# Multicriteria Decision-Making Problem via Weighted Cosine Similarity Measure and Several Characterizations of Hypergroup and (Weak) Polygroups under the Triplet Single-Valued Neutrosophic Structure

Authors: M. Shazib Hameed; Zaheer Ahmad; Shahbaz Ali; Augustine Larweh Mahu; Walid F. A Mosa
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Polygroups are an extended form of groups and a subclass of hypergroups that follow group-type axioms. In this paper, we define a triplet single-valued neutrosophic set, which is a generalization of fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets, and we combine this novel concept with hypergroups and polygroups. Firstly, the main goal of this paper is to introduce hypergroups, polygroups, and anti-polygroups under a triplet single-valued neutrosophic structure and then present various profound results. We also examine the interaction and properties of level sets of triplet single-valued neutrosophic polygroups and (normal) subpolygroups. Secondly, we rank the alternatives and select the best ones in a single-valued neutrosophic environment using the weighted cosine similarity measure between each alternative and the ideal alternative. Finally, we provide an example that clearly shows how the proposed decision-making method is applied.


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## Body

## 1. Introduction

The classical methods of mathematical analysis are unable to make sense of the ambiguities that exist in the universe. As a consequence of this, these structures need to be rethought in order to take into account the possibility of uncertainty. In 1965, Zadeh [1] proposed a fuzzy set. A fuzzy set is a mathematical model of ambiguity in which things belong to a specific set to some degree. This degree is generally a number that falls within the unit range of 0,1.In later years, as an extension of the fuzzy set, Sambuc [2] presented the notion of an interval-valued fuzzy set in 1975, Atanassov [3] provided the idea of an intuitionistic fuzzy set in 1984, Yager [4] initiated the concept of fuzzy multiset in 1986, Smarandache [5] presented the premise of a neutrosophic set (NS) in 1998, Molodstov [6] introduced the idea of soft sets in 1999, and Torra [7] developed a hesitant fuzzy set in 2010. Feng et al. [8] broadened soft sets by integrating them with fuzzy and rough sets, Aktas and Cagman [9] investigated soft groups, and Acar et al. [10] developed soft rings.Marty [11] was the first to propose algebraic hyperstructures, which are an overarching concept of classical algebraic structures. He broadened the definition of a group to include the concept of a hypergroup. The resultant of two elements in a classical algebraic structure is an element. However, the resultant of two elements in an algebraic hyperstructure is a set. Algebraic hyperstructures have been used in a wide range of subjects over the years, including hypergraphs, binary relations, cryptography, codes, median algebras, relation algebras, artificial intelligence, geometry, convexity, automata, combinatorial coloring problems, lattice theory, Boolean algebras, and logic probabilities. Hypergroups have mostly been used in the context of special subclasses.Polygroups, which are spectacular subclasses of hypergroups, are developed by Ioulidis in [12] and employed to examine color algebras by Comer in [13, 14]. Comer showed the effectiveness of polygroups by exploring their connections to graphs, relations, Boolean, and cylindric algebras. The theory of algebraic hyperstructures has since been investigated and expanded by a number of scholars. Many scholars working in these domains have been drawn to the combination of fuzzy sets and algebraic hyperstructures, as well as neutrosophic sets and algebraic hyperstructures, resulting in the creation of new branches of research, namely fuzzy algebraic hyperstructures and neutrosophic algebraic hyperstructures.Comer developed quasi-canonical hypergroups in [15] as an extension of canonical hypergroups, which were presented in [16]. In [17], Comer introduced a number of algebraic and combinatorial properties. In [18], Davvaz and Poursalavati introduced matrix representations of polygroups over hyperrings and the idea of a polygroup hyperring, which expanded the concept of a group ring. Davvaz devised permutation polygroups and topics connected to them, employing the notion of generalized permutation [19]. We refer to some important and recent innovative work relative to the fuzzy structures and polygroups in [20–42] for further information.Neutrosophy is a new subfield of philosophy that investigates the origin, nature, and multitude of neutralities, as well as their interactions with other ideological spectrums, which was first proposed by Smarandache in 1995. In the neutrosophic set, indeterminacy is quantified explicitly and truth-membership, indeterminacy membership, and falsity-membership are independent. In a neutrosophic set, truth (T), indeterminacy (I), and falsity (F) are the three types of membership functions. In this work, we develop set theoretic operators on a special kind of the neutrosophic set known as the single-valued neutrosophic set. A single-valued neutrosophic set (SVNS) is a type of NS that may be employed to address intellectual and technical problems in the real world. As a result, the study of SVNSs and their attributes is essential in terms of applications as well as comprehending the principles of uncertainty.In this article, first we define the generalized conceptη,ξ,φ-SVNS and then apply this concept to hypergroups and polygroups. For decision-making problems, a weighted cosine similarity measure (WCSM) is applied to each alternative, and the ideal alternative is used to rank the alternatives and choose the best option. In addition, we compared our strategy to current approaches and demonstrated its superiority. In conclusion, an example scenario illustrates how the suggested D-M technique may be implemented. In comparison, existing fuzzy multicriteria decision-making (M-CDM) strategies are incapable of tackling the decision-making difficulty stated in this paper. The suggested single-valued neutrosophic (SVN) decision-making technique has the benefit of being able to cope with ambiguous and inconsistent information, both of which are typical in real-world circumstances.The motivation of the proposed concept is explained as follows: to present a more generalized concept, i.e., (1)η,ξ,φ-single-valued neutrosophic hypergroups. (2) η,ξ,φ-single-valued neutrosophic polygroups. (3) η,ξ,φ-anti-single-valued neutrosophic polygroups. (4) Single-valued neutrosophic multicriteria decision-making method. Note that, clearly ϒΩ˜=Ω˜,ϒ∅˜=∅˜, which shows that our proposed definition can be converted into a single-valued neutrosophic set. The purpose of this paper is to present the study of single-valued neutrosophic hypergroups and single-valued neutrosophic polygroups, and anti-single-valued neutrosophic polygroups under the triplet structure as a generalization of hypergroups, polygroups, and anti-polygroups as a powerful extension of single-valued neutrosophic sets.This article is organized as follows: we offer some fundamental structure regarding single-valued neutrosophic sets,η,ξ,φ-single-valued neutrosophic hypergroup, and (weak) polygroups in Sections 2, 3, and 4, respectively. We present and analyze the idea of a η,ξ,φ-single-valued neutrosophic (weak) polygroup in Section 5. In Section 6, we explore the correlation between level sets of η,ξ,φ-single-valued neutrosophic polygroups (η,ξ,φ-SVNPs) and (normal) subpolygroups). Finally, in Section 7 we present the decision-making (D-M) procedure and for evaluation, we also offer an illustration example in Section 8.


## 2. Preliminaries

This section covers basic definitions related to SVNSs. In this section, we also present fundamental properties and relationships between SVNSs.Definition 1.
( [44]). On the universe set Ω a SVNS ϒ is stated as(1)Ω=u,τϒu,ιϒu,Ϝϒu,u∈Ω,where τ,ι,Ϝ:Ω⟶0,1, and 0≤τϒu+ιϒu+Ϝϒu≤3, ∀u∈Ω, τϒu,ιϒu,Ϝϒu∈0,1. τϒ, ιϒ, Ϝϒ indicates truth, indeterminacy, and falsity-membership function, in that order.Definition 2.
( [44]). Let Ω be a set of objects, with u denoting a generic entity belong to Ω. A SVNS ϒ on Ω is symbolized by truth τϒ, indeterminacy ιϒ, and falsity-membership function Ϝϒ, in that order. ∀u∈Ω, τϒu,ιϒu,Ϝϒu∈0,1. A SVNS ϒ can be written accordingly as(2)ϒ=∑inτui,ιui,Ϝuiui,ui∈Ω.Definition 3.
(see [44]). The complement of a SVNS ϒ is indicated by cϒ and is characterized by(3)τcϒu=Ϝϒu,ιcϒu=1−ιϒu,Ϝcϒu=τϒu,∀u∈Ω.Definition 4.
( [44]). Let ϒ and Θ be two SVNSs on Ω. Then(1)
(4)ϒ⊆Θ,⇔ϒu≤Θu.That is(5)τϒu≤τΘu,ιϒu≥ιΘu,Ϝϒu≥ϜΘu.Also(6)ϒ=Θ⇔ϒ⊆ΘΘ⊆ϒ.(2)
Δ=ϒ∪Θ such that(7)Δu=ϒu∨Θu,such that(8)ϒu∨Θu=τϒu∨τΘu,ιϒu∧ιΘu,Ϝϒu∧ϜΘu,∀u∈Ω.It means(9)τΔu=maxτϒu,τΘu,ιΔu=minιϒu,ιΘu,ϜΔu=minϜϒu,ϜΘu.(3)
Δ=ϒ∩Θ such that(10)Δu=ϒu∧Θu,
such that(11)ϒu∧Θu=τϒu∧τΘu,ιϒu∨ιΘu,Ϝϒu∨ϜΘu,∀u∈Ω.
It means(12)τΔu=minτϒu,τΘu,ιΔu=maxιϒu,ιΘu,ϜΔu=maxϜϒu,ϜΘu.Proposition 1.
( [44]). Let the SVNSs on the common universe Ω be ϒ, Θ, and Δ. Then the following conditions must hold the following:(1)
ϒ∪Θ=Θ∪ϒ,ϒ∩Θ=Θ∩ϒ.(2)
ϒ∪Θ∪Δ=ϒ∪Θ∪Δ,ϒ∩Θ∩Δ=ϒ∩Θ∩Δ.(3)
ϒ∪Θ∩Δ=ϒ∪Θ∩ϒ∪Δ,ϒ∩Θ∪Δ=ϒ∩Θ∪ϒ∩Δ.(4)
ϒ∩∅˜=∅˜,ϒ∪∅˜=ϒ,ϒ∪Ω˜=Ω˜,ϒ∩Ω˜=ϒ, where(13)τ∅˜=0,ι∅˜=Ϝ∅˜=1,τΩ˜=1,ιΩ˜=ϜΩ˜=0.(5)
cϒ∪Θ=cϒ∩cΘ,cϒ∩Θ=cϒ∪cΘ.


