*Now, we explain the practical meaning of alternatives X by taking the alternative z*1 *as an example: the single-valued neutrosophic fuzzy soft set* Ψˆ (*j*1 ) (*z*1) = (0.5, 0.6, 0.7, 0.4) *is the evaluation by four expert groups; the single-valued neutrosophic fuzzy soft value 0.5 (meaning 50*% *say yes in the first expert group) in* Ψˆ (*j*1 ) (*z*1)*, the single-valued neutrosophic fuzzy soft value 0.6 (meaning 60% say no in the second expert group) in* Ψˆ (*j*1 ) (*z*1), *the single-valued neutrosophic fuzzy soft value 0.7 (meaning 70*% *say yes in the third expert group) in* Ψˆ (*j*1 ) (*z*1)*, and fuzzy value 0.4 (meaning 40% say no in the fourth expert group) in* Ψˆ (*j*1 ) (*z*1). *Then, by computing* (Φˆ ∧Ψˆ )(*iq*,*jq*) (*q* = 1, 2, 3) *in the second step of Algorithm 2, we obtain the following:*

*By calculating in the third step of Algorithm 2, we get the truth membership* (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) *, the indeterminacy membership* (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) *, and the falsity membership* (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) , *respectively, as follows: (q* = 1, 2, 3*):*

*By calculating in the fourth step of Algorithm 2, we obtain the max-matrices of* (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) , (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) , *and* (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) (*p* = 1, 2, 3, 4, 5; *q* = 1, 2, 3), *respectively, for every zp* ∈ *X as follows:*

*By calculating in the fifth step of Algorithm 2, we obtain the max-decision τT*, *τI* , *and τF of elements zp*, *respectively, as follows (p* = 1, 2, 3, 4, 5*):*

$$\tau\_{\mathcal{T}}(z\_1) = 2, \ \tau\_{\mathcal{T}}(z\_2) = 0.3, \ \tau\_{\mathcal{T}}(z\_3) = 0.8, \ \tau\_{\mathcal{T}}(z\_4) = 2.05, \ \tau\_{\mathcal{T}}(z\_5) = 1.5;$$

$$\tau\_{\mathcal{I}}(z\_1) = 0.24, \ \tau\_{\mathcal{I}}(z\_2) = 0.68, \ \tau\_{\mathcal{I}}(z\_3) = 0.54, \ \tau\_{\mathcal{I}}(z\_4) = 1.06, \ \tau\_{\mathcal{I}}(z\_5) = 0.48;$$

$$\tau\_{\mathcal{F}}(z\_1) = 0, \ \tau\_{\mathcal{F}}(z\_2) = 0.25, \ \tau\_{\mathcal{F}}(z\_3) = 0, \ \tau\_{\mathcal{F}}(z\_4) = 0, \ \tau\_{\mathcal{F}}(z\_5) = 3.87.$$

*By calculating in the sixth step of Algorithm 2, the scores S*(*zp*) *of elements zp*(*p* = 1, 2, 3, 4, 5), *respectively, are as follows:*

 $S(z\_1) = 2.24$ ,  $S(z\_2) = 1.23$ ,  $S(z\_3) = 1.34$ ,  $S(z\_4) = 3.11$ ,  $S(z\_5) = 5.85$ .

*Finally, we know from the seventh step that z*5 *has a high value. Therefore, the experts should select z*5 *as the best choice.*

#### **Remark 2.**

*(1) By means of Algorithms 1 and 2, we can see that the final results are in agreement. Thus, x*5 *is the most accurate and refinable.*

*(2) By comparing the steps in Algorithms 1 and 2, we can see that step 4 and step 5 in Algorithm 2 are complicated in their process compared to step 2 and step 3 in Algorithm 1, respectively. So, if we take the complexity of these steps into consideration, Algorithm 2 gives its decision concisely.*

*(3) Algorithms 1 and 2 that we have elaborated here arrive at their decisions by combining the concept of single-valued neutrosophic fuzzy set theory and soft set theory. As result, we can apply Algorithm 1 to picture fuzzy soft sets [29], generalized picture fuzzy soft sets [13], and interval-valued neutrosophic soft sets [12]. Further, Algorithm 2 can be applied to possibility m-polar fuzzy soft sets [15] and possibility multi-fuzzy soft sets [17].*

