Theory, Algorithms, and Applications within Neutrosophic Modelling and OptimisationView this Special Issue
Neutrosophic Semiopen Hypersoft Sets with an Application to MAGDM under the COVID-19 Scenario
Hypersoft set is a generalization of soft sets, which takes into account a multiargument function. The main objective of this work is to introduce fuzzy semiopen and closed hypersoft sets and study some of their characterizations and also to present neutrosophic semiopen and closed hypersoft sets, an extension of fuzzy hypersoft sets, along with few basic properties. We propose two algorithms based on neutrosophic hypersoft open sets and topology to obtain optimal decisions in MAGDM. The efficiency of the algorithms proposed is demonstrated by applying them to the current COVID-19 scenario.
Fuzzy set theory  is an important tool for dealing with vagueness and incomplete data and is much more evolving and applied in different fields. Fuzzy set, which is an extension of general sets, has elements with membership function within the interval [0, 1]. In view of other options of human thinking, fuzzy set along with some conditions is extended to the intuitionistic fuzzy set . The intuitionistic fuzzy set assigns membership and nonmembership functions to each object which satisfies the constraint that the sum of both membership functions is between 0 and 1.
Fuzziness was improved and extended from intuitionistic sets to neutrosophic sets. Smarandache  proposed neutrosophic sets, an essential mathematical tool which deals with incomplete, indeterminant, and inconsistent information. Neutrosophic set is characterized by the elements with truth, indeterminacy, and false membership functions which assume values within the range of 0 and 1. Wang et al.  proposed the concept of single-valued neutrosophic sets, a generalization of intuitionistic sets and a subclass of neutrosophic sets, which comprise elements with three membership functions which they belong to interval [0, 1]. Under this neutrosophic environment, many researchers have worked on their extensions and developed many applications and results. A ranking approach based on the outranking relations of simplified neutrosophic numbers is developed in order to solve MCDM problems. Practical examples are provided to illustrate the proposed approach with a comparison analysis . A comparison analysis is performed for this method with two examples , and the developed single-valued neutrosophic TOPSIS extension is demonstrated on a numerical illustration of the evaluation and selection of e-commerce development strategies .
Molodtsov  introduced the idea of soft theory as a new approach to dealing with uncertainty, and now, there is a rapid growth of soft theory along with applications in various fields. Maji et al.  defined various basic concepts of soft theory, and the study of soft semirings by using the soft set theory has been initiated, and the notions of soft semirings, soft sub-semirings, soft ideals, idealistic soft semirings, and soft semiring homomorphisms with several related properties are investigated [10, 11]. Maji et al.  developed the fuzzy soft set theory, which is a combination of soft and fuzzy sets.
The idea of soft sets was generalized into hypersoft sets by Smarandache  by transforming the argument function F into a multiargument function. He also introduced many results on hypersoft sets. Saqlain et al.  utilized this notion and proposed a generalized TOPSIS method for decision-making. Neutrosophic sets , from their very introduction, have seen many such extensions and have been very successful in applications. A new hybrid methodology for the selection of offshore wind power station location combining the Analytical Hierarchy Process and Preference Ranking Organization Method for Enrichment Evaluations methods in the neutrosophic environment has been proposed , a neutrosophic preference ranking organization method for enrichment evaluation technique for multicriteria decision-making problems to describe fuzzy information efficiently was proposed and applied to a real case study to select proper security service for FMEC in the presence of fuzzy information , and a model is proposed based on a plithogenic set and is applied to differentiate between COVID-19 and other four viral chest diseases under the uncertainty environment .
In 2019, Rana et al.  introduced the plithogenic fuzzy hypersoft set (PFHS) in the matrix form and defined some operations on the PFHS. Single- and multivalued neutrosophic hypersoft sets were proposed by Saqlain et al. , who also defined tangent similarity measure for single-valued sets and an application of the same in a decision-making scenario. In another effort, Saqlain et al.  also introduced aggregation operators for neutrosophic hypersoft sets. A recent development in this area of research is the introduction of basic operations on hypersoft sets in which hypersoft points in different fuzzy environments are also introduced .
