A new approach for operations on neutrosophic soft sets based on the novel norms for constructing topological structures

: Neutrosophic sets have recently emerged as a tool for dealing with imprecise, indeterminate, inconsistent data, while soft sets may have the potential to deal with uncertainties that classical methods cannot control. Combining these two types of sets results in a unique hybrid structure, a neutrosophic soft set (NS-set), for working effectively in uncertain environments. This paper focuses on determining operations on NS-sets through two novel norms. Accordingly, the min −norm and max −norm are well-defined here for the first time to construct the intersection, union, difference, AND, OR operations. Then, the topology, open set, closed set, interior, closure, regularity concepts on NS-sets are introduced based on these just constructed operations. All the properties in the paper are stated in theorem form, which is proved convincingly and logically. In addition, we also elucidate the relationship between the topology on NS-sets and the fuzzy soft topologies generated by the truth, indeterminacy, falsity degrees by theorems and counterexamples


Introduction
Data is a valuable source of knowledge that contains helpful information if exploited effectively [1].One of the challenges facing data researchers is the ambiguity and uncertainty of the data they have access to, which makes it difficult for them to process information.But these challenges are, in a positive sense, opportunities for the development of new techniques and tools, such as they various approaches based on fuzzy set theory [2].The advent of fuzzy theory has prompted extensive work on ideas such as fuzzy sets [3], vague sets [4], soft sets [5], and neutrosophic sets [6].It was originally thought that the development of new theories would eclipse fuzzy theory, but that does not seem to be the case [7].This research field is becoming more and more active, with a number of fundamental contributions to the rapid development of new theories [8,9].One of the most prominent applications is the use of fuzzy set theory in emerging and vibrant fields like machine learning [10,11] or topological data analysis [12,13].
In recent years, the study of soft sets [5] and neutrosophic sets [14] has become an attractive research area.Neutrosophic sets recently emerged as a tool for dealing with imprecise, indeterminate, and inconsistent data [15].In contrast, soft sets show potential for dealing with uncertainties that classical methods cannot control [16].Combining these two types of sets results in a unique hybrid structure, a neutrosophic soft set (NS-set) [17], for working effectively in uncertain environments.Maji proposed this [17,18] in 2013 and it was modified by Deli and Broumi [19] in 2015.Furthermore, Karaaslan [20] redefined this concept and its operations to be more efficient and complete.Since then, this structure has proved to be quite effective when applied in real life in many fields, such as decision making [17], market prediction [21], and medical diagnosis [22,23].
The topology on NS-sets is one of the issues that needs more attention, alongside neutrosophic topology [24,25] and soft topology [26].This issue has emerged recently to help complete the overall picture for NS and aid its practical applications based on topology [27,28].In 2017, Bera and Mahapatra [29] gave general operations to construct a topology on NS-sets.They also presented concepts related to topological space such as interior, closure, neighborhood, boundary, regularity, base, subspace, separation axioms, along with specific illustrations and proofs.In 2018, these authors [30] continued to develop further studies on connectedness and compactness on NS -topological space.In 2019, Ozturk, Aras, and Bayramov [31] introduced a new approach to topology on NS-sets.This approach is quite different from the previous work [29], and was further developed by constructing separation axioms [32] in the same year, 2018.Recently, the continuum [33] or compactness [34] on the topological space generated on NS-sets has also been studied with the same properties as the normal space.Many variations [35] of the topological space on NS-sets have also attracted the attention of researchers, and most of the related works are inspired by topology on neutrosophic and soft sets with the idea of a hybrid structure [36,37].
In this work, we construct the topological space and related concepts on NS-sets through general operations in a way that is very different from the work of Bera and Mahapatra [29,30], but more general than the work of Ozturk, Aras, and Bayramov [31,32], with our operations based on the generality of min and max operations.This work begins by defining two new operations to create the relationships between NS-sets.These relations are then used as the kernel for forming topology and topological relations on NS-sets.One emphasis shown here is on elucidating the relationship between the topology on NS-sets and the component fuzzy soft topologies.All the ideas in this work are presented convincingly and clearly through definitions, theorems, and their consequences.
In summary, the significant contributions of this study are as follows: (1) Defining two novel concepts, called min −norm and max −norm, to provide a theoretical foundation for determining operations on NS-sets, including intersection, union, difference, AND, and OR.
(2) Constructing the topology, open set, closed set, interior, closure, and regularity concepts on NS-sets based on just determined operations.
(3) Elucidating the relationship between the topology on NS-sets and the fuzzy soft topologies generated by truth, indeterminacy, falsity degrees by the theorems and counterexamples.
(4) The concepts are well-defined, and the theorems are proved convincingly and logically.This work is organized as follows: Section 1 presents the motivation and introduces the significant contributions.Section 2 briefly introduces NS-sets and related concepts.The two new ideas, min −norm and max −norm, are provided in Section 3 as a theoretical foundation for determining operations on NS-sets, including intersection, union, difference, AND, and OR.In Section 4, the topology on NS-sets is defined with related concepts such as open set, closed set, interior, closure, and regularity.Furthermore, the relationship between the topology on NS-sets and the fuzzy soft topologies generated by truth, indeterminacy, and falsity functions by theorems and counterexamples in Section 5.The last section presents conclusions and future research trends in this area.