#### **5. Conclusions**

We introduced the notion of the single-valued neutrosophic fuzzy soft set as a novel neutrosophic soft set model. We discussed the five operations of the single-valued neutrosophic fuzzy soft set, such as subset, equal, union, intersection, and complement. The structure properties of the single-valued neutrosophic fuzzy soft set are explained. Then, a novel approach (i.e., Algorithm 1) is presented as a single-valued neutrosophic fuzzy soft set decision method. Lastly, an application (i.e., Algorithm 2) of a single-valued neutrosophic fuzzy soft set for fuzzy decision-making is constructed, and the two approaches (i.e., Algorithms 1 and 2) introduce an important contribution to further research and relevant applications. Therefore, in the future, we will provide a real application with a real dataset or we will apply the two approaches (i.e., Algorithms 1 and 2) to lung cancer disease [30] and coronary artery disease [31]. In addition, we will describe in more detail in order to clarify if the methods (i.e., Algorithms 1 and 2) converge or diverge from standard approaches such as fuzzy sets [1], intuitionistic fuzzy sets [2], picture fuzzy sets [3].

**Author Contributions:** Conceptualization and data curation, A.M.K.; Methodology, A.A.A. and W.A.; Writing—original draft preparation, F.S.; Funding acquisition, D.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors thank the editors and the anonymous reviewers for their insightful comments which improved the quality of the paper. The authors wish to thank the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia, for their support of their research.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Single-Valued Neutrosophic Set Correlation Coe**ffi**cient and Its Application in Fault Diagnosis**

#### **Shchur Iryna 1 , Yu Zhong 1 , Wen Jiang 1,2,\* , Xinyang Deng 1 and Jie Geng 1**

Received: 3 July 2020; Accepted: 11 August 2020; Published: 17 August 2020

**Abstract:** With the increasing automation of mechanical equipment, fault diagnosis becomes more and more important. However, the factors that cause mechanical failures are becoming more and more complex, and the uncertainty and coupling between the factors are getting higher and higher. In order to solve the given problem, this paper proposes a single-valued neutrosophic set ISVNS algorithm for processing of uncertain and inaccurate information in fault diagnosis, which generates neutrosophic set by triangular fuzzy number and introduces the formula of the improved weighted correlation coefficient. Since both the single-valued neutrosophic set data and the ideal neutrosophic set data are considered, the proposed method solves the fault diagnosis problem more effectively. Finally, experiments show that the algorithm can significantly improve the accuracy degree of fault diagnosis, and can better satisfy the diagnostic requirements in practice.

**Keywords:** neutrosophic set; fault diagnosis; triangle fuzzy number; weighted correlation coefficient

#### **1. Introduction**

With the development of automation technology, these mechanical machines gradually came into the stage of fully automated control operation [1–5]. In this way, people's hands are comparatively free, and machines are more intelligent and comprehensive; however, this kind of full automation greatly increases the probability of mechanical equipment failure as well [6–11]. If the mechanical equipment has faults, the quality of the manufactured products will not pass the standard, which will affect the economic benefits of the enterprise [12–15]; additionally, it will bring potential danger to personal safety [16–19]. In order to solve this problem, it is necessary to carry out fault diagnosis on mechanical equipment on a regular basis to detect and repair mechanical equipment and ensure its normal operation.

### *1.1. Research Status*

Therefore, the fault diagnosis of mechanical equipment has been widely concerned by many scholars, and has been applied in the military [20,22–24], medical [25–28], economic [29–32], and other fields. In ref. [33], looking at the problem of low efficiency of fault diagnosis of automobile exhaust system, based on a cold test, a fault diagnosis method is proposed for port vehicle exhaust system based on the principal component analysis. The variance contribution rate of principal component model is analyzed by the change of each variable of measurement data, and the fault diagnosis is achieved; In ref. [34], aiming at the problem of fault diagnosis of the data-driven system, a new diagnosis method based on Bayesian network (BN) combined with fault frequency is proposed to realize fault diagnosis; In ref. [35], based on the particle filter (PF) program, a dual estimation method is applied to fault diagnosis; In ref. [36], for the problem of bearing diagnosis under the condition

of variable speed, the method of support vector machine and neural network is used for bearing fault diagnosis; On the basis of machine learning technology, a new depth neural network model with domain self-adaptability is proposed in ref. [37], which realizes fault diagnosis, but the selection of the best parameters of the model is random, and the application of the model is limited.