Fuzzy topology, a collection of fuzzy sets fulfilling the axioms, was defined by Chang . A new definition of fuzzy space compactness and observed to have compactness along with a Tychonoff theorem for an arbitrary product of compact fuzzy spaces and a 1-point compactification , filters in the lattice , where I is the unit interval and X an arbitrary set, have all been studied and using this study the convergence is defined in fuzzy topological space which leads to characterise fuzzy continuity and compactness . Then, the basic concepts of intuitionistic fuzzy topological spaces were constructed, and the definitions of fuzzy continuity, fuzzy compactness, fuzzy connectedness, and fuzzy Hausdorff space and some characterizations concerning fuzzy compactness and fuzzy connectedness were defined . Neutrosophic topological spaces were introduced by Salama and Alblowi , and further concepts such as connectedness, semiclosed sets, and generalized closed sets  were developed.
The concept of fuzzy soft topology and some of its structural properties such as neighborhood of a fuzzy soft set, interior fuzzy soft set, fuzzy soft basis, and fuzzy soft subspace topology were studied . The soft topological spaces, soft continuity of soft mappings, soft product topology, and soft compactness, as well as properties of soft projection mappings, have all been defined , and a relationship between a fuzzy soft set's closure and its fuzzy soft limit points has been constructed on fuzzy soft topological spaces . Subspace, separation axioms, compactness, and connectedness on intuitionistic fuzzy soft topological spaces were defined along with some base theorems , some important properties of intuitionistic fuzzy soft topological spaces and intuitionistic fuzzy soft closure and interior of an intuitionistic fuzzy soft set were introduced, and an intuitionistic fuzzy soft continuous mapping with structural characteristics was studied . A topology on a neutrosophic soft set was constructed, neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, and neutrosophic soft boundary were introduced, some of their basic properties were studied, and the concept of separation axioms on the neutrosophic soft topological space was introduced . The concept of fuzzy hypersoft sets was applied to fuzzy topological spaces, and fuzzy hypersoft topological spaces were presented by Ajay and Charisma in . In the same work, fuzzy hypersoft topology has been extended to intuitionistic and neutrosophic hypersoft topological spaces along with their properties. In this paper, we define the idea of semiopen sets in fuzzy hypersoft topological spaces with their characterization and extend to semiopen sets in intuitionistic and neutrosophic hypersoft topological spaces.
The paper is structured as follows: Section 2 recalls few basic terminologies and definitions of fuzzy hypersoft topological spaces. In Section 3, we define semiopen sets in fuzzy hypersoft topological spaces along with some of their properties. Sections 4 and 5 elaborate the logical extension of fuzzy hypersoft semiopen sets to intuitionistic and neutrosophic hypersoft semiopen sets. In Section 6, we present an application of the neutrosophic hypersoft open set and topology in an MAGDM and conclude in Section 7.
Definition 1. (see ). Let be an element of (where with each a subset of ), and let the set of all fuzzy hypersoft (FH) subsets of be and , a subcollection of .(i)(ii)(iii)If the above axioms are satisfied, then is a fuzzy hypersoft topology (FHT) on which is called a fuzzy hypersoft topological space (FHTS). Every member of is called an open fuzzy hypersoft set (OFHS). A fuzzy hypersoft set is said to be a closed fuzzy hypersoft set (CFHS) if its complement is OFHS.
Example 1. Let and the attributes be , , and . Then, the fuzzy hypersoft set isLet us consider this fuzzy hypersoft as . Then, the subfamilyof is a FHT on .
Definition 2 (see ). Let be a FHT on and be a FH set in . A FH set in is a neighbourhood of the FH set of if and only if there exists an OFHS such that .
