#### Abstract

We introduce and develop the initial theory of vague soft hyperalgebra by introducing the novel concept of vague soft hypergroups, vague soft subhypergroups, and vague soft hypergroup homomorphism. The properties and structural characteristics of these concepts are also studied and discussed.

#### 1. Introduction

Vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of . Such a set is characterized by a truth-membership function and a false-membership function. Thus, a vague set is actually a form of fuzzy set, albeit a more accurate form of fuzzy set. Soft set theory has been regarded as an effective mathematical tool to deal with uncertainties. However, it is difficult to be used to represent the vagueness of problem parameters in problem-solving and decision-making contexts. Hence the concept of vague soft sets were introduced as an extension to the notion of soft sets, as a means to overcome the problem of assigning a suitable value for the grade of membership of an element in a set since the exact grade of membership may be unknown. Using the concept of vague soft sets, we are able to ascertain that the grade of membership of an element lies within a certain closed interval.

Hyperstructure theory was first introduced in 1934 by a French mathematician, Marty , at the 8th Congress of Scandinavian Mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Since the introduction of the notion of hyperstructures, comprehensive research has been done on this topic and the notions of hypergroupoid, hypergroup, hyperring, and hypermodule have been introduced. A recent book by Corsini and Leoreanu-Fotea  expounds on the applications of hyperstructures in the areas of geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automation, cryptography, combinatorics, codes, artificial intelligence, and probability theory.

The concept of -structures introduced by Vougiouklis  constitute a generalization of the well-known algebraic hyperstructures such as hypergroups, hyperrings, and hypermodules. Some axioms pertaining to the above-mentioned hyperstructures such as the associative law and the distributive law are replaced by their corresponding weak axioms.

The study of fuzzy algebraic structures started with the introduction of the concept of fuzzy subgroup of a group by Rosenfeld  in 1971. There is a considerable amount of work that has been done on the connections between fuzzy sets and hyperstructures (see ). In 1999,  applied the concept of fuzzy sets to the theory of algebraic hyperstructures and defined fuzzy subhypergroup (resp., -subgroup) of a hypergroup (resp., -group) which is a generalization of the concept of Rosenfeld’s fuzzy subgroup of a group. See also Davvaz et al. [7, 8].

The study on soft algebraic hyperstructure theory was initiated by Yamak et al.  as an extension to the theory of hyperstructures and soft set theory. They also defined the notion of soft hypergroupoids and soft subhypergroupoids as well as studied some of the basic properties of these concepts. Selvachandran and Salleh  on the other hand, introduced the notion of soft hypergroups and soft hypergroup homomorphism.

In this paper, we introduce and develop the initial theory of vague soft hyperalgebra in Rosenfeld’s sense. This is done by defining the notion of vague soft hypergroups, vague soft subhypergroups, and vague soft hypergroup homomorphism and using these definitions to develop the initial theory of vague soft hypergroups. The concepts defined in this paper are the combination of the theory of vague soft sets and the theory of fuzzy hypergroups that were introduced by [6, 11], respectively. Furthermore, some of the fundamental properties and structural characteristics of these concepts are studied and discussed. Lastly, we prove that there exists a one-to-one correspondence between some of these concepts and their corresponding concepts in soft hypergroup theory as well as classical hypergroup theory.

#### 2. Preliminaries

In this section, some well-known and useful definitions pertaining to the theory of soft sets, vague soft sets, hyperstructures, soft hyperstructures, and fuzzy hyperalgebra will be presented. These definitions and concepts will be used throughout this chapter.

Definition 1 (see ). A pair is called a soft set over , where is a mapping given by . In other words, a soft set over is a parameterized family of subsets of the universe . For , may be considered as the set of -elements of the soft set or as the -approximate elements of the soft set.

Example 2 (see ). Consider a soft set which describes the “attractiveness of houses” that Mr. is considering for purchase.
Suppose that there are six houses in the universe under consideration, and that is a set of decision parameters. The    stand for the parameters “expensive,” “beautiful,” “wooden,” “cheap,” and “in green surroundings,” respectively.
Consider the mapping given by “houses ,” where is to be filled in by one of the parameters . For instance, means “houses (expensive),” and its functional value is the set .
Suppose that , , , and . Then we can view the soft set as consisting of the following collection of approximations:
expensive houses , beautiful houses , wooden houses, cheap houses , in the green surroundings .
Each approximation has two parts: a predicate and an approximate value set.

