Picture Fuzzy Ideals of Near-Rings

Department of Math & Stats, International Islamic University Islamabad, Pakistan Nonlinear Analysis Research Group, Ton Duc $ang University, Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics, Ton Duc $ang University, Ho Chi Minh City, Vietnam Universit’e de Sousse, Institut Sup’erieur d’Informatique et des Techniques de Communication, Hammam Sousse 4000, Tunisia China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Engineering, Kyungpook National University, Daegu 41566, Republic of Korea Department of Computer Science, Nankai University Tianjin, Tianjin 300071, China


Introduction
Fuzzy set (FS) was first introduced by Zadeh [1]. It discusses the grade of truth values belonging to a unit interval. FS is a valuable technique to cope with vague and difficult information in a real-life decision. It has received extensive attention from researchers and has been utilized in decisionmaking in many ways (see [2,3]). However, in some cases, FS cannot work reliably. For coping such types of issues, Atanassov [4] explored the intuitionistic fuzzy set (IFS) to deal with more complex problems. IFS contains truth and falsity grades, whose sum is belonging to a unit interval. IFS is a more modified version of a FS, and many researchers have used it in decision-making [5], such as medical diagnosis [6] and pattern recognition [7]. Although FS and IFS are applicable in various fields, however, there exist problems when a decision maker face various kinds of opinions of human beings such as "yes," "abstinence," "no," and "refusal." To coping with such kinds of problems, Cuong [8] explored the concept of a picture fuzzy set (PFS), containing the grades of truth, abstinence, falsity, and refusal, whose sum is belonging to a unit interval. To deal with uncertain and difficult information in real-life decisions, PFS is more effective than the IFS and FS. Joshi [9] presented a new setting as criteria of fuzzy entropy for the PFS and proposed a picture fuzzy information model based on Tsallis-Havrda-Charvat entropy. Qyias et al. [10] established aggregation operators based on linguistic PFSs and their application in decision-making problems, while Ganie et al. [11] examined the new correlation coefficient for the PFS. e concept of an ideal in the frame of a near-ring is discussed by numerous scholars by using intuitionistic fuzzy sets.
To investigate complicated and awkward information, a picture fuzzy set is a more influential tool than the fuzzy set and intuitionistic fuzzy sets. us, the concept of a picture fuzzy ideal of near-rings is presented in this article. Our results clearly show that the idea of a picture fuzzy set is more proficient and reliable than the aforementioned concepts. e notion of fuzzy subnear-rings (FSNR) and fuzzy ideals (FI) of a NR was introduced by Abou-Zaid [12]. is concept was further discussed by Kim [13] and Dutta and Biswas [14]. Chinnadurai and Kadalarasi [15] discussed the direct product of n (n � 1, 2, . . ., k) FSNR, FI, and fuzzy R-subgroups. e advantages of a PFS became the motivation of our main theorems. e results presented are entirely new and more beneficial than the existing results in the literature. e aims of this manuscript are given as follows: (1) To explore the idea of a picture fuzzy near-ring (PFNR) and a picture fuzzy ideal (PFI) of a near-ring (NR) (2) To describe some basic properties such as union, intersection, homomorphic image, and preimage of PFIs of a NR (3) To elaborate the direct product of PFIs of a NR e summary of this manuscript is as follows: in Section 2, we review some notions such as groups, subgroups, ideals, and fuzzy sets. In Section 3, we explore the idea of a picture fuzzy near-ring (PFNR) and a picture fuzzy ideal (PFI) of a near-ring (NR). In Section 4, we investigate some basic properties such as union, intersection, homomorphic image, and preimage of a PFI of a NR. In Section 5, we discuss about the direct product of PFIs of a NR. e conclusion of this manuscript is discussed in Section 6.

Preliminaries
e aim of this study is to recall some basic notions such as near-rings, ideals of a near-ring, fuzzy sub-near-rings, fuzzy ideals, and picture fuzzy sets.
Definition 1 (see [13]). For any nonempty set E with two binary operations "+" and "·," we say that E is called a nearring if the following conditions hold: Definition 2 (see [14]). An ideal of a NR E is a subgroup A of E such that (i) (A, +) is a normal subgroup of (E, +), where A is a left ideal of E whenever A satisfies (i) and (ii) and a right ideal of E if A satisfies (i) and (iii).
Definition 3 (see [16]). A FS in a NR E is said to be a fuzzy sub-near-ring of E if Definition 4 (see [16]). A FS B in a NR E is said to be a fuzzy ideal of E if A fuzzy set satisfying (i)-(iii) is called a fuzzy left ideal of E, whereas a fuzzy set with (i) and (ii) is called a fuzzy right ideal of E.
Definition 5 (see [8]). A PFS P on the universe of discourse U characterized by a truth membership function B P (h), an indeterminacy function A P (h), and a falsity membership function F P (h) is defined as where B P , A P , F P : U ⟶ [0, 1] and 0 ≤ B P (h) + A P (h) + F P (h) ≤ 1.
Definition 6 (see [8]). Let P and Q be PFSs of U. en, where for all h ∈ U. where for all h ∈ U.