Definition 3 (see ). Let be a FHTS and be FH sets in such that . Then, is said to be an interior fuzzy hypersoft set (IFHS) of if and only if is a neighbourhood of . The union of the whole IFHS of is named the interior of and denoted as or .
Definition 4 (see ). Let be a FHTS and . The fuzzy hypersoft closure (FHC) of is the intersection of all CFH sets that contain which is denoted by or .
Definition 5 (see ). Let be an element of (where with each a subset of . Let the set of all neutrosophic hypersoft (NH) subsets of be and , a subcollection of .(i)(ii)(iii)If the above axioms are satisfied, then is a neutrosophic hypersoft topology (NHT) on which is called a neutrosophic hypersoft topological space (NHTS). Every member of is called an open neutrosophic hypersoft set (ONHS). A neutrosophic hypersoft set is called a closed neutrosophic hypersoft set (CNHS) if its complement is an ONHS.
For example, and are neutrosophic hypersoft topologies on and are called indiscrete NHT and discrete NHT, respectively.
Definition 6 (see ). Let be a NHT on and be a NH set in . A NH set in is a neighbourhood of the NH set of iff there exists an ONHS such that .
Definition 7 (see ). Let be a NHTS and be NH sets in such that . Then, is said to be an interior neutrosophic hypersoft set (INHS) of if and only if is a neighbourhood of . The union of the whole INHS of is named the interior of and denoted as or .
Definition 8 (see ). Let be a NHTS and . The neutrosophic hypersoft closure (NHC) of is the intersection of all CNH sets that contain which is denoted by or . Thus, is the smallest CNHS which has , and is the CNHS.
3. Fuzzy Semiopen and Closed Hypersoft Sets
Definition 9. Let be a FHTS and . If , then is called the fuzzy semiopen hypersoft set (FSOHS). We denote the set of all fuzzy semiopen hypersoft sets by .
Definition 10. A fuzzy hypersoft set in the FHST space is a fuzzy semiclosed hypersoft set (FSCHS) if and only if its complement is FSOHS. The class of FSCHS is denoted by .
Example 2. Let and the attributes be , , and .
The fuzzy hypersoft topological space is :The fuzzy hypersoft setis a FSOHS.
Theorem 1. Let be a FHTS and ; then,(i)Arbitrary fuzzy hypersoft union of FSOHS is FSOHS(ii)Arbitrary fuzzy hypersoft intersection of FSCHS is FSCHS
Proof. (i) Let .
Then, , .
Similarly, (ii) is proved.
Theorem 2. Let be a FHTS and . Then,(i) if and only if there exists such that (ii)If and , then
Proof. (i)Let . Then, . We know that ; thus, . Let; thus, we get . Conversely, let for some . Then, . . Thus, . Therefore, .(ii)Let . Then, for some . If , then .Hence, . Thus, by (i), .
Definition 11. Let be a FHTS and .
Then, the largest fuzzy semiopen hypersoft set contained in is called the fuzzy semi-hypersoft interior of and denoted by , i.e., .
And the smallest fuzzy semiclosed hypersoft set containing is called the fuzzy semi-hypersoft closure of and denoted by .
, and .
Theorem 3. Let be a FHTS and . Then, the following properties hold:(i) and (ii)(iii) is the largest fuzzy semiopen hypersoft set contained in (iv)If , then (v)(vi)(vii)
Theorem 4. Let be a FHTS and . Then, the following properties hold:(i), and (ii)(iii) is the smallest fuzzy semiclosed hypersoft set that contains (iv)If , then (v)(vi)(vii)
Theorem 5. Every fuzzy open (closed) hypersoft set in a FHTS is a fuzzy semiopen (closed) hypersoft set.
Proof. Let be a fuzzy open hypersoft set. Then, . Since , . Thus, .
Theorem 6. Let be a FHTS and . If either or , then .
Proof. Let .
Then, we haveThus, .
Theorem 7. Let be a FHTS, be a fuzzy hypersoft open set, and . Then, .