Definition 3 (see ). Let be a space of points (objects) with a generic element of denoted by . A vague set in is characterized by a truth-membership function and a false-membership function . The value is a lower bound on the grade of membership of derived from the evidence for and is a lower bound on the negation of derived from the evidence against . The values and both associate a real number in the interval with each point in , where . This approach bounds the grade of membership of to a subinterval of . Hence a vague set is a form of fuzzy set, albeit a more accurate form of fuzzy set.

Definition 4 (see ). A pair is called a vague soft set over where is a mapping given by and is the power set of vague sets over . In other words, a vague soft set over is a parametrized family of vague sets of the universe . Every set for all , from this family may be considered as the set of -approximate elements of the vague soft set . Hence the vague soft set can be viewed as consisting of a collection of approximations of the following form: for all and for all .

The concept of a support is defined in the literature for both fuzzy sets and formal power series. A similar notion of soft sets was defined by .

Example 5 (see ). Consider a vague soft set , where is a set of six houses under consideration of a decision maker to purchase, which is denoted by and is a parameter set, where expensive, beautiful, wooden, cheap, in the green surroundings}. The vague soft set describes the “attractiveness of the houses” to this decision maker.
Suppose that The vague soft set is a parameterized family of vague sets on , and

Definition 6 (see ). For a soft set , the set is called the support of the soft set . Thus a null soft set is indeed a soft set with an empty support and a soft set is said to be nonnull if .

Similarly, here we define the notion of the support of a vague soft set which is as given below.

Definition 7. Let be a vague soft set over . The support of denoted by is defined as for all .
It is to be noted that a null vague soft set is a vague soft set where both the truth and false membership functions are equal to zero. Therefore, a vague soft set is said to be nonnull if .

Definition 8 (see ). Let and be vague soft sets over . Set is called a vague soft subset of if and for every , and are the same approximation. In this case, is called the vague soft superset of and this relationship is denoted as .

Definition 9 (see ). If and are two vague soft sets over , the intersection of and , denoted as “” is defined by , where for all and for all .

Definition 10 (see ). If and are two vague soft sets over , the union of and , denoted by “” is defined by , where for all and for all .

Definition 11 (see ). Let and be vague soft sets over a universe of discourse . The restricted intersection of and , which is denoted by , is a vague soft set , where and for all and , where and are the truth membership function and false membership function of . This relationship can be written as .

Definition 12 (see ). Let and be vague soft sets over a universe of discourse . The restricted union of and , which is denoted by , is a vague soft set , where and for all and , where and are the truth membership function and false membership function of . This relationship can be written as .

Next we present some operations on vague soft sets, namely, the vague soft -cut and vague soft -cut and the notion of a vague soft characteristic set and vague soft groups.

Definition 13 (see ). Let be a vague soft set over . Then for all , where , the -cut or the vague soft -cut of , is a subset of which is as defined below: for all .
If , then it is called the vague soft -cut of or the -level set of , denoted by is a subset of which is as defined below: for all .

Definition 14 (see ). Let be a vague soft set over and be a nonnull subset of . Then is called a vague soft characteristic set over in and the lower bound and upper bound of are as defined below: where is a subset of , , and .

Definition 15 (see ). Let be a group and be a vague soft set over . Then is called a vague soft group over if and only if for all and for every , , the following conditions are satisfied.(i) and , that is .(ii) and , that is .That is, for every , is a vague subgroup of in Rosenfeld’s sense.

Theorem 16 (see ). Let be a vague soft set. Then is vague soft group if and only if for every and , , that is,

Definition 17 (see ). A hypergroup is a set equipped with an associative hyperoperation which satisfies the reproduction axiom given by for all in .

Example 18 (see ). Let be a set with hyperoperation “” as in Table 1.

Then is a hyperstructure; that is, it is a hypergroup.

Definition 19 (see ). A hyperstructure is called a -group if the following axioms hold:(i) for all , ,(ii) for all in .If only satisfies the first axiom, then it is called a -semigroup.

Definition 20 (see ). A subset of is called a subhypergroup if is a hypergroup.

