Research Article | Open Access

# A New Kind of Fuzzy -ary Hypergroups in the Framework of Soft Set Theory

**Academic Editor:**Jianming Zhan

#### Abstract

Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem. The aim of this paper is to apply the concept of fuzzy soft sets to -ary hypergroup theory. The concepts of -fuzzy soft (invertible) -ary subhypergroups over a commutative -ary hypergroup are introduced and some related properties and characterizations are obtained. The homomorphism properties of -fuzzy soft (invertible) -ary subhypergroups are also derived.

#### 1. Introduction

To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics, which we can consider as mathematical tools for dealing with uncertainties. However, all of these have their advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterization tools. To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. This theory has proven useful in many different fields such as decision making [2–6], data analysis [7, 8], and forecasting [9].

Up to the present, research on soft sets has been very active and many important results have been achieved in the theoretical aspect. Maji et al. [10] introduced several algebraic operations in soft set theory and published a detailed theoretical study on soft sets. Ali et al. [11] further presented and investigated some new algebraic operations for soft sets. Qin and Hong [12] further investigated the algebraic structure of soft sets and established the soft quotient algebra. Maji et al. [13] and Majumdar and Samanta [14] extended (classical) soft sets to fuzzy soft sets, respectively. Maji et al. [15] extended (classical) soft sets to intuitionistic fuzzy soft sets, which were further discussed by Maji et al. [16] and Jiang et al. [17]. Aktaş and Çağman [18] compared soft sets to the related concepts of fuzzy sets and rough sets. They also defined the notion of soft groups and derived some related properties. Aygünoğlu and Aygün [19] discussed the applications of fuzzy soft sets to group theory and investigated (normal) fuzzy soft groups. Feng et al. [20] investigated soft semirings by using the soft set theory. Jun [21] introduced and investigated the notion of soft BCK/BCI-algebras. Jun and Park [22] and Jun et al. [23] discussed the applications of soft sets in ideal theory of BCK/BCI-algebras and in -algebras, respectively. Acar et al. [24] introduced and studied soft rings. Zhan and Jun [25] characterized the (implicative, positive implicative, and fantastic) filteristic soft -algebras based on -soft sets and -soft sets.

On the other hand, the fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now extensively applied to many disciplines. The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. There is a considerable amount of work on the connections between fuzzy sets and hyperstructures. The reader is referred to [26–38]. As a generalization of the concepts of “belongingness ()” and “quasi-coincidence ()” of a fuzzy point with a fuzzy set introduced by Pu and Liu [39], Yin and Zhan [40] introduced the concepts of “-belongingness ()” and “-quasi-coincidence ()” of a fuzzy point with a fuzzy set. Using these new concepts, Yin and Zhan [40] introduced the concepts of -fuzzy (implicative, positive implicative, and fantastic) filters and -fuzzy (implicative, positive implicative, and fantastic) filters of -algebras, where , , , and , and some related properties were investigated. Special attention is paid to -fuzzy (implicative, positive implicative, and fantastic) filters and -fuzzy (implicative, positive implicative, and fantastic) filters which are generalizations of -fuzzy (implicative, positive implicative, and fantastic) filters and -fuzzy (implicative, positive implicative, and fantastic) filters studied by Ma et al. [41, 42]. This idea is continued and studied by Ma et al. [43], Yin and Zhan [44, 45], Yin et al. [46], and so on. The purpose of this paper is to deal with the algebraic structure of -ary hypergroups by applying fuzzy soft set theory. We introduce the concepts of -fuzzy soft (invertible) -ary subhypergroups over a commutative -ary hypergroup and focus on investigating their characterization and algebraic properties. We also discuss the homomorphism properties of -fuzzy soft (invertible) -ary subhypergroups.

#### 2. Preliminaries

In this section, we recall some basic notions and results about -ary hypergroups, fuzzy sets, and fuzzy soft sets (see [13, 28, 30, 47]).

##### 2.1. *n*-Ary Hypergroups

We will be concerned primarily with a basic nonempty set . Denote by the cartesian product , where appears times. An element of will be denoted by , where for all . In general, a mapping is called an *-ary hyperoperation* and is called the* arity* of the hyperoperation . Let be an -ary hyperoperation on and subsets in . Define
In the sequel, we will denote the sequence by . For , . Thus,
will be written as . Also means for and .

with an -ary hyperoperation is called an *-ary hypergroupoid* and will be denoted by . An -ary hypergroupoid is said to be an *-ary semihypergroup* if the following associative axiom holds:
for all and . An -ary semihypergroup is said to be an *-ary hypergroup* if, for all and fixed , there exists such that . An -ary hypergroup is said to be* commutative* if, for all and any permutation of , we have . An element of is called an* identity* of if, for any and , we have .

