# Onq-Rung Orthopair Fuzzy Subgroups

Authors: Asima Razzaque; Abdul Razaq
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Theq-rung orthopair fuzzy environment is an innovative tool to handle uncertain situations in various decision-making problems. In this work, we characterize the idea of a q-rung orthopair fuzzy subgroup and examine various algebraic attributes of this newly defined notion. We also present q-rung orthopair fuzzy coset and q-rung orthopair fuzzy normal subgroup along with relevant fundamental theorems. Moreover, we introduce the concept of q-rung orthopair fuzzy level subgroup and proved related results. At the end, we explore the consequence of group homomorphism on the q-rung orthopair fuzzy subgroup.


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## Body

## 1. Introduction

In classical fuzzy set theory, a fuzzy subset of a crisp setS is represented by a function from S to 0,1⊆ℝ. The inequalities and equations are used to define operations and characteristic. The original notion of the fuzzy set was proposed in 1965 by Zadeh [1]. Since then, it has been used in almost every field of science especially where mathematical logic and set theory are significantly involved. A fuzzy subset R of a crisp set S is an object s,μRs:s∈S such that μR:S⟶0,1 is called membership mapping of R and μRs is known as a degree of membership of s in R. One can see that fuzzy sets are the extensions of characteristic functions of classical sets, by expanding the range of the function from 0,1 to 0,1. After the proposal of fuzzy sets, a lot of theories have been put forward to handle uncertain and imprecision circumstances. Some of these theories are expansions of fuzzy sets, whereas others strive to cope with uncertainties in another appropriate manner. Atanassov [2] introduced an intuitionistic fuzzy set (IFS) which is the generalization of fuzzy set. An intuitionistic fuzzy subset R of a crisp set S is an object s,μRs,νRs:s∈S, where μR:S⟶0,1 and νR:S⟶0,1 are membership and nonmembership functions, respectively, such that μRs+νRs≤1 for all s∈S. Compared with classical fuzzy sets, the positive and negative membership functions of intuitionistic fuzzy sets ensure its effective handling of uncertain and vague situations in physical problem, especially in the field of decision-making [3–6]. In 2013, Yager [7] generalized intuitionistic fuzzy sets by presenting the idea of Pythagorean fuzzy set (PFS). The Pythagorean fuzzy subset R of a crisp set S is an object s,μRs,νRs:s∈S, where μR:S⟶0,1 and νR:S⟶0,1 are membership and nonmembership functions, respectively, such that μRs2+νRs2≤1 for all s∈S. This concept is designed to convert uncertain and vague environment in the form of mathematics and to find more effective solutions of such real-world problems [8–11]. Although Pythagorean fuzzy subsets solve different types of real-life problems in an efficient way but even then, there is a room for improvement because there exists so many cases where Pythagorean fuzzy subsets fail to work. For example, if positive and negative membership values proposed by a decision-maker are 0.75 and 0.85, respectively, then 0.752+0.852>1; therefore, Pythagorean fuzzy subsets fail to deal with such problems. In order to find a reasonable solution of such kinds of situations, Yager defines the notion of q-rung orthopair fuzzy set (q-ROFS), where q is a natural number [12]. The q-rung orthopair fuzzy subset R of a crisp set S is an object s,μRs,νRs:s∈S, where μR:S⟶0,1 and νR:S⟶0,1 are positive and negative membership functions, respectively, such that μRsq+νRsq≤1 for all s∈S. In order to find real-world applications of q-ROFSs, we suggest reading [13, 14].Theory of groups is one of the prominent branches of mathematics with numerous applications in physics [15], chemistry [16], cryptography [17–19], differential equations [20], and graph theory [21, 22]. Rosenfeld [23] initiated the study of fuzzy subgroups. Since then, many mathematicians studied classical group theoretic results in different fuzzy environments. In [24], Das presented a comprehensive study of level subgroups of a fuzzy subgroup. Sherwood defined products of fuzzy subgroups and proved important results related to this notion [25]. Bhattacharya and Mukherjee [26] presented the idea of fuzzy relations on fuzzy subgroups and presented some new results in this direction. Choudhury et al. presented a study on fuzzy subgroups and fuzzy homomorphism in [27]. Some new results on normal fuzzy subgroups have been proven in [28]. Kumar [29] discussed some properties of fuzzy cosets and fuzzy ideals. In [30], some problems related to equivalence relation on fuzzy subgroups have been studied. Biswas [31] initiated the work on intuitionistic fuzzy subgroups in 1989. Hur et al. [32] defined the notion of intuitionistic fuzzy coset and discussed some of its algebraic characteristics. In [33], Sharma defined direct product of intuitionistic fuzzy subgroups. To find more about intuitionistic fuzzy subgroups, we recommend reading [34–38]. Recently, Bhunia et al. [39] defined Pythagorean fuzzy subgroups and explored different attributes of this concept.Considering the above literature and the significance ofq-rung orthopair fuzzy sets and theory of groups, this article reveals the study of q-rung orthopair fuzzy subgroups (q-ROFSGs). The basic purpose and the principal contribution of this work are to
(1)
discuss various important algebraic attributes ofq-rung orthopair fuzzy subgroups(2)
define the concepts ofq-rung orthopair fuzzy coset and q-rung orthopair fuzzy normal subgroup along with the study of relevant fundamental theorems(3)
introduce the idea ofq-rung orthopair fuzzy level subgroup and prove some important results of this notion