## 3.η,ξ,φ- Single-Valued Neutrosophic Hypergroup

We define and investigate the basic properties and characterizations of a single-valued neutrosophic set, single-valued neutrosophic hypergroup, and single-valued neutrosophic subhypergroup over hypergroupH under the triplet structure in this section. We basically start with some introductory η,ξ,φ-SVNS, then define η,ξ,φ-SVN hypergroup, the t-level set on η,ξ,φ-SVNS, important operations and properties of η,ξ,φ-SVN hypergroups, and then study crucial results, propositions, theorems and remarks related to SVN hypergroup and SVN subhypergroup under the triplet structure. In this section, we present a very important result, that is intersection of two η,ξ,φ-SVN hypergroups over H is again η,ξ,φ-SVN hypergroup in 3.18, which shows that η,ξ,φ-SVN hypergroups are closed under intersection, and union of two η,ξ,φ-SVN hypergroups over H need not be η,ξ,φ-SVN hypergroup over H.Definition 5.
Ifϒ be a single-valued neutrosophic (SVN) subset of Ω, then η,ξ,φ-SVN subset ϒ of Ω is categorize as(14)ϒη,ξ,φ=e,τϒηu,ιϒξu,Ϝϒφu|u∈Ω,where(15)τϒηu=∧τϒu,η,ιϒξu=∨ιϒu,ξ,Ϝϒφu=∨Ϝϒu,φ,
such that(16)0≤τϒηu+ιϒξu+Ϝϒφu≤3,where η,ξ,φ∈0,1, also τ,ι,Ϝ:A⟶0,1, such that τϒη, ιϒξ, Ϝϒφ represents the functions of truth, indeterminacy, and falsity-membership, respectively.Definition 6.
LetΩ be a space of objects, with u denoting a generic entity belong to Ω. A η,ξ,φ-SVNS ϒ on Ω is symbolized by truth τϒη, indeterminacy ιϒξ, and falsity-membership function Ϝϒφ, respectively. For every u in Ω, τϒηu,ιϒξu,Ϝϒφu∈0,1, a η,ξ,φ-SVNS ϒ can be written accordingly as(17)ϒη,ξ,φ=∑inτηui,ιξui,Ϝφuiui,ui∈Ω.Definition 7.
Letϒ and Θ be two η,ξ,φ-SVNSs on Ω. The followings must hold the following:(1)
ϒη,ξ,φ⊆Θη,ξ,φ⇔ϒη,ξ,φu≤Θη,ξ,φu.That is,(18)τϒηu≤τΘηu,ιϒξu≥ιΘξu,Ϝϒφu≥ϜΘφu,(19)ϒη,ξ,φ=Θη,ξ,φ⇔ϒη,ξ,φ⊆Θη,ξ,φandΘη,ξ,φ⊆ϒη,ξ,φ.(2)
The union ofϒη,ξ,φ and Θη,ξ,φ is indicated by(20)Δη,ξ,φ=ϒη,ξ,φ∪Θη,ξ,φ,and defined as(21)Δη,ξ,φu=ϒη,ξ,φu∨Θη,ξ,φu,where(22)ϒη,ξ,φu∨Θη,ξ,φu=τϒηu∨τΘηu,ιϒξu∧ιΘξu,Ϝϒφu∧ϜΘφu,∀u∈Ω.That is,(23)τΔηu=maxτϒηu,τΘηu,ιΔξu=minιϒξu,ιΘξu,ϜΔφu=minϜϒφu,ϜΘφu.(3)
The intersection ofϒη,ξ,φ and Θη,ξ,φ is indicated by(24)Δη,ξ,φ=ϒη,ξ,φ∩Θη,ξ,φ,
and defined as(25)Δη,ξ,φu=ϒη,ξ,φu∧Θη,ξ,φu,
where(26)ϒη,ξ,φu∧Θη,ξ,φu=τϒηu∧τΘηu,ιϒξu∨ιΘξu,Ϝϒφu∨ϜΘφu,∀u∈Ω.
That is,(27)τΔηu=minτϒηu,τΘηu,ιΔξu=maxιϒξu,ιΘξu,ϜΔφu=maxϜϒφu,ϜΘφu.Proposition 2.
Letϒ,Θ, andΔbeη,ξ,φ-SVNSs on the common universeΩThen the following properties must hold the following:(1)
ϒη,ξ,φ∪Θη,ξ,φ=Θη,ξ,φ∪ϒη,ξ,φ.ϒη,ξ,φ∩Θη,ξ,φ=Θη,ξ,φ∩ϒη,ξ,φ.(2)
ϒη,ξ,φ∪Θη,ξ,φ∪Δη,ξ,φ=ϒη,ξ,φ∪Θη,ξ,φ∪Δη,ξ,φ. ϒη,ξ,φ∩Θη,ξ,φ∩Δη,ξ,φ=ϒη,ξ,φ∩Θη,ξ,φ∩Δη,ξ,φ.(3)
ϒη,ξ,φ∪Θη,ξ,φ∩Δη,ξ,φ=ϒη,ξ,φ∪Θη,ξ,φ∩ϒη,ξ,φ∪Δη,ξ,φ.ϒη,ξ,φ∩Θη,ξ,φ∪Δη,ξ,φ=ϒη,ξ,φ∩Θη,ξ,φ∪ϒη,ξ,φ∩Δη,ξ,φ.(4)
ϒη,ξ,φ∩∅˜=∅˜,ϒη,ξ,φ∪∅˜=ϒη,ξ,φ.ϒη,ξ,φ∪Ω˜=Ω˜,ϒη,ξ,φ∩Ω˜=ϒη,ξ,φ,where(28)τ∅˜η=0,ι∅˜ξ=Ϝ∅˜φ=1,τΩ˜η=1,ιΩ˜ξ=ϜΩ˜φ=0.(5)
cϒη,ξ,φ∪Θη,ξ,φ=cϒη,ξ,φ∩cΘη,ξ,φ⋅cϒη,ξ,φ∩Θη,ξ,φ=cϒη,ξ,φ∪cΘη,ξ,φ ? ?Definition 8.
The complement of aη,ξ,φ-SVNS ϒ is denoted by cϒη,ξ,φ and is defined by(29)cϒη,ξ,φ=u,τcϒηu,ιcϒξu,Ϝcϒφu,where(30)τcϒηu=Ϝϒφu,ιcϒξu=1−ιϒξu,Ϝcϒφu=τϒηu,∀u∈Ω.Definition 9.
The falsity-favorite of aη,ξ,φ-SVNS Θη,ξ,φ (i.e., Θη,ξ,φ=▽ϒη,ξ,φ) whose truth and falsity-membership functions are defined by(31)τΘηx=τϒηu,ιΘξu=0,ϜΘφu=minϜϒφu+ιϒξu,1.
Throughout this sectionH denotes the hypergroup H,°.Definition 10.
( [45]). A set H is called hypergroup H,° with an associative hyperoperation °:H∗H⟶PH, which satisfies x°±H=H°x=H, ∀x∈H (reproduction axiom).Definition 11.
( [46]). If the following properties satisfy, a hyperstructure H,° is called a Hv-group.(1)
x°y°z∩x°y°z≠∅, ∀x,y,z∈H, (Hv-semigroup).(2)
x°H=H°x=H, ∀x∈H.Definition 12.
( [45]). A subset K of H is called as subhypergroup if K,° is a hypergroup.Definition 13.
Letϒ be a η,ξ,φ-SVNS over H. Then ϒ is called a η,ξ,φ-SVN hypergroup over H, if the following conditions are satisfied:(i)
∀u,v∈H,(32)minτϒηu,τϒηv≤infτϒηw:w∈u°v,maxιϒξu,ιϒξv≥supιϒξw:w∈u°v,maxϜϒφu,Ϝϒφv≥supϜϒφw:w∈u°v.(ii)
∀l,u∈H, ∃v∈H such that u∈l°v and(33)minτϒηl,τϒηu≤τϒηv,maxιϒξl,ιϒξu≥ιϒξv,maxϜϒφl,Ϝϒφu≥Ϝϒφv.(iii)
∀l,u∈H, ∃w∈H such that u∈w°l and(34)minτϒηl,τϒηu≤τϒηw,maxιϒξl,ιϒξu≥ιϒξw,maxϜϒφl,Ϝϒφu≥Ϝϒφw.
Ifϒη,ξ,φ satisfies condition (i) then ϒ is a η,ξ,φ-SVN semihypergroup over H. Condition (ii) and (iii) represent the left and right reproduction axioms, respectively. Then ϒ is a η,ξ,φ-SVN subhypergroup of H.Example 1.
If the family of t-level sets ofη,ξ,φ-SVNS ϒ over H.(35)ϒtη,ξ,φ=u∈H|τϒηu≥t,ιϒξu≤tandϜϒφu≤t,is a subhypergroup of H. Then ϒ is a η,ξ,φ-SVN hypergroup over H.Theorem 1.
Letϒbe aη,ξ,φ-SVNS overH. Thenϒis aη,ξ,φ-SVN hypergroup overHif and only ifϒis aη,ξ,φ-SVN semihypergroup overHand alsoϒη,ξ,φsatisfies the left and right reproduction axioms.Proof 1.
The proof is obvious from Definition13.Theorem 2.
Letϒbe aη,ξ,φ-SVNS overH. Ifϒη,ξ,φis a SVN hypergroup overH, then∀t∈0,1ϒtη,ξ,φ≠∅is a subhypergroup ofH.Proof 2.
Letϒ be a η,ξ,φ-SVN hypergroup over H and let u,v∈ϒtη,ξ,φ, then(36)τϒηu,τϒηv≥t,ιϒξu,ιϒξv≤tandϜϒφu,Ϝϒφv≤t.
Then we have(37)infτϒηw:w∈u°v≥minτϒηu,τϒηv≥mint,t=t,supιϒξw:w∈u°v≤maxτϒηu,τϒηv≤maxt,t=t,supϜϒφw:w∈u°v≤maxτϒηu,τϒηv≤maxt,t=t.
This impliesw∈ϒtη,ξ,φ. Then ∀w∈u°v,u°v⊆ϒtη,ξ,φ.
Thus∀w∈ϒtη,ξ,φ, we obtain. w°ϒtη,ξ,φ⊆ϒtη,ξ,φ