Proof. Let be a FOHS and be a FSOHS.
Therefore, is a FSOHS.
Proposition 1. Let be a fuzzy hypersoft set in the FHTS . Then, is the FSCHS if and only if there exists an set such that .
Proposition 2. Every fuzzy hypersoft closed set is a FSCHS in a FHTS , but the converse need not be true.
Theorem 8. Let be a FHS in a FHTS . Then, is a FSCHS if and only if .
Proof. Suppose is a FSCHS. Then, there exists a such that . .
Conversely, let be a fuzzy hypersoft set in such that . Let . Then, . Thus, is a FSCHS.
Theorem 9. Let be a family of FSCHS in a FHTS . Then, the intersection is a FSCHS in .
Proof. Since each , is a FSCHS. Then, there exists a FCHS such that . Thus, . Consider . Then, is a FCHS, and hence, is a FSCHS.
Theorem 10. Let be a FSCHS and be a FCHS in . If , then is a FSCHS.
Proof. Since is a FSCHS, there exists a FCHS such that . Then, . Also, . Therefore, . Hence, is a FSCHS.
Remark 1. For any FCHS , . And for any FOHS , .
Remark 2. If is a fuzzy hypersoft set in , then .
Theorem 11. Let be a FHS in . Then,(i)(ii)(iii)(iv)
Proof. (i). Since is a FSCHS,. Conversely, . being FSCHS implies that is a FSOHS set. Thus, . And hence, .(ii)The proof is the same as that of (i).(iii) is FOHS implying that it is FSOHS. Therefore, . Now, . Thus, .(iv) is fuzzy closed hypersoft implying that is FSCHS. Therefore, . Now, .Hence, .
This implies .
4. Intuitionistic Semiopen and Closed Hypersoft Sets
Definition 12. Let be an IHTS and . If , then is called an intuitionistic semiopen hypersoft set (ISOHS). We denote the set of all intuitionistic semiopen hypersoft sets by .
Definition. 13An intuitionistic hypersoft set in the IHST space is an intuitionistic semiclosed hypersoft set (ISCHS) if and only if its complement is ISOHS. The class of ISCHS is denoted by .
Example 3. Let and the attributes be , , and .
The intuitionistic hypersoft topological space is :The intuitionistic hypersoft setis ISOHS.
Theorem 12. Let be an IHTS and ; then,(i)Arbitrary intuitionistic hypersoft union of ISOHS is an ISOHS(ii)Arbitrary intuitionistic hypersoft intersection of ISCHS is an ISCHS
Proof. (i) Let .
Then, , .
Similarly, (ii) is proved.
Theorem 13. Let be an IHTS and . Then,(i) if and only if there exists such that (ii)If and , then
Proof. (i)Let . Then, . We know that ; thus, . Let; thus, we get . Conversely, let for some . Then, . . Thus, . Therefore, .(ii)Let . Then, for some . If , then . Hence, . Thus, by (i), .
Definition 14. Let be an IHTS and .
Then, the largest intuitionistic semiopen hypersoft set contained in is called the intuitionistic semi-hypersoft interior of and denoted by , i.e., .
And the smallest intuitionistic semiclosed hypersoft set containing is called the intuitionistic semi-hypersoft closure of and denoted by . and .
Theorem 14. Let be an IHTS and . Then, the following properties hold:(i) and (ii)(iii) is the largest intuitionistic semiopen hypersoft set contained in (iv)If , then (v)(vi)(vii)
Theorem 15. Let be an IHTS and . Then, the following properties hold:(i) and (ii)(iii) is the smallest intuitionistic semiclosed hypersoft set that contains (iv)If , then (v)(vi)(vii)
Theorem 16. Every intuitionistic open (closed) hypersoft set in an IHTS is an intuitionistic semiopen (closed) hypersoft set.
Proof. Let be an intuitionistic open hypersoft set. Then, . Since