## 2. Theq-Rung Orthopair Fuzzy Subgroups

In this section, some important algebraic attributes ofq-rung orthopair fuzzy subgroups will be discussed. We start this section with the definition of q-rung orthopair fuzzy subgroup (q-ROFSG).Definition 1.
LetG be a group; then, a q-ROFS R=s,μRs,νRs of G is called a q-rung orthopair fuzzy subgroup (q-ROFSG) of G if the following conditions hold:
(i)
μRs1s2q≥minμRs1q,μRs1q and νRs1s2q≤maxνRs1q,νRs2q for all s1,s2∈G(ii)
μRs−1q≥μRsq and νRs−1q≤νRsq for all s∈GTheorem 2.
LetR=s,μRs,νRs:s∈G,μRsq+νRsq≤1 be a q-ROFSG of G. Then, the following conditions are true:
(i)
μReq≥μRsq and νReq≤νRsq for all s∈G(ii)
μRs−1q=μRsq and νRs−1q=νRsq for all s∈GProof.
(i)
Lets∈G; then, μRs−1sq≥minμRs−1q,μRsq⟹μReq≥minμRs−1q,μRsq=μRsq⟹μReq≥μRsq. Similarly, we can show that νReq≤νRsq(ii)
SinceμRs−1q≥μRsq and νRs−1q≤νRsq for all s∈G, therefore μRs−1−1q≥μRs−1q and νRs−1−1q≤νRs−1q which means that μRsq≥μRsq and νRsq≤νRs−1q. Thus, μRs−1q=μRsq and νRs−1q=νRsq for all s∈GThe following theorem shows that every Pythagorean fuzzy subgroup (PFSG) ofG is q-ROFSG of G.Theorem 3.
LetG be a group and P=s,μPs,νPs:s∈G,μPs2+νPs2≤1 be a PFSG of G. Then, P is a q-ROFSG of G.Proof.
Lets1,s2∈G; then
(1)μPs1s22≥minμPs12,μPs22,νPs1s22≤maxνPs12,νPs22,μs−12≥μs2,νs−12≤νs2.
This implies that(2)μPs1s2q≥minμPs1q,μPs2q,(3)νPs1s2q≤maxνPs1q,νPs2q,(4)μs−1q≥μsq,(5)νPs−1q≤νPsq.
SinceμPs12,μPs22,μPs2,νPs12,νPs22,νPs2∈0,1, therefore for all q>2, we have μPs1q≤μPs12,μPs2q≤μPs22,μPsq≤μPs2,νPs1q≤νPs12,νPs2q≤νPs22, and νPsq≤νPs2. Thus,
(6)μPs1q+νPs1q≤1,(7)μPs2q+νPs2q≤1,(8)μPsq+νPsq≤1.
The inequalities (2)–(8) reveal that P is a q-ROFSG of G.The next example provides evidence of invalidity of the converse of the above theorem.Example 1.
Consider dihedral groupD4, that is,
(9)D4=s1,s2:s12=s24=s1s22=e=e,s1,s2,s22,s23,s2s1,s22s1,s23s1.
One can easily verify that(10)R=e,0.95,0.10,s22,0.80,0.45,s2,0.80,0.75,s23,0.80,0.75,s2s1,0.80,0.78,s22s1,0.80,0.78,s23s1,0.80,0.78,s1,0.80,0.78
is 3-ROFSG ofD4, but it is not a PFSG of D4 as 0.802+0.782>1.Theorem 4.
Aq-ROFS R=s,μRs,νRs of group G is a q-ROFSG of G if and only if μPst−1q≥minμRsq,μRtq and νRst−1q≤maxνRsq,νRtq for all s,t∈G.Proof.
LetR=s,μRs,νRs:s∈G,μRsq+νRsq≤1 be a q-ROFSG of G. Then, for all s,t∈G, μPst−1q≥minμRsq,μRt−1q=minμRsq,μRtq and νRst−1q≤maxνRsq,νRt−1q=maxνRsq,νRtq.
Conversely, suppose thatμPst−1q≥minμRsq,μRtq and νRst−1q≤maxνRsq,νRtq for all s,t∈G. Then, μPstq=μPst−1−1q≥minμRsq,μRt−1q=minμRsq,μRtq. Thus,
(11)μPstq≥minμRsq,μRtq.
Similarly,(12)νRstq≤maxνRsq,νRtq.
Next,μPs−1q=μPes−1q≥minμReq,μRsq=μRsq, that is,
(13)μRs−1q≥μRsq.
Similarly,(14)νRs−1q≤νRsq.
The inequalities (11)–(14) show that R is a q-ROFSG of G.Theorem 5.
LetR1=s,μR1s,νR1s and R2=s,μR2s,νR2s be two q-ROFSGs of G; then, R1∩R2 is a q-ROFSG of G.Proof.
Suppose thatR1 and R2 are two q-ROFSGs of G. Then, for all s1,s2∈G, we have