Picture Fuzzy Ideals of Near-Rings
e aim of this study is to explore the idea of a PFNR and a PFI of a NR.
A picture fuzzy subset with the above conditions (i)-(iii) is called a picture fuzzy left ideal of E, whereas a picture fuzzy subset with (i), (ii), and (iv) is called a picture fuzzy right ideal of E.
Proof. Let P and Q be PFIs of E. Let h, k, t ∈ E; then, Similarly, for the abstinence degree, we write and for falsity grade, we have Next, we write For the abstinence degree, we obtain and for falsity grade, we get Furthermore, we deduce that At last, we obtain Proof. Let P and Q be PFIs of E. Let h, k, t ∈ E. en, the following are obtained.
For truth grade, we get Similarly, for abstinence grade, we write

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and for falsity grade, we obtain Next, we obtain Similarly, Furthermore, we deduce that Finally, we conclude that erefore, P ∩ Q is a PFI of E. □ Corollary 1. If (P 1 , P 2 , . . . , P n ) are PFIs of E, then P � ∩ n i�1 P i is a PFI of E.

Lemma 1. For all h, b ∈ A and i is any positive integer
Theorem 3. Let P be a PFI of E. en, is a PFI of E, where m is a positive integer and Proof. Let P be a PFI of E. Let h, k, t ∈ E. en, the following are observed.
For truth grade, we can write Similarly, for abstinence grade, we get and for falsity grade, we obtain the following: Next, it is obtained that Also, we examine that At last, we write that 4 Journal of Mathematics erefore, P m is a PFI of E m .

Direct Product of Picture Fuzzy Ideals of Near-Rings
is section throws light on some basic properties such as union, intersection, homomorphic image, and preimage of PFIs of a NR.

Definition 9.
Let P and Q be picture fuzzy subsets of nearrings (PFSSNR) E 1 and E 2 , respectively. en, the direct product of PFSSNRs is defined by P × Q: where } is a near-ring with binary operations "+" and "·," defined by , respectively. Suppose PFSs P � (α P , β P , c P ) and en, we say that P � (α P , β P , c P ) and Q � (α Q , β Q , c Q ) are PFI of R. Now, (31) Note that this example can be extended to construct an example for Definition 10.
Definition 10. Let P and Q be PFSSNRs E 1 and E 2 , respectively. en, P × Q is a PFI of E 1 × E 2 if it satisfies the following conditions: (32)

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Theorem 4. Let P and Q be PFIs of E 1 and E 2 , respectively. en, P × Q is a PFI of E 1 × E 2 .
For truth grade, we obtain the following: Similarly, for abstinence grade, we write and for falsity grade, we get Next, it is clear that 6

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Similarly, Moreover, we deduce the following inequalities: Similarly, Finally, we prove that Also,

Homomorphism of Picture Fuzzy Ideals of Near-Rings
is section is concerned with the direct product of PFIs of a NR.
Definition 11. Let R and S be two NRs. en, the mapping f: R ⟶ S is called a near-ring homomorphism (NRH) if for all h, k ∈ R, the following hold: Definition 12. Let U and Y be two nonempty sets and f: U ⟶ Y be a function.
, and so on.

Theorem 5.
Let E and E ′ be two NRs and f be a homo-

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Proof. Suppose h, k, t ∈ E.
en, we can deduce the following inequalities: For membership grade, Also, Furthermore, we write Finally, we get Similarly, we can deduce for abstinence grade that and for falsity grade, we deduce that Hence, f(P) is a PFI of E 2 . Note that, by choosing the value of nonmembership as zero, the main results of the article are obtained in the setting of fuzzy sets. However, taking the value of abstinence as zero, we acquire the mentioned results in intuitionistic fuzzy sets.
ese are the extensions of the presented work.

Conclusion
e theory of a picture fuzzy set is basically the extension of the fuzzy set, containing the grade of membership, the grade of abstinence, and the grade of nonmembership with a rule, provided the sum of all grades cannot exceed the unit interval. Keeping in mind the advantages of the picture fuzzy set, in this manuscript, we presented the PFSNR and PFIs of NRs and explained some of their algebraic properties. We have proved that the union of two PFIs of a NR is a PFI of that NR. Also, we have proved that the positive integral powers of a PFI of a NR are a PFI. We have defined the direct product on PFIs of NRs and proved that the direct product of any two PFIs of NRs is a PFI. Moreover, we showed that we can extend the result for a finite number of PFIs of NRs.
In future, we will investigate the decision-making problems based on Pythagorean fuzzy sets [17,18]. An investigation will be carried out about the Pythagorean fuzzy near-rings and Pythagorean ideals of a near-ring, and their algebraic properties will be discussed.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.