Now, Letl,u∈ϒtη,ξ,φ, then there exist v∈H such that u∈l°v and(38)τϒηv≥minτϒηl,τϒηu≥mint,t=t,ιϒξv≤maxτϒηl,τϒηu≤maxt,t=t,Ϝϒφv≤maxτϒηl,τϒηu≤maxt,t=t.
This implies thatv∈ϒtη,ξ,φ. This proves that ϒtη,ξ,φ⊆w°ϒtη,ξ,φ. As such. ϒtη,ξ,φ=w°ϒtη,ξ,φ
which proves thatϒtη,ξ,φ is a subhypergroup of H.Theorem 3.
Letϒbe aη,ξ,φ-SVNS overHThen the following are equivalent:(i)
ϒ is a η,ξ,φ-SVN hypergroup over H.(ii)
∀t∈0,1ϒtη,ξ,φ≠∅ is a subhypergroup of H.Proof 3.
(i)⇒ (ii) The proof is obvious from Theorem 2. (ii) ⇒ (i) Now assume that ϒtη,ξ,φ is a subhypergroup of H. Let u,v∈ϒtoη,ξ,φ and let. minτϒηu,τϒηv=maxιϒξu,ιϒξv=maxϜϒφu,Ϝϒφv=toSince u°v⊆ϒtoη,ξ,φ, then for every w∈u°v,τϒηw≥to,ιϒξw≤to,Ϝϒφw≤to.(39)minτϒηu,τϒηu≤infτϒηw:w∈u°v,maxιϒξu,ιϒξv≥supιϒξw:w∈u°v,maxϜϒφu,Ϝϒφv≥supϜϒφw:w∈u°v.
Condition (i) is verified.
Next, letl,u∈ϒt1η,ξ,φ, for every t1∈0,1 and
letminτϒηl,τϒηv=maxιϒξl,ιϒξv=maxϜϒφl,Ϝϒφv=t1.
Then there existv∈ϒt1η,ξ,φ such that u∈l°v⊆ϒt1η,ξ,φ. Since v∈ϒt1η,ξ,φ, then(40)τϒηv≥t1=minτϒηl,τϒηu,ιϒξv≤t1=maxιϒξl,ιϒξu,Ϝϒφv≤t1=maxϜϒφl,Ϝϒφu.
Condition (ii) is verified.
Next, letl,u∈ϒt1η,ξ,φ, for every t1∈0,1 and
letminτϒηl,τϒηl=maxιϒξl,ιϒξv=maxϜϒφl,Ϝϒφv=t1.
Then there existw∈ϒt1η,ξ,φ such that u∈w°l⊆ϒt1η,ξ,φ. Since w∈ϒt1η,ξ,φ, then(41)τϒηw≥t1=minτϒηl,τϒηu,ιϒξw≤t1=maxιϒξl,ιϒξu,Ϝϒφw≤t1=maxϜϒφl,Ϝϒφu.
Condition (iii) is verified.Theorem 4.
Letϒbe aη,ξ,φ-SVNS overH. Thenϒbe aη,ξ,φ-SVN hypergroup overHif and only if∀α,β,γ∈0,1,ϒα,β,γη,ξ,φis a subhypergroup ofH.Proof 4.
The proof is simple for readers.Theorem 5.
Letϒbe aη,ξ,φ-SVN hypergroup overHand∀t1,t2∈0,1,ϒt1η,ξ,φandϒt2η,ξ,φbe the t-level sets ofϒη,ξ,φwithϒt2η,ξ,φ, thenϒt1η,ξ,φis a subhypergroup ofϒt2η,ξ,φ.Proof 5.
∀t1,t2∈0,1,ϒt1η,ξ,φ and ϒt2η,ξ,φ be the t-level sets of ϒη,ξ,φ with t1≥t2.This implies that ϒt1η,ξ,φ⊆ϒt2η,ξ,φ. By Theorem 2, ϒt1η,ξ,φ is a subhypergroup of ϒt2η,ξ,φ.Proposition 3.
IfϒandΘbe twoη,ξ,φ-SVN subset of hypergroupH, then(42)ϒ∩Θη,ξ,φ=ϒη,ξ,φ∩Θη,ξ,φ.Proof 6.
Assume thatϒ and Θ are two η,ξ,φ-SVN subset of hypergroup H.(43)ϒ∩Θη,ξ,φu=minminτϒu,τΘu,η,maxmaxιϒu,ιΘu,ξ,maxmaxϜϒu,ϜΘu,φ=∧∧τϒu,η,∧τΘu,η,∨∨ιϒu,ξ,∨ιΘu,ξ,∨∨Ϝϒu,φ,∨ϜΘu,φ=minτϒηu,τΘηu,maxιϒξu,ιΘξu,maxϜϒφu,ϜΘφu=ϒη,ξ,φu∩Θη,ξ,φu,∀u∈H.Theorem 6.
LetϒandΘbeη,ξ,φ-SVN hypergroups overH. Thenϒ∩Θis aη,ξ,φ-SVN hypergroup overHif it is non-null.Proof 7.
Letϒ and Θ be two η,ξ,φ-SVN hypergroups over H. Let u∈H be any element,(44)ϒ∩Θη,ξ,φu=τϒ∧τΘηu,ιϒ∨ιΘξuϜϒ∨ϜΘφu.By using result of Proposition 3.,(45)ϒ∩Θη,ξ,φu=ϒη,ξ,φ∩Θη,ξ,φu,(46)Yη,ξ,φ∩Θη,ξ,φu=minτϒηuu,τΘηu,maxιϒξu,ιΘξu,maxϜϒφu,ϜΘφu,.By using (44), (45), and (46), we get(47)Y∩Θη,ξ,φu=τϒ∧τΘηu,ιϒ∨ιΘξuϜϒ∨ϜΘφu=minτϒηu,τΘηu,maxιϒξu,ιΘξu,maxϜϒφu,ϜΘφu.=τϒη∧τΘηu,ιϒξ∨ιΘξuϜϒφu∨ϜΘφu
Since,ϒ∩Θη,ξ,φu=u,τϒ∩Θηu,ιϒ∩Θξu,Ϝϒ∩Θφu:u∈H.
So by using (47), we get(48)τϒ∩Θηu=τϒηu∧τΘηu,ιϒ∩Θξu=ιϒξu∨ιΘξu,
(49)Ϝϒ∩Θφu=Ϝϒφu∨ϜΘφu.(i)
For allu,v∈H,(50)minτϒ∩Θηu,τϒ∩Θηv=minτϒηu∧τΘηu,τϒηv∧τΘηv=minτϒηu∧τΘηu,τϒηv∧τΘηv≤minτϒηu,τϒηv∧minτΘηu,τΘηv≤infτϒηw:w∈u°v∧infτΘηw:w∈u°v≤infτϒηw∧τΘηw:w∈u°v=infτϒ∩Θηw:w∈u°v.This impliesminτϒ∩Θηu,τϒ∩Θηv≤infτϒ∩Θηw:w∈u°v.Similarly for allu,v∈H, we get(51)maxιϒ∩Θξu,ιϒ∩Θξv=maxιϒξu∨ιΘξu,ιϒξv∨ιΘξv≥maxιϒξu,ιϒξv∨maxιΘξu,ιΘξv≥supιϒξw:w∈u°v∨supιΘξw:w∈u°v≥supιϒξw∨ιΘξw:w∈u°v=supιϒ∩Θξw:w∈u°v.This impliesmaxιϒ∩Θξu,ιϒ∩Θξv≥supιϒ∩Θξw:w∈u°v,Similarly we can show that(52)maxϜϒ∩Θφu,Ϝϒ∩Θφv≥supϜϒ∩Θφw:w∈u°v.(ii)
∀l,u∈H, ∃v∈H such that u∈l°v,(53)minτϒ∩Θηl,τϒ∩Θηu=minτϒηl∧τΘηl,τϒηu∧τΘηu=minτϒηl,τϒηu∧minτΘηl,τΘηu≤τϒηv∧τΘηv=τϒ∩Θηv.This impliesminτϒ∩Θηl,τϒ∩Θηu≤τϒ∩Θηv.Next, we get(54)maxιϒ∩Θξl,ιϒ∩Θξu=maxιϒξl∨ιΘξl,ιϒξu∨ιΘξu=maxιϒξl,ιϒξu∨maxιΘξl,ιΘξu≥ιϒξv∨ιΘξv=ιϒ∩Θξv.This impliesmaxιϒ∩Θξl,ιϒ∩Θξu≥ιϒ∩Θξv.Similarly, we can show thatmaxϜϒ∩Θφl,Ϝϒ∩Θφu≥Ϝϒ∩Θφv.(iii)
∀l,u∈H, ∃w∈H such that u∈w°u,(55)minτϒ∩Θηl,τϒ∩Θηu=minτϒηl∧τΘηl,τϒηu∧τΘηu=minτϒηl,τϒηu∧minτΘηl,τΘηu≤τϒηw∧τΘηw=τϒ∩Θηw.
This impliesminτϒ∩Θηl,τϒ∩Θηu≤τϒ∩Θηw.
Next, we get(56)maxιϒ∩Θξl,ιϒ∩Θξu=maxιϒξl∨ιΘξl,ιϒξu∨ιΘξu=maxιϒξl,ιϒξu∨maxιΘξl,ιΘξu≥ιϒξw∨ιΘξw=ιϒ∩Θξw.
This impliesmaxιϒ∩Θξl,ιϒ∩Θξu≥ιϒ∩Θξw.
Similarly, we can show thatmaxϜϒ∩Θφl,Ϝϒ∩Θφu≥Ϝϒ∩Θφw.
Therefore,ϒ∩Θ is a η,ξ,φ-SVN hypergroup over H.Remark 1.
Union of twoη,ξ,φ-SVN hypergroups over H need not be η,ξ,φ-SVN hypergroup over H.Theorem 7.
Letϒbe aη,ξ,φ-SVN hypergroup overH. Then the falsity-favorite ofϒη,ξ,φ(i.e.,Θ=▽ϒη,ξ,φ) is a SVN hypergroup overH.Proof 8.
By definition,Θ=▽ϒη,ξ,φ, where the membership values are τΘηu=τϒηu,ιΘξu=0, and ϜΘφu=minϜϒφu+ιϒξu,1,(i)
Then we have to prove forτϒη,ϜΘφ,∀u,v∈H.(57)minτΘηu,τΘηv=minτϒηu,τϒηvby Definition≤infτϒηw:w∈u°v.And we get(58)maxϜΘφu,ϜΘφv=maxϜϒφu+ιϒξu∧1,Ϝϒφv+ιϒξv∧1=maxϜϒφu+ιϒξu,Ϝϒφv+ιϒξv∧1≥maxϜϒφu,Ϝϒφv+maxιϒξu,ιϒξv∧1≥supϜϒφw:w∈u°v+supιϒξw:w∈u°v∧1=supϜϒφw+ιϒξw∧1:w∈u°v=supϜΘw:w∈u°v,⇒maxϜΘφu,ϜΘφv≥supϜΘw:w∈u°v.Similarly we can show thatmaxιΘξu,ιΘξv≥supιΘw:w∈u°v.(ii)
∀l,u∈H, ∃v∈H such that u∈l°v,(59)minτΘηl,τΘηu=minτϒηl,τϒηuby Definition≤τϒηv.And we get(60)maxϜΘφl,ϜΘφu=maxϜϒφl+ιϒξl∧1,Ϝϒφu+ιϒξu∧1=maxϜϒφl+ιϒξl,Ϝϒφu+ιϒξu∧1≥Ϝϒφl,Ϝϒφu+ιϒξl,ιϒξu∧1≥Ϝϒφv+ιϒξv∧1=Ϝϒφv+ιϒξv∧1=ϜΘv.⇒maxϜΘφl,ϜΘφu≥ϜΘv.Similarly we can show thatmaxιΘξl,ιΘξu≥ιΘv.(iii)
∀l,u∈H, ∃w∈H such that u∈w°l,(61)minτΘηl,τΘηu=minτϒηl,τϒηuby Definition≤τϒηw.
And we get(62)maxϜΘφl,ϜΘφu=maxϜϒφl+ιϒξl∧1,Ϝϒφu+ιϒξu∧1=maxϜϒφl+ιϒξl,Ϝϒφu+ιϒξu∧1≥Ϝϒφl,Ϝϒφu+ιϒξl,ιϒξu∧1≥Ϝϒφw+ιϒξw∧1=Ϝϒφw+ιϒξw∧1=ϜΘw.⇒maxϜΘφl,ϜΘφu≥ϜΘw.
Similarly we can show thatmaxιΘξl,ιΘξu≥ιΘw.
⇒Θ=▽ϒη,ξ,φ is a SVN hypergroup over H.