(15)μR1∩R2s1s2−1q=minμR1s1s2−1q,μR2s1s2−1q≥minminμR1s1q,μR1s2q,minμR2s1q,μR2s2q=minminμR1s1q,μR2s1q,minμR1s2q,μR2s2q=minμR1∩R2s1q,μR1∩R2s2q.
That is,(16)μR1∩R2s1s2−1q≥minμR1∩R2s1q,μR1∩R2s2q.
Similarly,(17)μR1∩R2s1s2−1q≤maxμR1∩R2s1q,μR1∩R2s2q.
The application of (16) and (17) together with the theorem give R1∩R2 which is a q-ROFSG of G.Theorem 6.
LetR=s,μRs,νRs be a q-ROFSG of G. Then, μRsmq≥μRsq and νRsmq≤νRsq for all s∈G and m∈ℕ.Proof.
We will use mathematical induction to prove this theorem. Supposes∈G; then, μRs2q=μRssq≥minμRsq,μRsq=μRsq.
Therefore, the inequality is valid form=2. Assume that the inequality holds for m=n−1, that is, μRsn−1q≥μRsq. Then, μRsnq=μRssn−1q≥minμRsq,μRsn−1q=μRsq. Thus, by mathematical induction, we have μRsmq≥μRsq for all m∈ℕ.
Similarly, we can showνRsmq≤νRsq for all m∈ℕ.Theorem 7.
LetR=s,μRs,νRs be a q-ROFSG of G. If μRs1≠μRs2 and νRs1≠νRs2 for some s1,s2∈G, then μRs1s2q=minμRs1q,μRs2q and νRs1s2q=maxνRs1q,νRs2q.Proof.
Suppose that for somes1,s2∈G, we have μRs1>μRs2; then, obviously μRs1q>μRs2q.
Consider(18)μRs2q=μRs1−1s1s2q≥minμRs1−1q,μRs1s2q=minμRs1q,μRs1s2q.
SinceμRs1q>μRs2q, therefore from relation (18), we obtain
(19)μRs2q≥μRs1s2q.
Also,μRs1s2q≥minμRs1q,μRs2q=μRs2q, that is,
(20)μRs1s2q≥μRs2q.
From (19) and (20), we have
(21)μRs1s2q=μRs2q=minμRs1q,μRs2q.
Similarly, the result can be proven ifμRs2>μRs1.
Next, assume thatνRs1<νRs2; therefore, νRs1q<νRs2q. Then,
(22)νRs2q=νRs1−1s1s2q≤maxνRs1−1q,νRs1s2q=maxνRs1q,νRs1s2q.
SinceνRs1q<νRs2q, therefore from relation (22), we obtain
(23)νRs2q≤νRs1s2q.
Also,νRs1s2q≤maxνRs1q,νRs2q=νRs2q, that is,
(24)νRs1s2q≤νRs2q.
From (23) and (24), we have
(25)νRs1s2q≤νRs2q=maxνRs1q,νRs2q.
Similarly,νRs1s2q=maxνRs1q,νRs2q, if νRs1<νRs2.Theorem 8.
Lete denote the identity element of G and R=s,μRs,νRs be a q-ROFSG of G. Then,
(i)
ifμRs1q=μReq for some s1∈G, then μRs1s2q=μRs2q for all s2∈G(ii)
ifνRs1q=νReq for some s1∈G, then νRs1s2q=νRs2q for all s2∈GProof.
Suppose thatR=s,μRs,νRs is a q-ROFSG of G.
(i)
LetμRs1q=μReqfor some s1∈G. Then,
(26)μRs2q=μRs1−1s1s2q≥minμRs1−1q,μRs1s2q=minμRs1q,μRs1s2q=minμReq,μRs1s2q.
SinceμReq≥μRs2q, therefore from relation (26), we obtain
(27)μRs2q≥μRs1s2q.
Also,μRs1s2q≥minμRs1q,μRs2q=μRs2q, that is,
(28)μRs1s2q≥μRs2q.
From (27) and (28), we have
(29)μRs1s2q=μRs2q.(ii)
The proof is similar to that of (i).Theorem 9.
Lete denote the identity element of G and R=s,μRs,νRs be a q-ROFSG of G. Then, H=s∈G:μRsq=μReqandνRsq=νReq is a subgroup of G.Proof.
By definition ofH, we have e∈H. Therefore, H is nonempty subset of G.
Lets1,s2∈H; then, μRs1q=μReq=μRs2qandνRs1q=νReq=νRs2q.
Now,(30)μRs1s2−1q≥minμRs1q,μRs2−1q=minμRs1q,μRs2q=minμReq,μReq=μReq.
Also, by Theorem2, we have μReq≥μRs1s2−1q. Therefore, μRs1s2−1q=μReq. Similarly, we can show that νRs1s2−1q=νReq. Thus, s1s2−1∈H, which completes the proof.