The concepts of hyperstructures were first introduced by Marty as an alternative mathematical tool to complement the concept of ordinary algebraic structures and overcome the shortcomings that is inherent in ordinary algebraic structures. It has been proven to be very useful in various fields. In the context of this paper, these definitions were heavily used to investigate the relationship between the concepts in vague soft hyperalgebraic structures that were introduced in this paper and the corresponding concepts in classical hyperalgebraic theory.

Definition 21 (see ). Let be a nonnull soft set over . Then is called a soft hypergroupoid over if is a subhypergroupoid of for all .

Definition 22 (see ). Let be a hypergroup (or -group) and let be a fuzzy subset of . Then is said to be a fuzzy subhypergroup (or fuzzy -subgroup) of if the following axioms hold:(i) for all , ;(ii)for all , , there exists such that and ;(iii)for all , , there exists such that and .

The study of fuzzy hyperalgebraic structures was further extended by Leoreanu-Fotea et al.  who introduced the notion of fuzzy soft hypergroups and categorized this concept into two classes, namely, fuzzy soft closed hypergroups and fuzzy soft ultraclosed hypergroups. The main definitions of these concepts are as given below.

Let be a hypergroup which will be denoted simply as for the sake of simplicity.

Definition 23 (see ). Suppose that is a fuzzy soft set over . Then is called a fuzzy soft hypergroup over if for all ,(i)For all , , ,(ii)For all , , there exists such that and ,(iii)For all , , there exists such that and .

Definition 24 (see ). Let be a fuzzy soft set over . Then is called a fuzzy soft closed hypergroup over if for all , (i)for all , , ,(ii)for all , and for all , such that , we have ,(iii)for all , and for all , such that , we have .

Definition 25 (see ). Let be a fuzzy soft set over . Then is called a fuzzy soft ultraclosed hypergroup over if for all , (i)for all , , ,(ii)for all , and for all , , if , then ,(iii)for all , and for all , if , then .

Beginning with the concept of classical groups, Rosenfeld  introduced the concept of a fuzzy subgroup of a group using the concept of fuzzy subsets that were introduced by Zadeh (1965). Early research by Zadeh (1965) and Rosenfeld  pertaining to the introduction of the concepts of fuzzy subsets of a set and fuzzy subgroup of a group, respectively, has led to the fuzzification of various algebraic structures (see Akgul, 1988; Bhakat and Das, 1992; Dib, 1994; Davvaz, 1998, 1999, 2009 and Liu, 1982). The notion of fuzzy subgroups of a group has inspired further research and development in the field of fuzzy algebra and subsequently led to the introduction of the notion of fuzzy subhypergroups of a hypergroup by Davvaz . Consequently, research in the area of fuzzy hyperalgebra saw rapid growth and development, mostly using the concept of fuzzy subhypergroups of a hypergroup. In this paper, the Rosenfeld approach is used to introduce the various vague soft hyperalgebraic concepts.

Next, the definition of a fuzzy subgroup of a group that was introduced by Rosenfeld  is given.

Definition 26 (see ). If is a classical group with ordinary binary operation and is a fuzzy subset of , then is called a fuzzy subgroup of if for all , the following axioms are satisfied:(i),(ii).

#### 3. Vague Soft Hypergroups

In this section, the concept of vague soft hypergroups in Rosenfeld’s sense is introduced. This is an extension to the concept of soft hypergroupoids that were introduced by Yamak et al. . Using this definition, some of the structural characteristics and fundamental properties of vague soft hypergroups are obtained.

From now on, let be a hypergroup, and let be a set of parameters and . For the sake of simplicity, a hypergroup will be denoted simply as .

Definition 27. Let be a nonnull vague soft set over . Then is called a vague soft hypergroup over if the following conditions are satisfied for all .(i)For all , , and , that is, .(ii)For all , , there exists a such that and and , that is .(iii)For all , , there exists a such that and and , that is .If only satisfies condition (i) of Definition 27 for all , then is called a vague soft semihypergroup over .
Conditions (ii) and (iii) represent the left and right reproduction axioms, respectively. Then is a nonnull vague subhypergroup of (in Rosenfeld’s sense) for all .

