then, Om divides IVIFOPm.Proof.
Assume thatOm=l and consider a subgroup K=<m:ml=e> of G. In view of Definition 6, we have m2∈Qm, and similarly, we can have m3,m4,m5,…,ml∈Qm. This indicates that K⊆Qm. Consequently, K forms subgroup of Qm and K divides Qm. This means that K divides IVIFOPm, andt therefore Om divides IVIFOPm.Definition 7.
TheIVIFO of IVIFSGP of G is denoted by IVIFOP and can be obtained by computing the greatest common divisor of the IVIFO of all elements of P.Example 2.
LetG=e,a,a2,b,ab,a2b be a symmetric group of order 6. An IVIFSGP of G is defined as(6)P=e,0.6,1,0,0.4>,<a,0.4,0.5,0.3,0.4>,<a2,0.4,0.5,0.3,0.4>,<b,0.3,0.35,0.5,0.55>,<ab,0.3,0.35,0.5,0.55>,<a2b,0.3,0.35,0.5,0.55.
Clearly,IVIFOPe=1,IVIFOPa=3,IVIFOPa2=3,IVIFOPb=IVIFOPab=IVIFOPa2b=6.
TheIVIFO of P in G is 1.
In the following result, we prove the condition thatμPUml≥μPUm,μPLml≥μPLmandνPUml≤νPUm,νPLml≤νPLm.Theorem 5.
LetP an IVIFSG of a group G and m∈G; then, μPUml≥μPUm,μPLml≥μPLmandνPUml≤νPUm,νPLml≤νPLm where l is an integer.Proof.
This result is clear forl=0 and 1. For l=2,(7)μPUm2≥μPUm⋅m≥minμPUm,μPUm=μPUm.
Assume the statement is true forn<l.
Now,(8)μPUmn+1=μPUmn⋅m≥minμPUmn,μPUm=μPUm,which completes the induction.
Ifl<0, then(9)μPUml=μPUml−1=μPUm−l≥μPUm.
Similarly,μPLml≥μPLm.
Therefore, we can easily proveνPUml≤νPUm,νPLml≤νPLm, for any integer l.Remark 1.
IfOm,l=1, then μPUml=μPUm,μPLml=μPLmandνPUml=νPUm,νPLml=νPLm, for any integer l.Theorem 6.
LetIVIFOPm=r and r,s=1,r,s∈Z and m∈G. Then, μPUms=μPUm,μPLms=μPLmandνPUms=νPUm,νPLms=νPLm.Proof.
We are aware that ifr,s=1, then ar+bs=1, for a,b∈Z. So,(10)μPUm=μPUmar+bs=μPUmra,msb≥minμPUmra,μPUmsb=minμPUe,μPUms≥μPUms.
ButμPUms≥μPUm.
Consequently,μPUms=μPUm.
Similarly, we can easily prove for the lower case.
Therefore, we can proveνPUms=νPUm,νPLms=νPLm.Theorem 7.
Letr,s∈Z such that μPUms=μPUe, μPLms=μPLeandνPUmr=νPUe,νPLmr=νPLe, for all m∈G. Then, both rands divide IVIFOPm.Proof.
Letm be a non-identity element and IVIFOPm=x. Suppose s does not divide x; then, s,x=1.
By Theorem6, we have μPUms=μPUm,μPLms=μPLm. But μPUms=μPUe, μPLms=μPLe, so m=e.
As such, it is a contradiction, and thuss divides IVIFOPm.
Similarly, we can easily prover divides IVIFOPm.Theorem 8.
IfIVIFOPm=r, then IVIFOPms=IVIFOPm/r,s for some integer s.Proof.
Suppose thatIVIFOPms=y.
Consider(11)μPUmsr/d=μPUmrs/d≥μPUes/d=μPUe.
Similarly, we can easily prove for the lower case.
Therefore,νPUmsr/d=νPUe. We can also prove for the lower limit.
By Theorem7, we have that r/d divides y.