## 3.q-Rung Orthopair Fuzzy Coset and q-Rung Orthopair Fuzzy Normal Subgroup

In this section, we defineq-rung orthopair fuzzy coset and q-rung orthopair fuzzy normal subgroup. Moreover, we prove some important results regarding q-rung orthopair fuzzy normal subgroup (q-ROFNS).Definition 10.
LetR=s,μRs,νRs be a q-ROFSG of G. Then, for x∈G,
(a)
aq-ROFS xR=s,μxRs,νxRs of G, where μxRsq=μRx−1sq and νxRsq=νRx−1sq, is called q-rung orthopair fuzzy left coset of R in G determined by x(b)
aq-ROFS Rx=s,μRxs,νRxs of G, where μRxsq=μRsx−1q and νRxsq=νRsx−1q, is called q-rung orthopair fuzzy right coset of R in G determined by xDefinition 11.
Aq-ROFSG R=s,μRs,νRs of G is called q-rung orthopair fuzzy normal subgroup (q-ROFNS) of G if xR=Rx for all x∈G.Theorem 12.
LetR=s,μRs,νRs be a q-ROFSG of G. Then, R is q-ROFNS of G if and only if μRs1s2q=μRs2s1q and νRs1s2q=νRs2s1q for all s1,s2∈G.Proof.
Assume thatR=s,μRs,νRs is a q-ROFNSG of G. Then, xR=Rx for all x∈G; it means that μxRsq=μRxsq and νxRsq=νRxsq for all s∈G. Therefore, μRx−1sq=μRsx−1q and νRx−1sq=νRsx−1q for all x,s∈G.
Now,(31)μRs1s2q=μRs1s2−1−1q=μRs2−1−1s1q=μRs2s1q,νRs1s2q=νRs1s2−1−1q=νRs2−1−1s1qνRs2s1q.
Conversely, suppose thatμRs1s2q=μRs2s1q and νRs1s2q=νRs2s1q for all s1,s2∈G. Then, μRs1s2−1−1q=μRs2−1−1s1q and νRs1s2−1−1q=νRs2−1−1s1q. Using s1=s and s2−1=x gives μRsx−1q=μRx−1sq and νRsx−1q=νRx−1sq. This means that μRxsq=μxRsq and νRxsq=νxRsq; therefore, xR=Rx. Hence, R is q-ROFNS of G.Theorem 13.
LetR=s,μRs,νRs be a q-ROFSG of G. Then, R is q-ROFNS of G if and only if μRsxs−1q=μRxq and νRsxs−1q=νRxq for all s,x∈G.Proof.
LetR be a q-ROFNS of G and s,x∈G. Then,
(32)μRsxs−1q=μRsxs−1q=μRs−1sxq=μRss−1xq=μRexq=μRxq
(by Theorem12).
Similarly, we can proveνRsxs−1q=νRxq.
Conversely, suppose thatμRsxs−1q=μRxq and νRsxs−1q=νRxq for all s,x∈G. Let s1,s2∈G; then,
(33)μRs1s2q=μRs2−1s2s1s2q=μRs2−1s2s1s2−1−1q=μRs2−1s2s1s2−1−1q=μRs2s1q.
Similarly,(34)νRs1s2q=νRs2s1q.
Then, the application of (33) and (34) together with Theorem 12 gives R which is q-ROFNS of G.Theorem 14.
LetR=s,μRs,νRs be a q-ROFNSG of G. Then, H=s∈G:μRsq=μReqandνRsq=νReq is a normal subgroup of G.Proof.
The application of Theorem9 gives H which is a subgroup of G. Let h∈H and s∈G; then,
(35)μRshs−1q=μRhqbyTheorem3.2=μReqsinceh∈H.
Similarly,(36)νRshs−1q=νReq.
Thus,shs−1∈H, which implies that H is a normal subgroup of G.


