Review Article | Open Access

A. Kandil, S. A. El-Sheikh, M. Hosny, M. Raafat, "Mappings and Connectedness on Hesitant Fuzzy Soft Multispaces", *Advances in Fuzzy Systems*, vol. 2020, Article ID 8894850, 15 pages, 2020. https://doi.org/10.1155/2020/8894850

# Mappings and Connectedness on Hesitant Fuzzy Soft Multispaces

**Academic Editor:**Antonin DvorÃ¡k

#### Abstract

In this paper, we introduce the concept of mapping on hesitant fuzzy soft multisets and present some results for this type of mappings. The notions of inverse image and identity mapping are defined, and their basic properties are investigated. Hence, kinds of mappings and the composition of two hesitant fuzzy soft multimapping with the same dimension are presented. The concept of hesitant fuzzy soft multitopology is defined, and certain types of hesitant fuzzy soft multimapping such as continuity, open, closed, and homeomorphism are presented in detail. Also, their properties and results are studied. In addition, the concept of hesitant fuzzy soft multiconnected spaces is introduced.

#### 1. Introduction

Since the introduction of fuzzy sets by Zadeh [1], several extensions of this concept have been introduced. The most agreed one may be Atanassovâ€™s intuitionistic fuzzy set (briefly, IFS or A-IFS) [2]. IFSs have the benefit that allows the user to model some uncertainties on the membership function of the elements. That is, fuzzy sets require a membership degree for each element in the universe set, whereas an IFS permits us to include some hesitation on this value. This is modeled with two functions that define an interval. Type 2 fuzzy sets [3, 4] are a generalization of the former, where the membership of a given element is presented as a fuzzy set. Other generalizations, such as type *n* fuzzy sets exist (see [3] for details about type *n* fuzzy sets). Dubois and Prade [3] report that Mizumoto and Tanaka [4] were the first to study type 2 fuzzy sets. Fuzzy multisets are another generalization of fuzzy sets. They are based on multisets (elements can be repeated in a multiset, for short, mset). In fuzzy multisets, the membership can be partial (instead of Boolean as for standard multisets). Tokat and Osmanoglu [5] introduced the concept of a soft mset as , where is a set of parameters and is a power set of an mset . In this paper, we adopt the notion of a soft mset in [5], since this way is better than the other [6, 7]. In 2013, Tokat et al. [7] introduced the concept of soft msets as a combination between soft sets and msets. Furthermore, in [7], the soft multitopology and its basic properties were given. Moreover, the soft multiconnectedness was studied in [5]. Additionally, the soft multicompactness on soft multitopological spaces was presented in [8]. In 2015, El-Sheikh et al. [9] introduced the concept of semicompact soft multispaces and the concept of soft multi-Lindelf spaces. Some other results and properties about soft multisets are presented in [10â€“12]. The concept of a generalized open soft mset is introduced in soft multitopological spaces, and its properties are presented in [10]. Several authors [13â€“15] discussed the concept of multisets, its generalizations, and its applications. In 2020, Hashmi et al. [16] introduced the notion of an m-polar neutrosophic set and m-polar neutrosophic topology and their applications to multicriteria decision-making (MCDM) in medical diagnosis and clustering analysis. They introduced a novel approach to census process by using Pythagorean m-polar fuzzy Dombis aggregation operators. Riaz and Hashmi [17] introduced the notion of linear Diophantine fuzzy set (LDFS) and its applications towards MCDM problem. Linear Diophantine fuzzy set (LDFS) is superior to IFSs, PFSs, and q-ROFSs. Riaz and Tehrim [18] introduced the concept of bipolar fuzzy soft mappings with application to bipolar disorders. Tehrim and Riaz [19] presented a novel extension of the TOPSIS method with bipolar neutrosophic soft topology and its applications to multicriteria group decision-making (MCGDM). Riaz et al. [20] presented the multiattribute group decision-making (MAGDM) methods to a hesitant fuzzy soft set. Moreover, Riaz et al. [21] developed the topological structure on a soft rough set by using pairwise soft rough approximations. The multicriteria group decision-making methods are introduced by using N-soft set and N-soft topology to deal with uncertainties in the real-world problems [22].