## 4. (Weak) Polygroups

This section contains basic definitions, remarks, propositions, and examples of (weak) polygroups (i.e., polygroup, commutative polygroup, and noncommutative polygroup).LetH be a nonempty set, and P∗H be the collection of all nonempty subsets of H. “∗” should be formulated as follows:∗:H×H⟶P∗Hu,vu∗vThenH,∗ becomes a hypergroupoid and “∗” is a hyperoperation.Definition 14.
( [13]). Let P,∗ be a hypergroupoid. Then P,∗ is a polygroup if the aforementioned conditions are fulfilled ∀u,v,w∈P.(1)
u∗v∗w=u∗v∗w,(2)
∃e in P with e∗u=u∗e=u, ∀u∈P,(3)
u∈v∗w implies v∈u∗w−1 and w∈v−1∗u.
Weak polygroups are generalization of polygroups and they are defined in the same way as polygroups but instead of (44) in Definition 14, we have u∗v∗w∩u∗v∗w≠∅.
In a (weak) polygroupP, u−1−1=u, ∀u∈P.Remark 2.
Every group is a (weak) polygroup.
We present examples on polygroups that are not groups.Example 2.
LetP1=e,ϑ1,ϑ2. Then P1,. defined in Table 1 is a polygroup with e serving as an identity.Table 1
The polygroupP1,..
eϑ1ϑ2eeϑ1ϑ2ϑ1ϑ1e,ϑ2ϑ1,ϑ2ϑ2ϑ2ϑ1,ϑ2e,ϑ1Example 3.
( [47]). let P2=e,ϑ1,ϑ2,ϑ3. Then P2,. defined in Table 2 is a commutative polygroup with e serving as an identity.Table 2
The polygroupP2,..
.eϑ1ϑ2ϑ3eeϑ1ϑ2ϑ3ϑ1ϑ1P2ϑ1,ϑ2,ϑ3ϑ1,ϑ2,ϑ3ϑ2ϑ2ϑ1,ϑ2,ϑ3P2ϑ1,ϑ2,ϑ3ϑ3ϑ3ϑ1,ϑ2,ϑ3ϑ1,ϑ2,ϑ3P2Example 4.
( [47]). Let P3=e,ϑ1,ϑ2,ϑ3. Then P3,. defined in Table 3 is a noncommutative polygroup with e serving as an identity.Table 3
The polygroupP3,..
.eϑ1ϑ2ϑ3eeϑ1ϑ2ϑ3ϑ1ϑ1ϑ1P3ϑ3ϑ2ϑ2e,ϑ1,ϑ2ϑ2ϑ2,ϑ3ϑ3ϑ3ϑ1,ϑ3ϑ3P3Definition 15.
( [47]). A subset Q of a polygroup P,∗ is a subpolygroup of P⇔Q,∗ is a polygroup.Proposition 4.
( [47]). A subsetQofPis a subpolygroup of polygroupP,∗⇔u∗v⊆Qandu−1∈Q,∀u,v⊆Q.Definition 16.
( [47]). A subset subpolygroup Q of a polygroup P,∗ is a normal subpolygroup of P if u−1∗Q∗u⊆P, ∀u∈P.