## 4.q-Rung Orthopair Fuzzy Level Subgroup

This section reveals the idea ofq-rung orthopair fuzzy level subgroup. We also prove some relevant results.Definition 15.
LetR=s,μRs,νRs be a q-ROFS of crisp set G and γ,δ∈0,1 such that 0≤γq+δq≤1. Then Rγ,δ=s∈G:μRsq≥γandνRsq≤δ is called q-rung orthopair fuzzy level subset (q-ROFLS) of q-ROFS R of G.Theorem 16.
LetR=s,μRs,νRs be a q-ROFS of G and γ,δ,γ′,δ′∈0,1. Then,
(i)
Rγ,δ⊆Rγ′,δ′ if γ′≤γ and δ≤δ′(ii)
Rγ,δ⊆Rγ,δ′ if R⊆R′Proof.
(i)
Lets∈Rγ,δ; then, μRsq≥γ and νRsq≤δ. Since γ′≤γ and δ≤δ′, therefore μRsq≥γ≥γ′ and νRsq≤δ≤δ′. It means that s∈Rγ′,δ′; hence, Rγ,δ⊆Rγ′,δ′(ii)
Lets∈Rγ,δ; then, μRsq≥γ and νRsq≤δ. Since R⊆R′, therefore μRsq≤μR′sq and νRsq≥νR′sq. So, μR′sq≥μRsq≥γ and νR′sq≤νRsq≤δ, which implies that s∈Rγ,δ′. Thus, Rγ,δ⊆Rγ,δ′Theorem 17.
Aq-ROFS R of a group G is q-ROFSG of G if and only if q-ROFLS Rγ,δ of G is a subgroup of G.Proof.
We knowRγ,δ=s∈G:μRsq≥γandνRsq≤δ. Since for all γ,δ∈0,1, we have μReq≥γ and νReq≤δ. Therefore, at least e∈Rγ,δ, which implies that Rγ,δ is nonempty.
Supposes,t∈Rγ,δ, which means that μRsq,μRtq≥γ and νRsq,νRtq≤δ. Since R is q-ROFSG of G, therefore
(37)μRst−1q≥minμRsq,μRt−1q=minμRsq,μRtq≥minγ,γ=γ,νRst−1q≤maxνRsq,νRt−1q=maxνRsq,νRtq≤maxδ,δ=δ.
Thus,st−1∈Rγ,δ; therefore, Rγ,δ is a subgroup of G.
Conversely, letR be a q-ROFS of G, and for all γ,δ∈0,1, Rγ,δ is a subgroup of G. Suppose s1,s2∈G such that μRs1q=γ1, μRs2q=γ2, νRs1q=δ1, and νRs2q=δ2. Then, s1,s2∈Rminγ1,γ2,minδ1,δ2; since Rminγ1,γ2,maxδ1,δ2 is a subgroup of G, therefore s1s2∈Rminγ1,γ2,maxδ1,δ2. This implies that μRs1s2q≥minγ1,γ2=minμRs1q,μRs2q and νRs1s2q≤maxδ1,δ2=maxνRs1q,νRs2q.
Next, lets∈G such that μRsq=γ and νRsq=δ. Then, s∈Rγ,δ; since Rγ,δ is a subgroup of G, therefore s1−1∈Rγ,δ. It means that μRs1−1q≥γ and νRs1−1q≤δ, which implies that μRs1−1q≥μRsq and νRs1−1q≤νRsq.
Hence,R is a q-ROFSG of G.Theorem 18.
IfR is a q-ROFNSG of G, then q-ROFLS Rγ,δ of G is a normal subgroup of G.Proof.
By Theorem17, Rγ,δ is a subgroup of G. Let x∈Rγ,δ and s∈G. Then, μRxq≥γ and νRxq≤δ. Since x∈Rγ,δ⊆G, therefore by using Theorem 13, we have μRsxs−1q=μRxq and νRsxs−1q=νRxq. Ultimately, it gives μRsxs−1q≥γ and μRsxs−1q≤δ, which means that sxs−1∈Rγ,δ. Thus, Rγ,δ is a normal subgroup of G.