Recently, the concept of hesitant fuzzy sets was introduced firstly in 2010 by Torra [23] which permits the membership to have a set of possible values and presents some of its basic operations in expressing uncertainty and vagueness. Torra et al. [24] established the similarities and differences with the hesitant fuzzy sets and the previous generalization of fuzzy sets such as intuitionistic fuzzy sets, type 2 fuzzy sets, and type *n* fuzzy sets. Therefore, other authors [25, 26] introduced the concept of hesitant fuzzy soft sets, and they presented some of the applications in decision-making problems. In 2015, Dey and Pal [27] presented the concept of hesitant multifuzzy soft topological space. In 2019, Kandil et al. [28] introduced some important and basic issues of hesitant fuzzy soft multisets and studied some of its structural properties such as the neighborhood hesitant fuzzy soft multisets, interior hesitant fuzzy soft multisets, hesitant fuzzy soft multitopological spaces, and hesitant fuzzy soft multibasis. Finally, they showed how to apply the concept of hesitant fuzzy soft multisets in decision-making problems.

The main goal of this paper is to introduce the definition of mapping on hesitant fuzzy soft multisets and present some results for this form of mappings. The notions of inverse image and identity mapping are introduced, and their basic properties are investigated in detail. The types of mappings are also given on hesitant fuzzy soft multisets, and their properties are established. Therefore, the composition of two hesitant fuzzy soft multimapping with the same dimension is presented. In addition, the concepts of hesitant fuzzy soft multitopologies and hesitant fuzzy soft multi-subspaces are introduced. Some types of hesitant fuzzy soft multimapping such as continuity, open, closed, and homeomorphism are presented in detail. Also, their properties and results are investigated. Finally, the concept of hesitant fuzzy soft multiconnected space is introduced.

#### 2. Preliminaries

The aim of this section is to present the basic concepts and properties of multisets, soft multisets, hesitant fuzzy sets, and hesitant fuzzy soft multisets.

*Definition 1. *(see [29]). An mset drawn from the set is represented by a count function defined as : , where represents the set of nonnegative integers.

Here, is the number of occurrences of the element in the mset . The mset is drawn from the set , and it is written as , where is the number of occurrences of the element , in the mset .

*Definition 2. *(see [29]). A domain is defined as a set of elements from which msets are constructed. The mset space is the set of all msets whose elements are in such that no element in the mset occurs more than times*.*

The mset space is the set of all msets over a domain such that there is no limit on the number of occurrences of an element in an mset. If , then .

*Definition 3. *(see [29]). Let be an mset drawn from the set . If , then is called an empty mset and denoted by , i.e., .

*Definition 4. *(see [5]). Let be a universal multiset, be a set of parameters, and . Then, an ordered pair is called a soft mset, where is a mapping given by ; is the power set of an mset . For all , mset is represented by count function , where represents the set of nonnegative integers and represents the support set of .

*Definition 5. *(see [5]). Let and be two soft msets over . Then,(1) is said to be a sub-soft mset of and denoted by if(i)(ii), for all , (2)Two soft msets and over are equal if is a sub-soft mset of and is a sub-soft mset of .(3)The union of two soft msets and over is the soft mset , where and , for all , . It is denoted by .(4)The intersection of two soft msets and over is the soft mset , where and, for all , . It is denoted by .(5)A soft mset over is said to be a null soft mset and denoted by if for all , .(6)A soft mset over is said to be an absolute soft mset and denoted by if for all , .

*Definition 6. *(see [5]). The complement of a soft mset is denoted by and is defined by , where is a mapping given by for all , where , for all .

*Definition 7. *(see [5]). Let be a universal mset and be a set of parameters. Then, the collection of all soft msets over with parameters from is called a soft multiclass and is denoted by .