Example 28. Let be a set, let be a subset of and , and let be a hypergroup with hyperoperation which is as defined below.
Let , . If , then . Next let be a group and be a vague soft group over . Consider a nonnull vague soft set over which is as defined below: for all .
Since is a vague soft group over , then for all , and . Then for all , , we obtain Therefore we have for all , . Thus condition (i) of Definition 27 is satisfied. As such, is a vague soft semihypergroup over . Next let , . Consider the two cases given below.
Case 1. . Then which implies that .
Case 2. . Then which implies that .
In both the cases given above, we obtain . By letting , we obtain and therefore . This implies that Then for all , , there exists such that and . This means that condition (ii) of Definition 27 is satisfied. Condition (iii) of Definition 27 can be verified in the same manner. Then is a vague subhypergroup of . As such, is a vague soft hypergroup over .

Example 29. A vague soft hypergroup for which is a singleton is a vague subhypergroup. Hence vague subhypergroups and also hypergroups are a particular type of vague soft hypergroups.

Definition 30. Let be a nonnull soft set over . Then is called a soft hypergroup over if is a subhypergroup of for all .

Example 31. Let be a vague subhypergroup of a hypergroup . This means that satisfies the axioms stated in Definition 27. Now consider the family of -level sets for given by for all and . Then for all , is a subhypergroup of . Hence is a soft hypergroup over .

Example 32. Any soft hypergroup is a vague subhypergroup since any characteristic function of a subhypergroup is a vague subhypergroup.

Theorem 33. Let be a vague soft set over . Then is a vague soft hypergroup over if and only if for all , is a soft hypergroup over .

Proof. () Let be a vague soft hypergroup over . Then for all , is a nonnull vague subhypergroup of . Now let , . Thus we obtain the following: that is, Furthermore, since is a vague subhypergroup of , we obtain the following: which means that . As such, for all , we obtain and therefore . Then for all , we have . Now let , . Then there exist such that and and that is, . Since , , we obtain and , that is, . Thus we obtain which means that and this implies that . This proves that . As such, which proves that is a subhypergroup of . Hence is a soft hypergroup over .
() Assume that is a soft hypergroup over . Then for all , is a nonnull subhypergroup of . For all , there are two cases that should be considered.
Case 1. Let and , which means that . Next let . Then and and therefore . Thus for all , we obtain which implies that . As such, condition (i) of Definition 27 is satisfied. To verify condition (ii) of Definition 27, assume that for all , , and , that is . Therefore and . Then there exists such that . On the contrary, since , therefore and thus which implies that . As such is a vague subhypergroup of . Hence is a vague soft hypergroup over .
Case 2. Assume that and , which means that . The subsequent proof is similar to that of the proof for Case 1 and in a similar manner it can be proven that is a vague subhypergroup of .
In both the cases, it has been proven that is a vague subhypergroup of and therefore it is proven that is a vague soft hypergroup over .

Theorem 33 proves that there exists a one-to-one correspondence between vague soft hypergroups and soft hypergroups of a hypergroup.

Corollary 34. Let be a vague soft hypergroup over . If , then if and only if there does not exist such that .

Proof. () Let and . Then there exists such that and . Since , we obtain . Therefore, there does not exist such that .
() Assume that there does not exist such that . From , we obtain and , that is, . However, this is not possible since we have because of . As such, since , then too for all . Thus and . For to satisfy both the conditions and , must satisfy the condition and . This implies that . Therefore it has been proven that .

Definition 35 (see ). Let be a set and be a subset of . The hyperoperation is as defined below.
Let , . If , then .

Proposition 36. Let be a group. Then is a vague soft group over if and only if is a soft hypergroup over .

Proof. () Let be a vague soft group over group and be a hypergroup with hyperoperation as defined in Definition 35. Then for all , , and . Therefore for all , , we obtain the following: Hence, condition (i) of Definition 27 has been verified. Now let , . Therefore if , then , which implies that (from Definition 35) and if , then which also implies that . Thus we obtain . As such, if we let , then for any case we obtain . Hence condition (ii) of Definition 27 has been verified. Then is a subhypergroup of . As such, it has been proven that is a soft hypergroup over .
() Let be a soft hypergroup over and let , . We consider two cases.
Case 1. Suppose that . Then we obtain the following: Therefore, is a vague subgroup of .
Case 2. Suppose that . Then we obtain the following: Therefore, is a vague subgroup of . In both the cases, it has been proven that is a vague subgroup of . Hence is a vague soft group over .

Theorem 37. Let be a vague soft set over . Then is a vague soft hypergroup over if and only if for all , , is a subhypergroup of .

Proof. The proof is straightforward.

As a result of Theorem 37, we obtain Corollary 38 which is as given below.