## 5. Homomorphism onq-Rung Orthopair Fuzzy Subgroups

This section is devoted to explore the impact of group homomorphism onq-rung orthopair fuzzy subgroups.Theorem 19.
Supposeθ:G⟶G′ is an onto group homomorphism and R=s,μRs,νRs:s∈G is a q-ROFAG of G. Then, θR=s′,μθRs′,νθRs′:s′∈G′ is a q-ROFSG of G′.Proof.
Sinceθ:G⟶G′ is an onto homomorphism, therefore θG=G′.
Lets1′,s2′∈G′; then, there exists s1,s2∈G such that θs1=s1′, θs2=s2′, and θs1s2=θs1θs2=s1′s2′.
(38)μθRs1′s2′q=maxμRz:z∈G,θz=s1′s2′q=maxμRzq:z∈G,θz=s1′s2′=maxμRs1s2q:s1s2=z∈G,θs1=s1′,θs2=s2′andθs1s2=θs1θs2=s1′s2′θishomomorphism≥maxminμRs1q,μRs2q:s1,s2∈G,θs1=s1′,θs2=s2′Risaq‐ROFSGofG=minmaxμRs1q:s1∈G,θs1=s1′,maxμRs2q:s2∈G,θs2=s2′=minμθRs1′q,μθRs2′q.
So,μθRs1′s2′q≥minμθRs1′q,μθRs2′q for all s1′,s2′∈G′. In a similar way, it can be shown that νθRs1′s2′q≥minνθRs1′q,νθRs2′q.
Again, suppose thats′∈G′; then,
(39)μθRs′−1q=maxμRz:z∈G,θz=s′−1q=maxμRz−1:z−1∈G,θz−1=s′q=μθRs′q.
Similarly, we haveνθRs′−1q=νθRs′q for all s′∈G′.
Thus,θR=s′,μθRs′,νθRs′:s′∈G′ is a q-ROFSG of G′.Theorem 20.
Supposeθ:G⟶G′ is a bijective homomorphism and T=s′,μTs′,νTs′:s′∈G′ is a q-ROFSG of G′. Then, θ−1T=s,μθ−1Ts,νθ−1Ts:s∈G is a q-ROFSG of G.Proof.
Lets1,s2∈G; then, s1s2∈G.
Next,(40)μθ−1Ts1s2q=μTθs1s2q=μTθs1θs2qθishomomorphism≥minμTθs1q,μTθs2qTisaq‐ROFSGofG′=minμθ−1Ts1q,μθ−1Ts2q.
In a similar fashion, we can showνθ−1Ts1s2q≤maxνθ−1Ts1q,νθ−1Ts2q, for all s1,s2∈G.
Again, suppose thats∈G; then,
(41)μθ−1Ts−1q=μTθs−1q=μTθsq=μθ−1Tsq.
Similarly, we can proveνθ−1Ts−1q=νθ−1Tsq.
Thus,θ−1T is a q-ROFSG of G.Theorem 21.
Supposeθ:G⟶G′ is an onto group homomorphism and R=s,μRs,νRs:s∈G is a q-ROFNSG of G. Then, θR=s′,μθRs′,νθRs′:s′∈G′ is a q-ROFNSG of G′.Proof.
The application of Theorem19 yields that θR is a q-ROFSG of G′. Let s1′,s2′∈G′; then, there exists s1,s2∈G such that θs1=s1′, θs2=s2′, and θs1s2=θs1θs2=s1′s2′.