## 5.η,ξ,φ- Single-Valued Neutrosophic (Weak) Polygroups

In this section, we present some fundamental definitions, characteristics, theorems, propositions, and examples in relation to the SVNPs,η,ξ,φ-SVNPs, η,ξ,φ-SVNWPs, and η,ξ,φ-ASVNPs. In addition to this, we provide an example of a η,ξ,φ-SVN subpolygroup that is not normal.Definition 17.
( [48]). Let P,∗ be a polygroup and ϒ be a fuzzy set with a degree of membership m over P. Then, ϒ is considered a fuzzy polygroup over P if the followings conditions are satisfied ∀u,v∈P.(1)
mw≥minmu,mv, ∀w∈u∗v,(2)
mu−1≥mu.Remark 3.
( [44]). Intersection of fuzzy polygroups over P is a fuzzy polygroup.Definition 18.
Ifϒ be a single-valued neutrosophic (SVN) subset of Ω, then a η,ξ,φ-SVN subset ϒ of Ω is categorize as(63)ϒη,ξ,φ=e,τϒηu,ιϒξu,Ϝϒφu|u∈Ω,where(64)τϒηu=∧τϒu,η,ιϒξu=∧ιϒu,ξ,Ϝϒφu=∨Ϝϒu,φ,
such that(65)0≤τϒηu+ιϒξu+Ϝϒφu≤3.Here, η,ξ,φ∈0,1, also τ,ι,Ϝ:A⟶0,1, such that τϒη, ιϒξ, Ϝϒφ represents the functions of truth, indeterminacy, and falsity-membership, respectively.Definition 19.
Letϒ and Θ be two η,ξ,φ-SVNSs on Ω. The followings must hold the following:(1)
ϒη,ξ,φ⊆Θη,ξ,φ⇔ϒη,ξ,φu≤Θη,ξ,φu.That is,(66)τϒηu≤τΘηu,ιϒξu≤ιΘξu,Ϝϒφu≥ϜΘφu,and(67)ϒη,ξ,φ=Θη,ξ,φ⇔ϒη,ξ,φ⊆Θη,ξ,φandΘη,ξ,φ⊆ϒη,ξ,φ.(2)
The union ofϒη,ξ,φ and Θη,ξ,φ is indicated by(68)Δη,ξ,φ=ϒη,ξ,φ∪Θη,ξ,φ,and defined as(69)Δη,ξ,φu=ϒη,ξ,φu∨Θη,ξ,φu,where(70)ϒη,ξ,φu∨Θη,ξ,φu=τϒηu∨τΘηu,ιϒξu∨ιΘξu,Ϝϒφu∧ϜΘφu,∀u∈Ω.That is,(71)τΔηu=maxτϒηu,τΘηu,ιΔξu=maxιϒξu,ιΘξu,ϜΔφu=minϜϒφu,ϜΘφu.(3)
The intersection ofϒη,ξ,φ and Θη,ξ,φ is indicated by(72)Δη,ξ,φ=ϒη,ξ,φ∩Θη,ξ,φ,
and defined as(73)Δη,ξ,φu=ϒη,ξ,φu∧Θη,ξ,φu,where(74)ϒη,ξ,φu∧Θη,ξ,φu=τϒηu∧τΘηu,ιϒξu∧ιΘξu,Ϝϒφu∨ϜΘφu,∀u∈Ω.
That is,(75)τΔηu=minτϒηu,τΘηu,ιΔξu=minιϒξu,ιΘξu,ϜΔφu=maxϜϒφu,ϜΘφu.Definition 20.
The complement of aη,ξ,φ-SVNS ϒ is denoted by cϒη,ξ,φ and is defined by(76)cϒη,ξ,φ=u,τcϒηu,ιcϒξu,Ϝcϒφu,where(77)τcϒηu=Ϝϒφu,ιcϒξu=1−ιϒξu,Ϝcϒφu=τϒηu,∀u∈Ω.Definition 21.
LetP,∗ be a (weak) polygroup and ϒ a η,ξ,φ-SVNS over P. Then ϒ is called a η,ξ,φ-SVNP over P (η,ξ,φ-SVN weak polygroup (η,ξ,φ-SVNWP) over P) if for all ∀u,v∈P, the following conditions are satisfied.(1)
τϒηw≥minτϒηu,τϒηv, ιϒξw≥minιϒξu,ιϒξv and Ϝϒφw≤maxϜϒφu,Ϝϒφvfor allz∈x∗y,(2)
τϒηu−1≥τϒηu, ιϒξu−1≥ιϒξu and Ϝϒφx−1≤Ϝϒφu.Example 5.
LetP4=0,1. Then P4,∗ defined in Table 4 is a polygroup with 0 serving as an identity.
Let(78)ϒ=0.6,0.7,0.20+0.1,0.3,0.71.
Considerη=0.4,ξ=0.5,φ=0.5.
Thenϒη,ξ,φ=0.4,0.5,0.5/0+0.1,0.3,0.7/1.
⇒ϒ is a η,ξ,φ-SVNP over P4.Table 4
The polygroupP4,∗.
∗0100111P1Example 6.
LetP5=e,ϑ1,ϑ2,ϑ3. Then P5,° defined in Table 5 is a weak polygroup with e serving as an identity. Moreover, it is not a polygroup because ° is not associated, i.e.,(79)ϑ1°ϑ2°ϑ3=ϑ3°ϑ3=e,ϑ3,ϑ1°ϑ2°ϑ3=ϑ1°ϑ1=e,ϑ1.
Let(80)ϒ=0.8,0.7,0.3/e+0.1,0.3,0.7/ϑ1+0.1,0.25,0.9/ϑ2+0.1,0.25,0.9/ϑ3.
Considerη=0.4,ξ=0.5,φ=0.5.
Thenϒη,ξ,φ=0.4,0.5,0.5/e+0.1,0.3,0.7/ϑ1+0.1,0.25,0.9/ϑ2+0.1,0.25,0.9/ϑ3.
⇒ϒ is a η,ξ,φ-SVNWP over P5.Table 5
The polygroupP5,°.
°eϑ1ϑ2ϑ3eeϑ1ϑ2ϑ3ϑ1ϑ1e,ϑ1ϑ3ϑ2ϑ2ϑ2ϑ3e,ϑ2ϑ1ϑ1ϑ1ϑ2ϑ1e,ϑ3Remark 4.
All the theorems and results in this paper that are valid forη,ξ,φ-SVNP are also valid for η,ξ,φ-SVNWP. So, we restrict our results to η,ξ,φ-SVNP.Proposition 5.
Letϒaη,ξ,φ-SVNP over polygroupP,∗. Then the preceding holds true∀u∈P.(1)
τϒηu−1=τϒηu, ιϒξu−1=ιϒξu, andϜϒφu−1=Ϝϒφu;(2)
τϒηe≥τϒηu,ιϒξe≥ιϒξu, andϜϒφe≤Ϝϒφuwhereeis the identity inP.Proof 9.
Letu∈P.(1)
By Definition21 implies that τϒηu−1≥τϒηu, ιϒξu−1≥ιϒξu, and Ϝϒφu−1≤Ϝϒφu. Also we have u−1−1=u implies that τϒηu≥τϒηu−1, ιϒξu≥ιϒξu−1, and Ϝϒφu≤Ϝϒφu−1. Thus, τϒηu−1=τϒηu, ιϒξu−1=ιϒξu, and Ϝϒφu−1=Ϝϒφu.(2)
Sincee∈u∗u−1, it follows by Definition 21 (1) that τϒηe≥minτϒηu,τϒηu−1=τϒηu, ιϒξe≥minιϒξu,ιϒξu−1=ιϒξu, and Ϝϒφe≤maxϜϒφu,Ϝϒφu−1=Ϝϒφu.Example 7.