Moreover, sincer,s=d, ar+bs=d, for some a,b∈Z. Now,(12)μPUmyd=μPUmyar+bs=μPUmyarmybs≥minμPUmray,μPUmysb≥minμPUmr,μPUmys≥minμPUmr,μPUmsy=minμPUe,μPUe=μPUe.
We know thatμPUmnxy≤μPUe, and hence μPUmnxy=μPUe.
Similarly, we can easily prove for the lower case.
Therefore,νPUmyd=νPUe,νPLmyd=νPLe. By applying Theorem 7, we get yd/r.
Consequently,y=r/d.Theorem 9.
LetP be an IVIFSG of a group G and m∈G; then, IVIFOPm−1=IVIFOPm.Proof.
SinceP is IVIFSG of G, μPUm=μPUm−1,μPLm=μPLm−1 and νPUm=νPUm−1,νPLm=νPLm−1, for all m∈G. This means that Qm=Qm−1; as such, Qm=Qm−1. In addition, we know that IVIFOPa=Qa, for all a. Therefore, IVIFOPm−1=IVIFOPm.
In the following theorem, we illustrate another form ofIVIFO of elements of IVIFNSG.Theorem 10.
LetP be an IVIFSGP of a group G and n∈G be any fixed element; then, IVIFOPmnm−1=IVIFOPn for all m∈G.Proof.
By Definition5, we have μPUmnm−1=μPUn, μPLmnm−1=μPLn and νPUn=νPUmnm−1,νPLn=νPLmnm−1. So, Qn=Qmnm−1.
Consequently,IVIFOPmnm−1=IVIFOPn.Theorem 11.
LetP be an IVIFSG of a group G; then, IVIFOPmn=IVIFOPnm, for all m,n∈G.Proof.
SinceIVIFOPmn=IVIFOPn−1nmn=IVIFOPn−1nmn, by Theorem 10, IVIFOPn−1nmn=IVIFOPnm.
So, we haveIVIFOPmn=IVIFOPnm.Theorem 12.
LetIVIFOPm=x, for all m∈G. If i≡jmodx, where i,j∈Z, then IVIFOPmi=IVIFOPmj.Proof.
Assume thatIVIFOPmi=s and IVIFOPmj=r. Since i=j+qx for some q∈Z,(13)μPUmir=μPUmj+qxr=μPUmjrmxqr≥minμPUmjr,μPUmxqr≥minμPUe,μPUe=μPUe.
As such,s/r. Similarly, we can prove μPLmir=μPLe and r/s. Hence, IVIFOPmi=IVIFOPmj.Theorem 13.
Assume that for allm,n∈GIVIFOPm,IVIFOPn=1,mn=nmandPmn=Pe. Then, Pm=Pn=Pe.Proof.
Proof. Suppose thatIVIFOPm=x and IVIFOPn=y. By Theorem 5, we have μPUmyny=μPUe,μPLmyny=μPLe. By Theorem 7, we have μPUmy=μPUny=μPUe,μPLmy=μPLny=μPLe.
Similarly, we can easily prove for the non-membership function.Theorem 14.
IfIVIFOPm,IVIFOPn=1 and mn=nm for all m,n∈G, then IVIFOPmn=IVIFOPm×IVIFOPn.Proof.
Proof. SupposeIVIFOPmn=z, IVIFOPm=x and IVIFOPn=y. Now consider(14)μPUmnxy=μPUmxynxy≥minμPUmxy,μPUnyx,=minμPUe,μPUe≥μPUe.
We know that(15)μPUmnxy≤μPUe.
From (14) and (15), we have μPUmnxy=μPUe.
Similarly,μPLmnxy=μPLe.
Likewise,νPUmnxy=νPUe,νPLmnxy=νPLe.
By Theorem7, we have the relation(16)xyz.
Sincex,y=1, x/z or y/z.
Assume thatx/z; then,(17)IVIFOPmz=xx,z.