*Definition 1. *Let be an -ary hypergroup and a nonempty subset of . One says that is an -*ary subhypergroup* of if the following conditions hold:(i)is closed under the -ary hyperoperation ;(ii)for all and fixed , there exists such that .

In what follows, denotes a commutative -ary hypergroup with an identity unless otherwise stated.

An -ary subhypergroup of is said to be* invertible* if, for any , implies .

##### 2.2. Fuzzy Sets

Let be a nonempty set. A fuzzy subset of is defined as a mapping from into , where is the usual interval of real numbers. We denote by the set of all fuzzy subsets of . For , by , we mean for all . And the* union* and* intersection* of and , denoted by and , are defined as the fuzzy subsets of by and for all .

For and , the sequence is denoted by , and if . Moreover, for , the sequence is denoted by , and denotes the zero fuzzy set if .

For any and , define a fuzzy subset of by
for all . In particular, when , is said to be the* characteristic function* of , denoted by , and when, is said to be a* fuzzy point* with support and value and is denoted by .

In what follows, let be such that . For a fuzzy point and a fuzzy subset of , we say that(1) if ;(2) if ;(3) if or [40].

Let us now introduce a new relation on , denoted by “,” as follows.

For any , by , we mean that implies for all and . Moreover, and are said to be -equal, denoted by , if and .

In the sequel, unless otherwise stated, means that does not hold, where .

Lemma 2 (see [40]). *Let . Then if and only if for all .*

Lemma 3 (see [40]). *Let . If and , then .*

*Proof. *It is straightforward by Lemma 2

Lemmas 2 and 3 give that “” is an equivalence relation on . It is also worth noticing that if and only if for all by Lemma 2

##### 2.3. Fuzzy Soft Sets

Let be an initial universe set and the set of all possible parameters under consideration with respect to . As a generalization of soft set introduced by Molodtsov [1], Maji et al. [13] defined fuzzy soft set in the following way.

*Definition 4. *A pair is called a* fuzzy soft set* over , where and is a mapping given by .

In general, for every , is a fuzzy set of and it is called* fuzzy value set* of parameter . The set of all fuzzy soft sets over with parameters from is called* fuzzy soft classes*, and it is denoted by . A fuzzy soft set in is said to* have sup-property* if, for any and , there exists such that .

Let and . Define to be a fuzzy soft set by for all . For any fuzzy point in , define to be a fuzzy soft set by for all .

*Definition 5 (see [6]). *Let and be two fuzzy soft sets over. One says that is a* fuzzy soft subset* of and writes if(i);(ii)for any , .

and are said to be* fuzzy soft equal* and write if and .

Let us now introduce some new concepts on fuzzy soft sets analogous to the concepts introduced by Ali et al. [11].

*Definition 6. *The* extended intersection* of two fuzzy soft sets and over is a fuzzy soft set denoted by , where and
for all . This is denoted by .

*Definition 7. *Let and be two fuzzy soft sets over such that . The* restricted intersection* of and is defined to be the fuzzy soft set , where and for all . This is denoted by .

*Definition 8. *Let and . A fuzzy soft set over is said to be a* relative whole *-*fuzzy soft set* (with respect to universe set , parameter set , and value ), denoted by , if for all .

*Definition 9. *Let and be two fuzzy soft sets over . One says that is -*fuzzy soft subset* of and writes if(i);(ii)for any , .

and are said to be -*fuzzy soft equal* and write if and .

Clearly, implies by Lemma 2 and Definition 9.

Lemma 10. *Let , , and be fuzzy soft sets over . If and , then .*

*Proof. *It is straightforward by Lemma 3 and Definition 9.

Lemmas 2 and 10 and Definition 9 give that “” is an equivalence relation on .

#### 3. A New Operation on Fuzzy Soft Sets over an -Ary Hypergroupoid

We first introduce a fuzzy -ary hyperoperation on an -ary hypergroupoid as follows.

*Definition 11. *Let be an -ary hypergroupoid and . Define a fuzzy -ary hyperoperation by
for all . In particular, for any and , define
for all .

Clearly, for any and , . Note that, for any and , by Definition 11, .

*Definition 12. *Let be an -ary hypergroupoid and , where and is an index set, such that . One defines a fuzzy soft set over , denoted by , where and
for all . This is denoted by .

In the following, we will denote the sequence by . For , is the empty set. The following elementary facts follow easily from the definition.