Corollary 38. Let be a vague soft set over . Then is a vague soft hypergroup over if and only if for all , is a subhypergroup of .

Theorem 37 and Corollary 38 prove that there exists a one-to-one correspondence between vague soft hypergroups and subhypergroups of a hypergroup.

Theorem 39. Let be a non-null subset of and be a vague soft characteristic set over . If is a vague soft hypergroup over , then is a subhypergroup of .

Proof. The proof is straightforward.

Theorem 40. Let and be vague soft hypergroups over . Then is a vague soft hypergroup over if it is nonnull.

Proof. The proof can be easily obtained using Definition 9.

Remarks/Note. Theorem 40 also holds true for other operations in vague soft set theory such as union, restricted intersection, restricted union, “AND,” and “OR”.

#### 4. Vague Soft Hypergroup Homomorphism

In the first part of this section, the notion of vague soft function as well as the image and preimage of a vague soft set under a vague soft function are introduced and defined. These definitions will then be used in the context of vague soft hypergroups to introduce the notion of vague soft hypergroup homomorphism. Lastly, it is proven that the vague soft hypergroup homomorphism preserves vague soft hypergroups.

Definition 41. Let and be two functions, where and are parameter sets for the classical sets and , respectively. Let and be vague soft sets over and , respectively. Then the ordered pair is called a vague soft function from to , and it is denoted as .

Definition 42. Let and be vague soft sets over and , respectively. Let be a vague soft function.(i)The image of under the vague soft function , which is denoted as , is a vague soft set over , which is defined as where for all , , and .(ii)The preimage of under the vague soft function , which is denoted as , is a vague soft set over , which is defined as where for all , , and .

If and are injective (surjective), then the vague soft function is said to be injective (surjective).

Definition 43 (see ). Let and be hypergroups. The map is called a good homomorphism if and it is called an inclusion homomorphism if for all , .

Definition 44. Let and be hypergroups and and be vague soft hypergroups over and , respectively. Let be a vague soft function. Then we obtain the following.(a) is called a vague soft inclusion hypergroup homomorphism if the following conditions are satisfied:(i) is an inclusion homomorphism,(ii) is a function,(iii) for all .(b) is called a vague soft good hypergroup homomorphism if the following conditions are satisfied:(i) is a good homomorphism,(ii) is a function,(iii) for all .

Theorem 45. Let and be vague soft hypergroups over and , be a vague soft good hypergroup homomorphism and be a bijective mapping. Then is a vague soft hypergroup over .

Proof. Since is a vague soft good hypergroup homomorphism, it can be observed that . Next let , . Then there exist , such that and . Therefore for all , and for all , where such that and .
Furthermore for all , , there exist , such that and . As such, for all , , and it follows that where such that and and also where such that and .
Hence is a vague soft hypergroup over by Definition 27.

Theorem 46. Let and be vague soft hypergroups over and , respectively, and be a vague soft inclusion hypergroup homomorphism. Then is a vague soft hypergroup over .

Proof. The proof is similar to that of Theorem 45.

Theorems 45 and 46 prove that the homomorphic image and preimage of a vague soft hypergroup are also vague soft hypergroups.

Theorem 47. Let , , and be nonnull hypergroups and , , and be vague soft hypergroups over , , and , respectively. Let and be vague soft good (inclusion) hypergroup homomorphisms. Then is a vague soft good (inclusion) hypergroup homomorphism.

Proof. The proof is straightforward.

#### 5. Vague Soft Subhypergroups

In this section, the concept of vague soft subhypergroups is introduced. This concept is an extension of the concept of vague soft hypergroups that was introduced in Section 3. Some of the basic properties of this concept are then studied and discussed.

Definition 48. Let and be vague soft hypergroups over . Then is called a vague soft subhypergroup of , which is denoted as , if the following conditions are satisfied: (i),(ii) is a non-null vague subhypergroup of for all .

Theorem 49. Let and be vague soft hypergroups over and be a vague soft subset of . Then .

Proof. Let and be vague soft hypergroups over and be a vague soft subset of , that is, . Since , by Definition 6, it can be concluded that . Furthermore, since and are vague soft hypergroups over and , is a nonnull vague subhypergroup of for all . As such, is a vague soft hypergroup over . Hence is a vague soft subhypergroup of , that is, .

Theorem 50. Let and