Preliminaries
This section recalls the NS-set proposed in 2013 by Maji [17,18], then modified and improved in 2015 by Deli and Broumi [19].This concept is based on combining soft [5] and neutrosophic [6] sets.Some background related to NS-sets is briefly presented below so that readers can better understand the following sections.
Without loss of generality, we consider  to be a universal set, ℰ to be a parameter set, and () to denote the collection of all neutrosophic sets on .Definition 1. ( [18,19]).The pair (, ℰ) is a NS-set on  where : ℰ ⟶ () is a set valued function determined by  ⟼ () ≔   with for all  ∈ ℰ, and the real function triples    ,    ,    :  ⟶] − 0; 1 + [ indicate truth, indeterminacy, and falsity degrees, respectively, with no restriction on their sum.In other words, the NS-set can be described as a set of ordered tuples as follows: ) :  ∈ ℰ,  ∈ }.
If nothing changes, the symbol () indicates the collection of all NS-sets on .Besides, if the NS-sets consider the same parameter set ℰ, then it is not mentioned repeatedly in order to simplify the notations.Moreover, because the values of , ,  belong to the unit interval [0; 1], the integral part of the values is almost zero.Typically, it may occur that the integer part is omitted (for example, . 1 instead of 0.1).Therefore, if it does not lead to confusion, this omitted format of a decimal is always used in all the tables used in this paper.
Example 2. The NS-set  and its complement  ̅ are represented according to Eq (10) in Table 2 as follows: Proof.These properties are directly inferred from the definitions of the null, semi-null, absolute, semiabsolute NS-sets and the complement operation.

Another novel approach for operations on NS-sets
In this section, we focus on defining two novel norms, called  − and  −, as the foundations for determining operations on NS-sets in general.Each operation is well-defined along with its well-proven properties.

𝑚𝑖𝑛 −𝑛𝑜𝑟𝑚 and 𝑚𝑎𝑥 −𝑛𝑜𝑟𝑚
Some commonly used min −norm and max −norm are shown in Table 3.On the other hand, all of these norms satisfy De Morgan's law in pairs.Table 3.Some commonly used min −norm • and max −norm ∘ satisfying the De Morgan's law.

Intersection
Definition 7. The intersection of the two NS-sets  and , written as  ∩ , is determined by Example 3. Let two NS-sets  and  be represented in Table 4 as follows: If using min −norms • = max{ +  − 1,0} and max −norms  ∘  = min{ + , 1} , the intersection  ∩  of the two above NS-sets is described according to Eq (13) in Table 5 as follows: Proof.These properties are directly inferred from the definitions of norms and union operation.Proof. ( and Moreover, due to De Morgan's law of the min −norm and max −norm.Therefore, Moreover, The distributive properties between intersection and union operations are not satisfied in the case of these general operations.Counterexamples are shown in Example 5.
Example 5. Let the NS-set  be represented in Table 7 as follows:  Proof.

AND and OR
Definition 12.The AND operation of the two NS-sets  and B with the same parameter set ℰ, written as  ∧ , is determined over the same parameter set ℰ × ℰ by Proof. ( and Moreover, due to De Morgan's law of the min −norm and max −norm.Therefore, and Moreover, due to De Morgan's law of the min −norm and max −norm.Therefore,

Topology on NS-sets
This section uses the operations just constructed above as the core to build the topology and related concepts on NS-sets.It is important to note that the norms used must satisfy De Morgan's law.Volume 7, Issue 6, 9603-9626.

NS-topological space
Definition 14.A collection  ⊆ () is NS-topology on  if it obeys the following properties: (a) ∅ ℰ and  ℰ belongs to , (b) The intersection of any finite collection of 's elements belongs to , (c) The union of any collection of 's elements belongs to .Then, the pair (, ) is a NS-topological space and each element of  is a NS-open set.Example 8. Let three NS-sets  1 ,  2 ,  3 be represented in Table 12 as follows: If using the min −norm • = min{, } , max −norm  ∘  = max{, } , the collection  = {∅ ℰ ,  ℰ ,  1 ,  2 ,  3 } is a NS-topology.

4. 3 .Example 11 .
NS-closure Definition 17.The NS-closure of a NS-set , written as cl(), is the intersection of all NS-closed supersets of .The cl() is the smallest NS-closed set which contains .For the NS-topology  given in Example 10, let NS-set  be represented in

Table 1 .
Let two NS-sets  and  be represented in Table1as follows: NS-sets  and .

Table 11 .
Definition 13.The OR operation of the two NS-sets  and B with the same parameter set ℰ, written as  ∧ , is determined over the same parameter set ℰ × ℰ by NS-sets  ∧  and  ∨ .

Table 16 .
NS-sets .Theorem 11.A NS-set  is a NS-closed set if and only if  = cl().

Table 17 .
For the NS-topology  given in Example 10 and the NS-set  given in Example 11, the complement of  is represented in Table17as follows: NS-sets  ̅ .