Lemma 13. *Let be an -ary hypergroupoid and . Then*(1)*if is an -ary semihypergroup and , then
for all ;*(2)*if is commutative and , then the value of does not depend on the permutation of ;*(3)*if for all and , then ;*(4)*if and, then and if .*

#### 4. -Fuzzy Soft (Invertible) -Ary Subhypergroups over a Commutative -Ary Hypergroup

In this section, we will introduce and investigate the concept of -fuzzy soft -ary subhypergroups over a commutative -ary hypergroup. Let us start by giving the following concept.

*Definition 14. *Let . Then is called -*fuzzy soft *-*ary subhypergroup* over if it satisfies the following conditions:(F1a)for all and , ;(F2a)for all and , ;(F3a)for all and , there exists such that
An -fuzzy -ary subhypergroup over is said to be* invertible* if it satisfies the following:(F4a)for all and ,

*Example 15. *Let be the residue class of modulo 8 with an -ary hyperoperation as follows:
Then, is a commutative -ary hypergroup. Let and . Define a fuzzy soft set over by
for all and . Then is an -fuzzy soft invertible -ary subhypergroup over .

Next let us first provide some auxiliary lemmas as follows.

Lemma 16. *Let . Then (F2a) holds if and only if one of the following conditions holds:*(F2b)*;*(F2c)*for all and , ,;*(F2d)*for all and , .*

*Proof. *(F2a)(F2b) Assume that (F2a) holds. Then, for any and , we have
and so by Lemma 2. Therefore, .

(F2b)(F2c) Let and . Since , for all , we have
and so .

(F2c)(F2d) This is straightforward.

(F2d)(F2a) Let , , and be such that . Then, by Lemma 2 and (F2d), we have
This implies for all and . Hence, (F2a) holds.

Lemma 17. *Let . Then, one has the following.*(1)*(F3a) holds if and only if the following condition holds: (F3b) for all and , there exists such that
*(2)*If (F3a) holds, then the following condition holds:*(F3c)* for all and ,
**Moreover, (F3c) implies (F3a) if has the sup-property or is finite.*

*Proof. *(F3a)(F3b) Let and . By (F3a), there exists such that
This implies ,, and so
It follows that , ,, and so (F3b) holds.

(F3b)(F3a) Let . By (F3b), there exists such that
It follows from Lemma 2 that
This implies and ,,,. Hence,
and so (F3a) holds.

(F3a)(F3c) This is straightforward by (F3b).

In the following, assume that has the sup-property or is finite. We prove (F3c)(F3a). Let . If, for any such that , we have
then since has the sup-property or is finite, we have
which contradicts , ,, by Lemma 2. Hence (F3a) holds.

Lemma 18. *Let . Then (F4a) holds if and only if the following condition holds:**() for all , , and , implies .*

*Proof. *(F4a)(F4c) Let and . If but . Then
but , and ,. Then
gives and so ,; that is, , . Thus we have , a contradiction. Hence (F4c) holds.

(F4c)(F4a) Let . If there exists such that , then
and . This implies that , but ,, a contradiction. Hence ,,, , and so , ,,, ,,. In a similar way, we have ,,,,,,. Thus (F4a) holds.

For any fuzzy soft set over , , and, denote, , and . The next theorem presents the relationships between -fuzzy soft (invertible) -ary subhypergroups over and crisp (invertible) -ary subhypergroups of .

Theorem 19. *Let . Then,*(1)* is an -fuzzy soft (invertible) -ary subhypergroup over if and only if nonempty subset is an (invertible) -ary subhypergroup of for all and ;*(2)*if , then is an -fuzzy soft (invertible) -ary subhypergroup over if and only if nonempty subset is an (invertible) -ary subhypergroup of for all and ;*(3)* is an -fuzzy soft (invertible) -ary subhypergroup over if and only if nonempty subset is an (invertible) -ary subhypergroup of for all and .*

*Proof. *We only prove (2) and (3). (1) can be easily proved.

(2) Assume that . Let be an -fuzzy soft -ary subhypergroup over and assume that for some and . Let . Then ; that is, . Since is an -fuzzy -ary subhypergroup over , we have for all . Hence, by ,
From , that is, , we have and so . Similarly we can show that and, for all , there exists such that . Therefore, is an -ary hypergroup of .

Conversely, assume that the given condition holds. Let and . If there exists such that , take . Then , , , that is, , but, a contradiction. Hence and so . Similarly we can show that conditions (F1a) and (F3a) hold. Therefore, is an -fuzzy soft -ary subhypergroup over .

(3) Let be an -fuzzy soft -ary subhypergroup over and assume that for some and . Let . Then ; that is, or , or . Since is an -fuzzy -ary subhypergroup over , we have for all and so since in any case. Now we consider the following cases.*Case 1 *. In this case, . (1)If or or , then . Hence .(2)If , for all , then