By using Theorem7,(18)IVIFOPnz=yy,z.
By equations (17) and (18), we have IVIFOPmz,IVIFOPnz=1.
From Theorem13 and equations (17) and (18), we have Pm=Pn=Pe. This means that(19)zxy.
From (16) and (19), we get the result.Remark 2.
LetP and Q be any two IVIFSG of group G. If P⊆Q and Pe=Qe, then IVIFOPm/IVIFOQm for all m∈G.Theorem 15.
IfP and Q are any two IVIFSG of a group G such that P⊆Q and Pe=Qe, then IVIFOPm/IVIFOQm.Proof.
AsIVIFOP and IVIFOQ are finite, the IVIFO of every element of P and Q is finite. Let M and N be the sets consisting of IVIFOs of elements in P and Q, respectively. Remark 2 gives that IVIFOPm divides IVIFOQm for all m∈G. Then, gcd of every elements of M divides the gcd of every elements of N. As a result, IVIFOPm/IVIFOQm.


## 3. Properties ofIVIFO of Elements in IVIFSG in a Finite Cyclic Group

This section examines theIVIFO of elements of IVIFSG in cyclic groups and their elementary properties.Lemma 1.
Assume that there exists anIVIFSGP of a cyclic group G and m,n are any two generators of G; then, IVIFOPm=IVIFOPn.Proof.
Assume thatOG=r. Since mandn are two generators of G, mr=nr=e.
Since for somes∈Z we have n=ms, r,s=1. Next, by Theorem 6, IVIFOPm=IVIFOPn=IVIFOPms.Theorem 16.
LetP be an IVIFSG on a finite cyclic group G. The following results hold for all m,n∈G.(1)
IfOm=On, then IVIFOPm=IVIFOPn.(2)
IfOm divides On, then IVIFOPm divides IVIFOPn.Proof.
Letx be a generator of G; then, m=xr,n=xs and IVIFOPx=u where r,s,u∈Z. We have Om=n/n,r and On=n/n,s. In view of Theorem 8, we have IVIFOPm=u/u,r and IVIFOPn=u/u,s. From Theorem 3, we have n/u.(1)
SinceOm=On, then Oxr=Oxs. This shows that r,n=s,n. From the above relation, we have r,u=s,u. Consequently, IVIFOPm=IVIFOPn.(2)
SinceOm divides On, s,n/r,n. This implies that s,u/r,u In addition, as n/u, IVIFOPm divides IVIFOPn.Corollary 2.
LetP be an IVIFSG of a finite cyclic group G of order q. If IVIFOPn=IVIFOPm, then Pm=Pn for all m,n∈G.Corollary 3.
LetP an IVIFSG of a group G of order q. If IVIFOPn divides IVIFOPm, then μPUn≥μPUm,μPLn≥μPLmandνPUn≤νPUm,νPUn≤νPUm.Theorem 17.
LetP be an IVIFSG of a group G and K=<x> be a cyclic subgroup of G. For all m,n∈K, if Om divides On, then μPUn≥μPUm,μPLn≥μPLmandνPUn≤νPUm,νPLn≤νPLm.Proof.
SupposeOm=r and On=qr for some q∈N. Let m=xuandn=xv for some u,v∈N. It follows that xur=e=xvqr. Thus, m=nq. As such, μPUm=μPUnq≥μPUn. Similarly, we can prove for the lower limit of a non-membership function.
Likewise,νPUn≤νPUm,νPUn≤νPUm.
In the following example, we show that Theorem17 is not valid for all m,n∈G.Example 3.
LetG=e,1,2,3,4,5 be a group of order 6. Then, and IVIFSGP of G is defined as(20)P=e,0.6,1,0,0.4>,<1,0.4,0.5,0.3,0.4>,<2,0.4,0.5,0.3,0.4>,<3,0.3,0.35,0.5,0.55>,<4,0.3,0.35,0.5,0.55>,<5,0.3,0.35,0.5,0.55.
