Abstract

The -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 -rung orthopair fuzzy subgroup and examine various algebraic attributes of this newly defined notion. We also present -rung orthopair fuzzy coset and -rung orthopair fuzzy normal subgroup along with relevant fundamental theorems. Moreover, we introduce the concept of -rung orthopair fuzzy level subgroup and proved related results. At the end, we explore the consequence of group homomorphism on the -rung orthopair fuzzy subgroup.

1. Introduction

In classical fuzzy set theory, a fuzzy subset of a crisp set is represented by a function from to . 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 of a crisp set is an object such that is called membership mapping of and is known as a degree of membership of in . One can see that fuzzy sets are the extensions of characteristic functions of classical sets, by expanding the range of the function from to . 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 of a crisp set is an object , where and are membership and nonmembership functions, respectively, such that for all . 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 [36]. In 2013, Yager [7] generalized intuitionistic fuzzy sets by presenting the idea of Pythagorean fuzzy set (PFS). The Pythagorean fuzzy subset of a crisp set is an object , where and are membership and nonmembership functions, respectively, such that for all . 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 [811]. 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 ; 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 -rung orthopair fuzzy set (q-ROFS), where is a natural number [12]. The -rung orthopair fuzzy subset of a crisp set is an object , where and are positive and negative membership functions, respectively, such that for all . In order to find real-world applications of -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 [1719], 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 [3438]. Recently, Bhunia et al. [39] defined Pythagorean fuzzy subgroups and explored different attributes of this concept.

Considering the above literature and the significance of -rung orthopair fuzzy sets and theory of groups, this article reveals the study of -rung orthopair fuzzy subgroups (q-ROFSGs). The basic purpose and the principal contribution of this work are to (1)discuss various important algebraic attributes of -rung orthopair fuzzy subgroups(2)define the concepts of -rung orthopair fuzzy coset and -rung orthopair fuzzy normal subgroup along with the study of relevant fundamental theorems(3)introduce the idea of -rung orthopair fuzzy level subgroup and prove some important results of this notion

2. The -Rung Orthopair Fuzzy Subgroups

In this section, some important algebraic attributes of -rung orthopair fuzzy subgroups will be discussed. We start this section with the definition of -rung orthopair fuzzy subgroup (-ROFSG).

Definition 1. Let be a group; then, a -ROFS of is called a -rung orthopair fuzzy subgroup (q-ROFSG) of if the following conditions hold: (i) and for all ,(ii) and for all

Theorem 2. Let be a -ROFSG of Then, the following conditions are true: (i) and for all (ii) and for all

Proof. (i)Let ; then, . Similarly, we can show that (ii)Since and for all , therefore and which means that and . Thus, and for all

The following theorem shows that every Pythagorean fuzzy subgroup (PFSG) of is -ROFSG of .

Theorem 3. Let be a group and be a PFSG of . Then, is a -ROFSG of .

Proof. Let ,; then This implies that Since , therefore for all , we have ,, and . Thus, The inequalities (2)–(8) reveal that is a -ROFSG of .

The next example provides evidence of invalidity of the converse of the above theorem.

Example 1. Consider dihedral group , that is, One can easily verify that is 3-ROFSG of , but it is not a PFSG of as .

Theorem 4. A -ROFS of group is a -ROFSG of if and only if and for all ,.

Proof. Let be a -ROFSG of . Then, for all , and .
Conversely, suppose that and for all ,. Then, . Thus, Similarly, Next, , that is, Similarly, The inequalities (11)–(14) show that is a -ROFSG of .

Theorem 5. Let and be two -ROFSGs of ; then, is a -ROFSG of .

Proof. Suppose that and are two -ROFSGs of . Then, for all ,, we have That is, Similarly, The application of (16) and (17) together with the theorem give which is a -ROFSG of .

Theorem 6. Let be a -ROFSG of Then, and for all and .

Proof. We will use mathematical induction to prove this theorem. Suppose ; then, .
Therefore, the inequality is valid for Assume that the inequality holds for , that is, . Then, . Thus, by mathematical induction, we have for all .
Similarly, we can show for all .

Theorem 7. Let be a -ROFSG of If and for some ,, then and .