LetP6=e,ϑ1,ϑ2,ϑ3. Then P6,. defined in Table 6 is a polygroup with e serving as an identity.
Let(81)ϒ=0.4,0.6,0.2e+0.8,0.3,0.7ϑ1+0.7,0.3,0.7ϑ2+0.1,0.2,0.8ϑ3.
Considerη=0.5,ξ=0.5, and φ=0.5.
Then(82)ϒη,ξ,φ=0.4,0.5,0.5e+0.8,0.3,0.7ϑ1+0.5,0.3,0.7ϑ2+0.1,0.2,0.8ϑ3.
⇒ϒη,ξ,φ is not a η,ξ,φ-SVNP over P6 as τϒηe≥τϒηϑ1 does not hold.Table 6
The polygroupP6,..
.eϑ1ϑ2ϑ3eeϑ1ϑ2ϑ3ϑ1ϑ1eϑ2ϑ3ϑ2ϑ2ϑ2e,ϑ1,ϑ3ϑ2,ϑ3ϑ3ϑ3ϑ3ϑ2,ϑ3e,ϑ1,ϑ2Example 8.
LetP6,. be the polygroup in example 7. Then e and e,ϑ1 are subpolygroups of P6 that are not normal.Proposition 6.
Letϒbe aη,ξ,φ-SVNS over polygroupP,∗, andϒη,ξ,φ−1=τϒηu−1,ιϒξu−1,Ϝϒφu−1/u:u∈P. Ifϒis aη,ξ,φ-SVNP overPthenϒη,ξ,φ−1=ϒη,ξ,φ.Proof 10.
It is simple by using Proposition5.Proposition 7.
LetP,∗be a polygroup,τ1,τ2,τ3be numbers in the unit interval0,1.Ifϒη,ξ,φ=τ1,τ2,τ3/u:u∈P. Thenϒis aη,ξ,φ-SVNP overP.Proof 11.
The proof is simple for readers.Remark 5.
Theη,ξ,φ-SVNP present in Proposition 6 is called the constant η,ξ,φ-SNVP.Theorem 8.
Letϒaη,ξ,φ-SVNS over polygroupP,∗. Thenϒandcϒareη,ξ,φ-SVNP overPif and only ifϒis the constantη,ξ,φ-SVNP.Proof 12.
Ifϒ is the constant η,ξ,φ-SVNP over P then cϒ is also the constant η,ξ,φ-SVNP over P. Let ϒ and cϒ be η,ξ,φ-SVNP. Then ∀u∈P, we have(83)τϒηe≥τϒηu,ιϒξe≥ιϒξu,Ϝϒφe≤Ϝϒφu(84)Ϝϒφe≥Ϝϒφu,1−ιϒξe≥1−ιϒξu,τϒηe≤τϒηu
Equations (83) and (84) implies that τϒηe=τϒηu, ιϒξe=ιϒξu, Ϝϒφe=Ϝϒφu. Thus, ϒ is the constant η,ξ,φ-SVNP over P.Definition 22.
( [48]). Let ϒ be a fuzzy set over a polygroup P,∗ with membership function m. Then ϒ is called the anti-fuzzy polygroup over P if ∀u,v∈P, the following conditions are fulfilled.(1)
mw≤maxmu,mv, ∀w∈u∗v,(2)
mu−1≤mu.Remark 6.
( [48]). Union of anti-fuzzy polygroups over P is an anti-fuzzy polygroup.Definition 23.
Letϒ be a η,ξ,φ-SVNS over polygroup P,∗. Then ϒ is called an η,ξ,φ-anti SVNP (η,ξ,φ-ASVNP) over P if ∀u,v∈P, the following conditions are satisfied.(1)
τϒηw≤maxτϒηu,τϒηv, ιϒξw≤maxιϒξu,ιϒξv and Ϝϒφw≥minϜϒφu,Ϝϒφv for all w∈u∗v,(2)
τϒηu−1≤τϒηu, ιϒξu−1≤ιϒξu, Ϝϒφu−1≥Ϝϒφu.Proposition 8.
LetP,∗be a polygroup andϒanη,ξ,φ-ASVNP overPThen following holds true:∀u∈P.(1)
τϒηu−1=τϒηu, ιϒξu−1=ιϒξuandϜϒφu−1=Ϝϒφu.(2)
τϒηe≤τϒηu, ιϒξe≤ιϒξu, andϜϒφe≥Ϝϒφu where e is the identity element in P.Proof 13.
The proof is similar to that of 5.10.Example 9.
ConsiderP4,∗ be the polygroup present in example 5.
Let(85)ϒ=0.4,0.7,0.90+0.6,0.8,0.21.
Considerη=0.5,ξ=0.5,φ=0.7.
Thenϒη,ξ,φ=0.4,0.5,0.9/0+0.5,0.5,0.7/1.
⇒ϒ is a η,ξ,φ-ASVNP over P4.Theorem 9.
Letϒbe aη,ξ,φ-SVNS over polygroupP,∗. Thenϒis aη,ξ,φ-SVNP overPif and only ifτϒηandιϒξare fuzzy polygroups overPandϜϒφis an anti-fuzzy polygroup overP.Proof 14.
It follows from the definition ofη,ξ,φ-SVNP, fuzzy polygroups, and anti-fuzzy polygroups.Theorem 10.
Letϒbe aη,ξ,φ-SVNS over polygroupP,∗. Thenϒis acϒ-ASVNP overPif and only ifτϒηandιϒξare anti-fuzzy polygroups overPandϜϒφis an fuzzy polygroup overP.Proof 15.
It follows from the definition ofη,ξ,φ-ASVNP, fuzzy polygroups, and anti-fuzzy polygroups.Theorem 11.
Letϒbe aη,ξ,φ-SVNS over polygroupP,∗. Thenϒis aη,ξ,φ-SVNP overPif and only ifcϒis anη,ξ,φ-ASVNP overP.Proof 16.
Letϒ be a η,ξ,φ-SVNP. Theorem 9 asserts that τϒη and ιϒξ are fuzzy polygroups over P and Ϝϒφ is an anti-fuzzy polygroup over P. We get now that τcϒη=Ϝϒφ and ιcϒξ=1−ιϒξ are anti-fuzzy polygroups over P and Ϝcϒφ=τϒη is a fuzzy polygroup over P. Using Theorem 10, it completes the proof. Similarly, we can prove that if cϒ is an η,ξ,φ-ASVNP over P then ϒ is a η,ξ,φ-SVNP.Corollary 1.
Letϒλbe aη,ξ,φ-SVNS over polygroupP,∗. Ifϒλis aη,ξ,φ-SVNP overPthen∩λ∈φϒλisη,ξ,φ-SVNP overP.Corollary 2.
Letϒλbe aη,ξ,φ-SVNS over polygroupP,∗. Ifϒλis aη,ξ,φ-ASVNP overPthen∩λ∈φϒλis anη,ξ,φ-ASVNP overP.