We know thatO4=3andO1=6 in G.
Clearly,O4 divides O1, but μPU1≥μPU4,μPL1≥μPL4andνPU1≤νPU4,νPL1≤νPL4.


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

This part recapitulates the concept pertaining to the index ofIVIFSG. In addition, interval-valued intuitionistic fuzzification of Lagrange’s theorem of IVIFSG is studied.Theorem 18.
Assume that there exists anIVIFSGP of a group G and ϖ is the set of all interval-valued intuitionistic fuzzy left cosets IVIFLC of G by P. Then, ϖ forms a group with(21)mPonP=mnP,for allm,n∈G.
Define a mappingP:ϖ⟶0,1 by(22)PmP=Pm,for allm∈G.
Then,P∧ is an IVIFSG of ϖ.Proof.
Letm,n,mo,no∈G such that(23)mP=moPandnP=noP.
Then, we must show that(24)mPonP=moPonoP,mnP=monoP.
By Definition4,(25)μmnPUu=μPUun−1m−1,for allu∈G,μmonoPUu=μPUuno−1mo−1,for allu∈G.
Now,(26)μPUun−1m−1=μPUuno−1non−1m−1=μPUuno−1mo−1monon−1m−1≥minμPUuno−1mo−1,μPUmonon−1m−1.
Using Definition4 in (23) gives(27)μPUum−1=μPUumo−1,for allu∈G,and(28)μPUun−1=μPUun0−1,for allu∈G.
Now, replaceu by monon−1 in (27), and we have(29)μPUmonon−1m−1=μPUmonon−1mo−1.
Substituteu with no in (28), and we have(30)=μPUnon−1=μPUe.
ButμPUe≥μPUuno−1mo−1. Since P is IVIFSG, μPUu≥μPUe and νPUe≥νPUu for all u∈G. Thus, (26) now yields(31)μPUun−1m−1≥μPUuno−1mo−1.
Similarly,μPUun−1m−1≤μPUuno−1mo−1.
This shows thatμPUun−1m−1=μPUuno−1mo−1.
Consequently,μmnPUu=μmonoPUu for all u∈G.
The lower case can be proved in the same way.
Similarly, we can show that(32)νmnPUu=νmonoPUu,νmnPLu=νmonoPLu,for allu∈G.
This shows that this is a well-defined composition.
We can view that the inverse ofmP is m−1P, for m∈G.
Hence,ϖ is a group.
Now, letP∧mP,P∧nP∈P∧ where mP,nP∈ϖ.
Consider(33)μPU∧μmPUoμnPU=μPU∧μmnPU=μPUmn≥minμPUm,μPUn=minμPU∧μmPU,μPU∧μnPU.
Similarly, the lower case can be established.
As such,(34)νPU∧νmPUoνnPU≤maxνPU∧νmPU,νPU∧νnPU,νPL∧νmPLoνnPL≤maxνPL∧νmPL,νPL∧νnPL.
Moreover,(35)μPU∧μm−1PU=μPUm−1=μPUm=μPU∧μmPU.
The lower case can be proved in the same way.
Similarly,νPU∧νm−1PU=νPU∧νmPUandνPL∧νm−1PL=νPL∧νmPL.
This shows thatP∧ is a IVIFSG of ϖ.Definition 8.
Assume that there exists anIVIFSGP of a finite group G; define a mapping P∧:ϖ⟶0,1 by
P∧mP=Pm, for all m∈G, which is called an interval-valued intuitionistic fuzzy quotient group.Theorem 19.
LetP be an IVIFNSG. Then, establish a homomorphism t from G to ϖ defined by tm=mP for all m∈G with kernel G=x∈G:μPUx=μPUe,μPLx=μPLeandνPUx=νPUe,νPLx=νPLe.Proof.
Letm,n∈G. Then,(36)tmn=mnP=mPonP=tmtn,which indicates that t is a natural homomorphism.
