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.

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 .

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 [2042] 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 concept -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) -single-valued neutrosophic hypergroups. (2) -single-valued neutrosophic polygroups. (3) -anti-single-valued neutrosophic polygroups. (4) Single-valued neutrosophic multicriteria decision-making method. Note that, clearly , 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, -single-valued neutrosophic hypergroup, and (weak) polygroups in Sections 2, 3, and 4, respectively. We present and analyze the idea of a -single-valued neutrosophic (weak) polygroup in Section 5. In Section 6, we explore the correlation between level sets of -single-valued neutrosophic polygroups (-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 a SVNS is stated aswhere , and , , . , , indicates truth, indeterminacy, and falsity-membership function, in that order.

Definition 2. ( [44]). Let be a set of objects, with denoting a generic entity belong to . A SVNS on is symbolized by truth , indeterminacy , and falsity-membership function , in that order. , . A SVNS can be written accordingly as

Definition 3. (see [44]). The complement of a SVNS is indicated by and is characterized by

Definition 4. ( [44]). Let and be two SVNSs on . Then(1)That isAlso(2) such thatsuch thatIt means(3) such thatsuch thatIt means

Proposition 1. ( [44]). Let the SVNSs on the common universe be , , and . Then the following conditions must hold the following:(1).(2).(3).(4), where(5).

3. - 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 hypergroup under the triplet structure in this section. We basically start with some introductory -SVNS, then define -SVN hypergroup, the t-level set on -SVNS, important operations and properties of -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 -SVN hypergroups over is again -SVN hypergroup in 3.18, which shows that -SVN hypergroups are closed under intersection, and union of two -SVN hypergroups over need not be -SVN hypergroup over .

Definition 5. If be a single-valued neutrosophic (SVN) subset of , then -SVN subset of is categorize aswheresuch thatwhere , also , such that , , represents the functions of truth, indeterminacy, and falsity-membership, respectively.

Definition 6. Let be a space of objects, with denoting a generic entity belong to . A -SVNS on is symbolized by truth , indeterminacy , and falsity-membership function , respectively. For every in , , a -SVNS can be written accordingly as

Definition 7. Let and be two -SVNSs on . The followings must hold the following:(1).     That is,(2)The union of and is indicated byand defined aswhereThat is,(3)The intersection of and is indicated byand defined aswhereThat is,

Proposition 2. Let,, andbe-SVNSs on the common universeThen the following properties must hold the following:(1)..(2). .(3) .(4),where(5) ? ?

Definition 8. The complement of a -SVNS is denoted by and is defined bywhere

Definition 9. The falsity-favorite of a -SVNS (i.e., ) whose truth and falsity-membership functions are defined byThroughout this section denotes the hypergroup .

Definition 10. ( [45]). A set is called hypergroup with an associative hyperoperation , which satisfies , (reproduction axiom).

Definition 11. ( [46]). If the following properties satisfy, a hyperstructure is called a -group.(1), , (-semigroup).(2), .

Definition 12. ( [45]). A subset of is called as subhypergroup if is a hypergroup.

Definition 13. Let be a -SVNS over . Then is called a -SVN hypergroup over , if the following conditions are satisfied:(i),(ii), such that and(iii), such that andIf satisfies condition (i) then is a -SVN semihypergroup over . Condition (ii) and (iii) represent the left and right reproduction axioms, respectively. Then is a -SVN subhypergroup of .

Example 1. If the family of t-level sets of -SVNS over .is a subhypergroup of . Then is a -SVN hypergroup over .

Theorem 1. Letbe a-SVNS over. Thenis a-SVN hypergroup overif and only ifis a-SVN semihypergroup overand alsosatisfies the left and right reproduction axioms.

Proof 1. The proof is obvious from Definition 13.

Theorem 2. Letbe a-SVNS over. Ifis a SVN hypergroup over, thenis a subhypergroup of.

Proof 2. Let be a -SVN hypergroup over and let , thenThen we haveThis implies . Then .
Thus , we obtain.
Now, Let , then there exist such that andThis implies that . This proves that . As such.
which proves that is a subhypergroup of .

Theorem 3. Letbe a-SVNS overThen the following are equivalent:(i) is a -SVN hypergroup over .(ii) is a subhypergroup of .

Proof 3. (i) (ii) The proof is obvious from Theorem 2. (ii) (i) Now assume that is a subhypergroup of . Let and let. Since , then for every .Condition (i) is verified.
Next, let , for every and
let .
Then there exist such that . Since , thenCondition (ii) is verified.
Next, let , for every and
let .
Then there exist such that . Since , thenCondition (iii) is verified.

Theorem 4. Letbe a-SVNS over. Thenbe a-SVN hypergroup overif and only ifis a subhypergroup of.

Proof 4. The proof is simple for readers.

Theorem 5. Letbe a-SVN hypergroup overandandbe the t-level sets ofwith, thenis a subhypergroup of.

Proof 5. and be the t-level sets of with .This implies that . By Theorem 2, is a subhypergroup of .

Proposition 3. Ifandbe two-SVN subset of hypergroup, then

Proof 6. Assume that and are two -SVN subset of hypergroup .

Theorem 6. Letandbe-SVN hypergroups over. Thenis a-SVN hypergroup overif it is non-null.

Proof 7. Let and be two -SVN hypergroups over . Let be any element,By using result of Proposition 3.,By using (44), (45), and (46), we getSince, .
So by using (47), we get(i)For all ,This implies .Similarly for all , we getThis implies ,Similarly we can show that(ii), such that ,This implies .Next, we getThis implies .Similarly, we can show that .(iii), such that ,This implies .
Next, we getThis implies .
Similarly, we can show that .
Therefore, is a -SVN hypergroup over .

Remark 1. Union of two -SVN hypergroups over need not be -SVN hypergroup over .

Theorem 7. Letbe a-SVN hypergroup over. Then the falsity-favorite of(i.e.,) is a SVN hypergroup over.

Proof 8. By definition, , where the membership values are , and ,(i)Then we have to prove for .And we getSimilarly we can show that .(ii), such that ,And we get