Proof. Suppose that for some ,, we have ; then, obviously .
Consider Since , therefore from relation (18), we obtain Also, , that is, From (19) and (20), we have Similarly, the result can be proven if .
Next, assume that ; therefore, . Then, Since , therefore from relation (22), we obtain Also, , that is, From (23) and (24), we have Similarly, , if .

Theorem 8. Let denote the identity element of and be q-ROFSG of Then, (i)if for some , then for all (ii)if for some , then for all

Proof. Suppose that is a -ROFSG of . (i)Let for some . Then, Since , therefore from relation (26), we obtain Also, , that is, From (27) and (28), we have (ii)The proof is similar to that of (i).

Theorem 9. Let denote the identity element of and be a -ROFSG of . Then, is a subgroup of .

Proof. By definition of , we have . Therefore, is nonempty subset of .
Let ; then, .
Now, Also, by Theorem 2, we have . Therefore, . Similarly, we can show that . Thus, , which completes the proof.

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

In this section, we define -rung orthopair fuzzy coset and -rung orthopair fuzzy normal subgroup. Moreover, we prove some important results regarding -rung orthopair fuzzy normal subgroup (-ROFNS).

Definition 10. Let be a -ROFSG of Then, for , (a)a -ROFS of , where and , is called -rung orthopair fuzzy left coset of in determined by (b)a -ROFS of , where and , is called -rung orthopair fuzzy right coset of in determined by

Definition 11. A -ROFSG of is called -rung orthopair fuzzy normal subgroup (-ROFNS) of if for all .

Theorem 12. Let be a -ROFSG of Then, is -ROFNS of if and only if and for all .

Proof. Assume that is a -ROFNSG of . Then, for all ; it means that and for all . Therefore, and for all .
Now, Conversely, suppose that and for all . Then, and . Using and gives and . This means that and ; therefore, . Hence, is -ROFNS of .

Theorem 13. Let be a -ROFSG of Then, is -ROFNS of if and only if and for all .

Proof. Let be a -ROFNS of and . Then, (by Theorem 12).
Similarly, we can prove .
Conversely, suppose that and for all . Let ; then, Similarly, Then, the application of (33) and (34) together with Theorem 12 gives which is -ROFNS of .

Theorem 14. Let be a -ROFNSG of . Then, is a normal subgroup of .

Proof. The application of Theorem 9 gives which is a subgroup of . Let and ; then, Similarly, Thus, , which implies that is a normal subgroup of .

4. -Rung Orthopair Fuzzy Level Subgroup

This section reveals the idea of -rung orthopair fuzzy level subgroup. We also prove some relevant results.

Definition 15. Let be a -ROFS of crisp set and such that . Then is called -rung orthopair fuzzy level subset (-ROFLS) of -ROFS of .

Theorem 16. Let be a -ROFS of and . Then, (i) if and (ii) if

Proof. (i)Let ; then, and . Since and , therefore and . It means that ; hence, (ii)Let ; then, and . Since , therefore and . So, and , which implies that . Thus,

Theorem 17. A -ROFS of a group is -ROFSG of if and only if -ROFLS of is a subgroup of .

Proof. We know . Since for all , we have and . Therefore, at least , which implies that is nonempty.
Suppose , which means that and . Since is -ROFSG of , therefore Thus, ; therefore, is a subgroup of .
Conversely, let be a -ROFS of , and for all , is a subgroup of . Suppose such that , , , and . Then, ; since is a subgroup of , therefore . This implies that and .
Next, let such that and . Then, ; since is a subgroup of , therefore . It means that and , which implies that and .
Hence, is a -ROFSG of .

Theorem 18. If is a -ROFNSG of , then -ROFLS of is a normal subgroup of .

Proof. By Theorem 17, is a subgroup of . Let and . Then, and . Since , therefore by using Theorem 13, we have and . Ultimately, it gives and , which means that . Thus, is a normal subgroup of .

5. Homomorphism on -Rung Orthopair Fuzzy Subgroups

This section is devoted to explore the impact of group homomorphism on -rung orthopair fuzzy subgroups.

Theorem 19. Suppose is an onto group homomorphism and is a -ROFAG of . Then, is a -ROFSG of .

Proof. Since is an onto homomorphism, therefore .
Let ; then, there exists , such that , , and . So, for all . In a similar way, it can be shown that .
Again, suppose that ; then,