(42)μθRs1′s2′q=maxμRz:z∈G,θz=s1′s2′q=maxμRzq:z∈G,θz=s1′s2′=maxμRs1s2q:s1s2=z∈G,θs1=s1′,θs2=s2′andθs1s2=θs1θs2=s1′s2′θishomomorphism=maxμRs2s1q:s2s1=z′∈G,θs1=s1′,θs2=s2′andθs2s1=θs2θs1=s2′s1′Risaq‐ROFNSGofG=maxμRz′q:z′∈G,θz′=s2′s1′=maxμRz′:z′∈G,θz′=s2′s1′q=μθRs2′s1′q.
So,μθRs1′s2′q=μθRs2′s1′q for all s1′,s2′∈G′. In a similar way, it can be shown that νθRs1′s2′q=νθRs2′s1′q for all s1′,s2′∈G′.
Hence, by Theorem12, θR=s′,μθRs′,νθRs′:s′∈G′ is a q-ROFNSG of G′.Theorem 22.
Supposeθ:G⟶G′ is a bijective homomorphism and T=s′,μTs′,νTs′:s′∈G′ is a q-ROFNSG of G′. Then, θ−1T=s,μθ−1Ts,νθ−1Ts:s∈G is a q-ROFNSG of G.Proof.
By using Theorem13, we have θ−1T is a q-ROFSG of G. Let s1,s2∈G; then, s1s2∈G.
Now,(43)μθ−1Ts1s2q=μTθs1s2q=μTθs1θs2qθishomomorphism=μTθs2θs1qTisaq‐ROFNSGofG′=μTθs2s1q=μθ−1Ts2s1q.
Therefore,μθ−1Ts1s2q=μθ−1Ts2s1q for all s1,s2∈G. In a similar way, it can be shown that νθ−1Ts1s2q=νθ−1Ts2s1q for all s1,s2∈G.
Hence, by Theorem12, θ−1T=s,μθ−1Ts,νθ−1Ts:s∈G is a q-ROFNSG of G.


## 6. Conclusion

This article aims to initiate the study ofq-rung orthopair fuzzy group theory.We have introduced the notion ofq-ROFSG and many algebraic characteristics of this newly defined concept. We have proven that every PFSG is q-ROFSG, but the converse is not true. We presented the idea of q-rung orthopair fuzzy coset and q-ROFNSG and found a necessary and sufficient condition for ROFSG to be a ROFNSG. In addition, we found that q-ROFLS q-rung orthopair fuzzy level subset of a group G is a normal subgroup of G. Lastly, we have explored the impact of group homomorphism on q-ROFSG. We are working on some other classical group theoretic topics like quotient groups, Lagrange’s theorem, isomorphism theorems, conjugate subgroups, Caley’s theorem, subgroups of nilpotent, solvable, Hamiltonian, and P-Hall groups under q-rung orthopair fuzzy environment. We will share this work in our upcoming papers.



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*Source: 8196638-2022-06-06.xml*