## 6. Level Sets ofη,ξ,φ-Single-Valued Neutrosophic (Weak) Polygroups

This section defines level sets ofη,ξ,φ-SVNPs and relate them with (normal) subpolygroups.Definition 24.
Letξ be any set t=τ1,τ2,τ3, where 0≤τ1,τ2<1 and 0<τ3≤1, and ϒ be a η,ξ,φ-SVNS over ξ. Then ϒtη,ξ,φ=u∈ξ:τϒηu≥τ1,ιϒξu≥τ2,Ϝϒφu≤τ3 is named a t-level set of ϒη,ξ,φ.Theorem 12.
Letϒbe aη,ξ,φ-SVNS over polygroupP,∗. Thenϒis aη,ξ,φ-SVNP overP⇔ϒtη,ξ,φ≠∅is a subpolygroup ofHTML translation failedfor everyt=τ1,τ2,τ3, where0≤τ1,τ2<1and0<τ3≤1.Proof 17.
Letϒ be a η,ξ,φ-SVNP over P and u,v∈ϒtη,ξ,φ≠∅. ∀w∈u∗v, we have(86)τϒηw≥minτϒηu,τϒηv≥τ1,ιϒξw≥minιϒξu,ιϒξv≥τ2,
(87)Ϝϒφw≤maxϜϒφu,Ϝϒφv≤τ3.
Thusu∗v⊆ϒtη,ξ,φ.
Furthermore, we have(88)τϒηu−1≥τϒηu≥τ1,ιϒξu−1≥ιϒξu≥τ2,
(89)Ϝϒφu−1≤Ϝϒφu≤τ3.
This implies thatu−1∈ϒtη,ξ,φ. Thus, ϒtη,ξ,φ is a subpolygroup of P.
Conversely, letϒtη,ξ,φ≠∅ be a subpolygroup of P and u,v∈P.
Setτ1=minτϒηu,τϒηv, τ2=minιϒξu,ιϒξv, τ3=maxϜϒφu,Ϝϒφv and t=τ1,τ2,τ3.
So it illustrates thatu∗v⊆ϒtη,ξ,φ and u−1∈ϒtη,ξ,φ.
This indicates∀w∈u∗v,(90)τϒηw≥τ1=minτϒηu,τϒηv,ιϒξw≥τ2=minιϒξu,ιϒξv,
(91)Ϝϒφw≤τ3=maxϜϒφu,Ϝϒφv.
As a result, condition (1) of Definition17 is achieved.
Moreover,(92)τϒηu−1≥τ1=τϒηu,ιϒξu−1≥τ2=ιϒξu,
(93)Ϝϒφu−1≤τ3=Ϝϒφu.
Thus, condition (2) of Definition17 is satisfied. Therefore, ϒ become η,ξ,φ-SVNP over P.Corollary 3.
Letϒ be a η,ξ,φ-SVNP over polygroup P,∗. Then P has no non-trivial proper subpolygroups if and only if the constant η,ξ,φ-SNVP and ϒη,ξ,φ=τ1,τ2,τ3/u+τ1′,τ2′,τ3′/e:u≠e∈P, where τ1≥τ1′,τ2≥τ2′ and τ3≤τ3′ are the only η,ξ,φ-SVNP over P.Example 10.
LetP4=0,1 and P4,∗ be the polygroup referred in example 5. Then the constant η,ξ,φ-SNVP and ϒη,ξ,φ=τ1,τ2,τ3/1+τ1′,τ2′,τ3′/0, where τ1≤τ1′, τ2≤τ2′, and τ3≥τ3′ are the only η,ξ,φ-SVNP over P4.Notation 1.
Lett=τ1,τ2,τ3 and let ϒ is a η,ξ,φ-SVNS of P. Then by Au=t, we means that τϒηu=τ1, ιϒξu=τ2 and Ϝϒφu=τ3. And by ϒη,ξ,φu≤t, we mean that τϒηu≤τ1, ιϒξu≤τ2, and Ϝϒφu≥τ3.Theorem 13.
Each subpolygroup of polygroupP,∗is a level set of aη,ξ,φ-SVNP overP.Proof 18.
LetQ be a subpolygroup of P, consider t=τ1,τ2,τ3, where 0<τ1,τ2<1, and 0<τ3<1. Define the η,ξ,φ-SVNS over P as follows:(94)ϒη,ξ,φu=τ1,τ2,τ3,ifx∈Q,0,0,1,otherwise.
Lett′=τ1′,τ2′,τ3′. Then(95)At′η,ξ,φ=Qifτ1≥τ1′,τ2≥τ2′,τ3≤τ3′,Pifτ1′=0,τ2′=0,τ3′=1,is either∅orasubpolygroup ofP.∅otherwise.
Using Theorem12, we get that ϒ is a η,ξ,φ-SVNP over P.Definition 25.
Letϒ be a η,ξ,φ-SVNP over polygroup P,∗. Then ϒ is said to be a normal η,ξ,φ-SVNP over P if ϒη,ξ,φw=ϒη,ξ,φw′, ∀w∈u∗v, and w′∈v∗u.Example 11.
Letϒ be a η,ξ,φ-SVNP over polygroup P,∗. Then the constant η,ξ,φ-SNVP is a normal η,ξ,φ-SNVP over P.Theorem 14.
Letϒis aη,ξ,φ-SVNS over polygroupP,∗. Thenϒis a normalη,ξ,φ-SVNP overP⇔ϒtη,ξ,φ≠∅is a normal subpolygroup ofPfor everyt=τ1,τ2,τ3, where0≤τ1,τ2<1, and0<τ3≤1.Proof 19.
Letϒ be a normal η,ξ,φ-SVNP over P and u,v∈ϒtη,ξ,φ≠∅. Theorem 12 argues that ϒtη,ξ,φ≠∅ is a subpolygroup of P. Let u∈P. We need to show that u−1∗ϒtη,ξ,φ∗u⊆ϒtη,ξ,φ. Let w∈u−1∗ϒtη,ξ,φ∗u. Then ∃v in ϒtη,ξ,φ such that w∈u−1∗v∗u, hence w∈u−1∗p, where p∈v∗u. The latter implies that v∈p∗u−1. And since ϒ is a normal η,ξ,φ-SVNP over P. Accordingly, ϒη,ξ,φw=ϒη,ξ,φv. Hence, w∈ϒtη,ξ,φ.
Conversely, supposeϒtη,ξ,φ≠∅ be a normal subpolygroup of P. Theorem 12 argues that ϒ is a η,ξ,φ-SVNP over P. To show that ϒ is a normal η,ξ,φ-SVNP over P, it is sufficient to enhance that ϒη,ξ,φw=ϒη,ξ,φw′∀w∈u∗v, w′∈v∗u.
Letw∈u∗v, w′∈v∗u with ϒη,ξ,φw′=t. Having w′∈v∗u implies that v∈w′∗u−1. The latter reveals that let w∈u∗v′∗u−1. Since w′∈ϒtη,ξ,φ and ϒtη,ξ,φ≠∅ is a normal subpolygroup of P, it pursues that w∈ϒtη,ξ,φ and therefore, ϒη,ξ,φw≥ϒη,ξ,φw′=t. Similarly, we get that ϒη,ξ,φw′≥ϒη,ξ,φw. ⇒ϒη,ξ,φw′=ϒη,ξ,φw.Corollary 4.
Letϒbe aη,ξ,φ-SVNP over polygroupP,∗. ThenPhas no proper normal subpolygroups if and only if the constantη,ξ,φ-SNVP is the only normalη,ξ,φ-SNVP overP.Example 13.
LetP6=e,ϑ1,ϑ2,ϑ3 and P6,. be the polygroup illustrated in example 7. Then the constant η,ξ,φ-SNVP is the only normal η,ξ,φ-SVNP over P6.Theorem 15.
Every normal subpolygroup of polygroupP,∗is a level set of a normalη,ξ,φ-SVNP overP.Proof 20.
The result is identical to that of Theorem13.Corollary 5.
Letϒ be a η,ξ,φ-SVNP over polygroup P,∗. Then ϒη,ξ,φ∗=u∈P:ϒη,ξ,φu=ϒη,ξ,φe is a subpolygroup of P. Moreover, if ϒ is a normal η,ξ,φ-SVNP over P, then ϒη,ξ,φ∗ is a normal subpolygroup of P.Proof 21.
Lett=ϒη,ξ,φe. Then ϒtη,ξ,φ=u∈P:τϒηu≥τϒηe,ιϒξu≥ιϒξe,ϜϒφuϜϒφe. Proposition 5 and Proposition 6 asserts that(96)ϒtη,ξ,φ=u∈P:τϒηu=τϒηe,ιϒξu=ιϒξe,Ϝϒφu≤Ϝϒφe=ϒη,ξ,φ∗.
Theorem12 and Theorem 14 complete the proof.


## 7. Single-Valued Neutrosophic Multicriteria Decision-Making Method

Multiple-criteria decision-making is an operations research subdiscipline that explicitly assesses multiple competing criteria in decision-making (both in everyday life and in settings as well as in situations like as the business, government, and medicine). M-CDM offers a basis for choosing, categorizing, and ranking items and aids in the overall evaluation. M-CDM is a useful tool that may be used to a variety of complicated/sophisticated or when the materials are novel. It is especially beneficial in circumstances involving a decision between options. It helps us to focus on the real issues and it is logical and consistent and is easy to use; it has all the qualities of an excellent decision-making tool.A SVNS is a stereotype of a classic set, a fuzzy set, a paraconsistent set, and an intuitionistic fuzzy set. It is more broad and can handle not only partial information but also equivocal and unreliable information, both of which are typical in real-world situations. As a result, SVN D-M is more suited for real-world scientific and technical applications.In this section, we present strategies for resolving M-CDM issues in a SVN environment by using the WCSM between SVNSs.Assumeℜ1,ℜ2,ℜ3,…,ℜr resemble the alternatives and Y1,Y2,Y3,…,Yy represent the set of criteria. Consider the weight of the criterion Yjj=1,2,…,y enters by decision-makers is wj,wj∈0,1 and∑j=1ywj=1. The preceding SVNS indicate the feature of alternative ℜii=1,2,…,r in this case:(97)ℜi=∑j=1yτℜiYj,ιℜiYj,ϜℜiYjYj,Yj∈Y,where τℜiYj,ιℜiYj,ϜℜiYj∈0,1,j=1,2,…,y and i=1,2,…,r.We represent a SVNS byδij=ρij,ϱij,σij. An SVNS is often synthesized from the evaluation of an alternative ℜi with regard to a criteria Yj in implementation using a score law and data processing. As a result, we may derive a SVN decision matrix D=δijr×y.The notion of ideal point has been intended to assist discover the optimal option in a M-CDM scenario. Although the perfect alternative does not exist in the real world, it does give a valuable theoretical framework against which alternatives may be evaluated.The notion of optimum point has been achieved by involving to discover the optimal option in a M-CDM context. Although the perfect alternative somehow does not persist in the everyday life, it does give a valuable theoretical framework against which alternatives may be evaluated.As a reason, the ideal alternativeℜ@ is defined as the SVNS δj=ρj@,ϱj@,σj@=1,0,0 for j=1,2,…y. The WCSM between an alternative ℜi and the ideal alternative ℜ∗ represented by the SVNSs is defined by(98)Qiℜi,ℜ@=∑j=1ywjρij.ρj∗+ϱij.ϱj∗+σij.σj∗ρij2+ϱij2+σij2.ρj@2+ϱj@2+σj@2Then, the higher the WCSM value, the better the option. The measure values can produce the ranking order of all alternatives and the best option by using (98).