*Definition 8. *(see [23]). Let be a reference set, then a hesitant fuzzy (briefly, an ) set is defined in terms of a function from into the power set of .

*Definition 9. *(see [23]). Let , and be hesitant fuzzy sets over a set . Then, the following operations are defined:(1)Full set : for all (2)Null set : for all (3)Lower bound: (4)Upper bound: (5)-Upper bound: (6)-Lower bound: (7)Complement: (8)Union: or, equivalently, for (9)Intersection: or, equivalently, for

*Definition 10. *(see [28]). A hesitant fuzzy multiset of dimension (briefly, set) on a nonempty mset is denoted by and is defined in terms of when applied to , and is a set of some distinct values in sorting into increasing order, indicating the possible membership degrees of the elements to the multiset .

*Definition 11. *(see [28]). Let and be two sets on a nonempty mset . is called a hesitant fuzzy submset of if for each , and denoted by .

*Definition 12. *(see [28]). A pair is a hesitant fuzzy soft mset of dimension if is a mapping from to , where is the set of all hesitant fuzzy msets of dimension defined over an mset and , i.e., for all , and is the membership function of .

*Definition 13. *(see [28]). An set over is said to be(1)A relative null set and is denoted by , if for all , (2)A relative absolute set and is denoted by , if for all ,

*Definition 14. *(see [28]). Let and be two hesitant fuzzy soft multisets of dimension , then is called a hesitant fuzzy soft multi-subset (briefly, ) of of dimension if(1)(2) is a hesitant fuzzy submset of , for every , i.e., for all , , Hence, this relationship is denoted by , and is called an superset of .

*Definition 15. *(see [28]). Let be a hesitant fuzzy soft multitopological space and , be two sets over a hesitant fuzzy soft mset (for short, ). A hesitant fuzzy soft mset is called neighborhood of if there exists an open hesitant fuzzy soft mset such that *.*

*Definition 16. *(see [28]). Let be a hesitant fuzzy soft multitopological space and and be two sets over such that . Then, is called an interior hesitant fuzzy soft mset of if is a neighborhood of . Additionally, the union of all interior hesitant fuzzy soft mset of is called the interior of , and it is denoted by .

Theorem 1 (see [28]). *Let be a hesitant fuzzy soft multitopological space and , be two sets over , then*(1)* is the largest open hesitant fuzzy soft mset contained in *(2)* is an open hesitant fuzzy soft mset if and only if *(3)*(4)**If , then *

#### 3. Mappings in Hesitant Fuzzy Soft Multisets

The purpose of this section is to present a concept of mapping in hesitant fuzzy soft multisets. The main properties of the current branch are studied, and some results of this type of sets are established. Also, the concept of inverse mapping in hesitant fuzzy soft multisets is defined. Therefore, the composition of two hesitant fuzzy soft multimappings is introduced. Finally, some examples are used to explain the current definitions in a friendly way.

It should be noted that, in this section, let be a universal set, be a set of parameters, and be a multiset over . The union and intersection of hesitant fuzzy sets are defined by Torra [23], but these definitions did not preserve the dimension, so we introduce the following definitions.

*Definition 17. *Union of two sets and over is the set , where , for all ,where . It is written as .

*Example 1. *Let , , and . The hesitant fuzzy soft msets of dimension , , are defined as , , . , , . Hence, such that , , .

*Definition 18. *Intersection of two sets and over is the set , where , for all ,It is written as .

*Example 2. *Let , and . The hesitant fuzzy soft msets of dimension , , are defined as , , . , , . Hence, such that , , .

Theorem 2. *Let , and be three elements in . Then,*(1)*(2)**(3)**(4)**(5)**(6)**(7)**(8)**(9)**(10)**(11)**(12)**(13)** if and only if *(14)* if and only if *(15)*If , then *(16)* if and only if *

*Proof. *Straightforward.

*Example 3. *From Example 2, the complements of are defined as , , . , , . Then, such that , , .

Also, such that , , . Then, the complement of is , , . Hence,