Moreover,(37)Ker.t=x∈G:tx=P=x∈G:xP=P.=x∈G:xPy=Py,for ally∈Gx∈G:μxPUy=μPUy,μxPLy=μPLyandνxPUy=νPUy,νxPLy=νPLy,for ally∈G.
In view of Definition4, we have(38)Ker.t=x∈G:μPUyx−1=μPUy,μPLyx−1=μPLyandνPUyx−1=νPUy,νPLyx−1=νPLy,for  ally∈G.
Using Theorem2 in the above relation yields μPUx=μPUe,μPLx=μPLeandνPUx=νPUe,νPLx=νPLe.
Consequently,(39)Ker.t=x∈G:μPUx=μPUe,μPLx=μPLeandνPUx=νPUe,νPLx=νPLe.Remark 3.
Note thatKer.t=IVIFOP.Definition 9.
LetP be an IVIFSG of finite group G. Then, ϖ is called the index of IVIFSGP and is denoted by G:P.Theorem 20 (interval-valued intuitionistic fuzzification of Lagrange’s theorem).
IfP is an IVIFSG of a finite group G, then the index of IVIFSG of G divides the order of G.Proof.
By Theorem19, we have a homomorphism t from G to ϖ, where ϖ=mP,for allm∈G.
AsG is finite, it is trivial that ϖ is also finite.
Define(40)H∧=x∈G:xP=eP.
By Theorem19, we have H∧=x∈G:Px=Pe.
Now we partition groupG into disjoint union of cosets.
Consider(41)G=m1H∧∪m2H∧∪…∪mkH∧,where m1H∧=H∧. Now we prove that for each coset mjH∧ in relation (41), there exists an IVIF coset in ϖ; also, its corresponding counterpart is injective. Take a coset mjH∧. Let ℏ∈H∧; then,(42)tmjℏ=mjℏP=mjPℏP=mjPeP=mjP.
Thus,ϖ maps every element of mjH∧ into the interval-valued intuitionistic fuzzy coset mjP.
Now, we give a relation ƫ between setmjH∧:1≤j≤k and set ϖ by(43)TmjG=mjP,1≤j≤k.
The correspondence ƫ is injective.
For this, letmiP=mlP; then, ml−1miP=eP.
By using (40), we have ml−1mi∈H∧; this means that miH∧=mlH∧, and hence ƫ is injective.
It is clear from the above discussion thatG:H∧andG:P are equal, since G:H∧ divides OG.Corollary 4.
Assume that there exists anIVIFSGP of a finite group G; then, IVIFOP divides the order of G.The index ofIVIFSGP of a finite group G can be obtained from the following relation.Remark 4.
G:P=G/IVIFOP.
The algebraic information can be observed in the following examples.Example 4.
LetG=e,1,2,3,4,5 be a group of order 6. The IVIFSGP of G is defined as(44)P=e,0.6,1,0,0.4>,<1,0.4,0.5,0.3,0.4>,<2,0.4,0.5,0.3,0.4>,<3,0.3,0.35,0.5,0.55>,<4,0.3,0.35,0.5,0.55>,<5,0.3,0.35,0.5,0.55.
The set of all interval-valued intuitionistic fuzzy left cosets ofGbyP is given by(45)ϖ=Peg,P1g,P2g,P3g,P4g,P5g.
This means thatG:P=Cardϖ=6.Example 5.
LetG=1,i,−i,−1 be a cyclic group of order 4. The IVIFSGP of G is defined as(46)P=1,0.8,0.9,0,0.4>,<−1,0.7,0.8,0.3,0.4>,<i,0.5,0.6,0.5,0.55>,<−i,0.5,0.6,0.5,0.55.
The set of all interval-valued intuitionistic fuzzy left cosets ofGbyP is given by(47)ϖ=P−1g,Pig.
This means thatG:P=Cardϖ=2.


## 5. Conclusion

In this article, we have fostered the idea ofIVIFO 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.



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