## 8. Application

This section demonstrates an overview of a M-CDM issue with choices to exemplify the relevance and efficacy of the offered D-M strategy. Consider the paradox of D-M. There is an investment firm that wants to put money into the finest choice.There is a panel with four potential financing options:(1)
ℜ1 is a manufacturer of automobiles;(2)
ℜ2 is a manufacturer of electronics;(3)
ℜ3 is a vacation rentals; and(4)
ℜ4 is an industrial 3D printing builder company.The investment firm must make a judgement based on the three criteria listed below:(1)
Y1 is the financial, risk, and sensitivities;(2)
Y2 is the progress assessment; and(3)
Y3 is the environmental and location assessment.The criteria’s weight vector is hence specified byw=0.30,0.25,0.45.The questionnaire of a professional expert is used to appraise an alternativeℜii=1,2,3,4 in relation to a criteria Yjj=1,2,3.When asked to experts of their opinion on a potential alternativeℜ1 corresponding to Y1, for instance, an expert might respond that there is a 0.6 chance that the statement is superb, a 0.2 chance that it is low, and a 0.1 chance that they are unsure. It may be written as δ11=0.6,0.2,0.1 using the neutrosophic notation. The following SVN decision matrix D may be obtained when the expert evaluates the four potential options in light of the aforementioned three criteria:(99)D=0.6,0.2,0.10.5,0.4,0.30.7,0.5,0.40.3,0.2,0.30.8,0.3,0.50.5,0.3,0.20.9,0.5,0.40.7,0.6,0.50.6,0.5,0.40.8,0.7,0.30.4,0.1,0.10.9,0.2,0.2.By employing (98), we can also give the following values of WCSM Qiℜi,ℜ@i=1,2,3,4 as(100)Q1ℜ1,ℜ@=0.7899;Q2ℜ2,ℜ@=0.7589;Q3ℜ3,ℜ@=0.7190;Q4ℜ4,ℜ@=0.8823:The four options are thus ranked as follows:ℜ4,ℜ1,ℜ2, and ℜ3.According to the order described by the rank matrix, industrial 3D printing builder company is turn out to be the best investment firm to put money into the finest choice whereas vacation rentals is the worst as per the criteria described.


### 8.1. Superiority of the Proposed Approach

Through this analysis and comparison, it was possible to conclude that the proposed procedure has produced more frequent results than either of the alternatives. In general, the D-M approach associated with prevalent D-M methods permits additional data to alleviate hesitancy. In the D-M process, it is thus acceptable to propagate false and unclear information. Therefore, the proposed method is reasonable, modest, and ahead of the fuzzy set’s characteristic structures. The general information associated with the object could be stated precisely and analytically, as shown in Table7.Table 7
Comparison between SVNS and some existing approaches.
SetTruthIndeterminacyFalsityAttributesZadeh [1]Fuzzy set√××Atanassov [3]Intuitionistic fuzzy set√×√√Yager [49]Pythagorean fuzzy set√×√√Chen et al. [50]m-polar fuzzy set√××√Naeem et al. [51]Pythagorean m-polar fuzzy set√×√√Maji et al. [52]Fuzzy soft set√××√Maji et al. [53]Intuitionistic fuzzy soft set√×√√Peng et al. [54]Pythagorean fuzzy soft set√×√√Zulqarnain et al. [55]Intuitionistic fuzzy hyper soft set√×√√Zulqarnain et al. [56]Pythagorean fuzzy hyper soft set√×√√Proposed techniqueSVNS√√√√


## 8.1. Superiority of the Proposed Approach

Through this analysis and comparison, it was possible to conclude that the proposed procedure has produced more frequent results than either of the alternatives. In general, the D-M approach associated with prevalent D-M methods permits additional data to alleviate hesitancy. In the D-M process, it is thus acceptable to propagate false and unclear information. Therefore, the proposed method is reasonable, modest, and ahead of the fuzzy set’s characteristic structures. The general information associated with the object could be stated precisely and analytically, as shown in Table7.Table 7
Comparison between SVNS and some existing approaches.
SetTruthIndeterminacyFalsityAttributesZadeh [1]Fuzzy set√××Atanassov [3]Intuitionistic fuzzy set√×√√Yager [49]Pythagorean fuzzy set√×√√Chen et al. [50]m-polar fuzzy set√××√Naeem et al. [51]Pythagorean m-polar fuzzy set√×√√Maji et al. [52]Fuzzy soft set√××√Maji et al. [53]Intuitionistic fuzzy soft set√×√√Peng et al. [54]Pythagorean fuzzy soft set√×√√Zulqarnain et al. [55]Intuitionistic fuzzy hyper soft set√×√√Zulqarnain et al. [56]Pythagorean fuzzy hyper soft set√×√√Proposed techniqueSVNS√√√√


## 9. Conclusion

This paper presented an algebraic hyperstructure ofη,ξ,φ-SVNSs in the form of η,ξ,φ-SVN hypergroup, η,ξ,φ-SVNPs, and η,ξ,φ-ASVNPs. Several intriguing properties of the newly defined notions were discussed. The findings of this article can be thought of as a generalization of prior research on fuzzy hypergroups and fuzzy polygroups. We also discussed in this section a M-CDM system developed in an SVN environment using WCSM. WCSM between each option and the ideal alternative may be used to establish the ranking order of all alternatives and to readily identify the greatest alternative. Finally, an instructive example demonstrated how the new technique may be used. As a result, the proposed SVN M-CDM technique is more suited for real-world scientific and engineering applications since it can manage not only inadequate information but also indeterminate and inconsistent information, both of which are typical in real-world scenarios. The strategy suggested in this study enhances previous D-M methods and offers decision-makers with an useable method.This work provided an algebraic hyperstructure ofη,ξ,φ-SVNSs as η,ξ,φ-SVN hypergroup, η,ξ,φ-SVNPs, and η,ξ,φ-ASVNPs. Several remarkable characteristics of the newly formed concepts were addressed. The results of this article can be seen as a generalization of previous research on fuzzy hypergroups and fuzzy polygroups. In this part, we also described an M-CDM system constructed in an SVN environment utilizing WCSM. WCSM between each option and the best option may be used to define the ranking order of all options and quickly discover the best choice. Finally, an illustrative illustration explained how the new method may be implemented. Consequently, the suggested SVN M-CDM approach is more suitable for real-world scientific and engineering applications, since it can handle not only insufficient information but also indeterminate and inconsistent information, both of which are characteristic of real-world settings. This research proposes an approach that advances earlier D-M methods and provides decision-makers with a practical method.(i)
Researchers will continue to work on complex D-M issues with uncertain weights of criteria, as well as other disciplines such as expert systems, information fusion systems, biochemistry, epidemiology, geology, entomology, and biomedical engineering. In the realm of algebraic structure theory, it possesses a fantastic novel idea that has the potential to be utilized in the future for the solution of a variety of algebraic issues.(ii)
Using the algebraic structure of multi-polygroup in terms of intuitionistic fuzzy set theory, this method may be readily extended to the intuitionistic fuzzy multi-polygroups. Connecting intuitionistic fuzzy multiset theory, set theory, and polygroup theory may provide a novel notion of polygroup that may be used to illustrate the effect of intuitionistic fuzzy multisets on a polygroup’s structure. Using this concept, researchers may study intuitionistic fuzzy normal multi-subpolygroups along with their characterizations and algebraic characteristics. Additionally, the homomorphisms of intuitionistic fuzzy multi-polygroups and some of their structural properties may be addressed. Additionally, this idea may be used to investigate intuitionistic fuzzy quotient multi-polygroups.(iii)
Researchers may expand this concept to include various neutrosophic multi-topological group structures. For this, they can introduce the definition of semi-open neutrosophic multiset, semi-closed neutrosophic multiset, neutrosophic multi-regularly open set, neutrosophic multi-regularly closed set, neutrosophic multi-continuous mapping. In addition, since the idea of the almost topological group is so novel, they may utilize the definition of neutrosophic multi almost topological group to define neutrosophic multi almost topological group.(iv)
This idea can be used to the development of the neutrosophic multi almost topological group of the neutrosophic multi-vector spaces, etc. This notion can be expanded to soft neutrosophic polygroups, weak soft neutrosophic polygroups, strong soft neutrosophic polygroups, soft neutrosophic polygroup homomorphism, and soft neutrosophic polygroup isomorphism. Furthermore, scholars might explore the homological properties of these polygroups.



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