In the actual operating environment of mechanical equipment, fault information is usually inaccurate, incomplete and uncertain. It is difficult to use the above fault diagnosis method for deterministic analysis and processing of fault information. In order to deal with the uncertain and inaccurate information in the fault information, so as to better handle the fault information and get more accurate fault diagnosis results, Smarandache [38] proposed the theory of neutrosophic set from a philosophical point of view. It describes the uncertainty, imprecision, and inconsistent information in the objective world much better. The literature [39] introduced the theory of interval neutrosophic set and single-valued neutrosophic set; The literature [40] proposed the theory of simplified neutrosophic set; Literature [41] proposed a single-valued neutrosophic set SVNS method with a weighted correlation coefficient to realize fault diagnosis. However, the correlation coefficient does not comprehensively consider the single-valued neutrosophic set and ideal neutrosophic set data under various faults [42–46], only the maximum value between them is considered [47–49]. This does not completely deal with the uncertain and inaccurate information in the fault information, and may lead to an incorrect diagnosis.

#### *1.2. Contribution of This Work*

Based on the problems above, properly handling the uncertain information in the fault diagnosis process is an important goal to be achieved, however, complicated and changeable environmental information, the mutual influence between the factors causing the failure are difficult to handle. Due to neutrosophic set's outstanding performance in handling uncertain information issues, this paper proposes a single-valued neutrosophic set ISVNS algorithm, which generates neutrosophic set by triangular fuzzy number and introduces the formula of the improved weighted correlation coefficient. In addition, the ISVNS algorithm comprehensively considers the single-valued neutrosophic set and ideal neutrosophic set data of various faults, so make it possible to analyze the data more comprehensively and make more accurate judgments. Finally, an example was used to diagnose the fault; the degree of accuracy of the fault diagnosis was calculated; the excellent productivity of the improved method, proposed in this paper was obtained by comparison. For the current difficulties in dealing with some uncertain issues, this method may have some enlightenment.

Due to the Overall Equipment Effectiveness (OEE) and the Overall labor effectiveness (OLE) are simple and practical production management tool, which has been widely used in European and American manufacturing and Chinese multinational companies. The global equipment efficiency index has become an important standard for measuring the production efficiency of enterprises, so it is also important to consider the proposed method's impact on OEE and OLE. During the simulation process in the laboratory, the ratio of the operating hours and the planned working hours is relatively high; therefore, the fault can be repaired in a more timely manner based on the diagnosis result, and the OEE and the OLE can be improved.

The remainder of this paper is organized as follows. Section 2 briefly introduced triangular fuzzy numbers and single-valued neutrosophic sets. Section 3 proposed improved correlation coefficient between single-valued neutrosophic sets. In Section 4, a numerical example is given to fault diagnosis and fault diagnosis accuracy based on the proposed approach. Some conclusions are shown in Section 5.

#### **2. Preliminaries**

#### *2.1. Triangular Fuzzy Numbers*

About data sets *D* = {*d*1, *d*2, · · · , *dn*}, take its minimum value *l* = min{*d*1, *d*2, · · · , *dn*}, average value *m* = *mean*{*d*1, *d*2, · · · , *dn*}, maximum value *u* = max{*d*1, *d*2, · · · , *dn*}, so *s* = [*l*; *m*; *u*] named as triangular fuzzy number [9] of this data set D, as shown in Figure 1: 1 2 {, , , } min{ , , , } 1 2 1 2 {, , , } max{ , , , } 1 2 [; ; ]

**Figure 1.** Geometric Interpretation of Triangular Fuzzy Number.

It can be seen from the geometric interpretation of the triangular fuzzy number in the figure above: If *x* = *m*, so *x* completely belongs to *D*;

*x* If *l* ≤ *x* ≤ *u*, so *x* takes a certain degree of data set *D*;

*x* If *x* < *l* or *x* > *u*, so *x* completely does not belong to the data set *D*.

*x* 1 1 11 (, , ) 2 2 22 (, , ) And, for two triangle fuzzles *M*1 = (*l*1, *m*1, *u*1) and *M*2 = (*l*2, *m*2, *u*2), have the following counting method:

$$\begin{array}{c} M\_1 \oplus M\_2 = (l\_1 + l\_{2\prime}m\_1 + m\_{2\prime}\mu\_1 + \mu\_2) \\ M\_1 \otimes M\_2 \approx (l\_1 l\_{2\prime} m\_1 m\_{2\prime} \mu\_1 \mu\_2) \\ \lambda \otimes M\_1 \approx (\lambda l\_{1\prime} \lambda m\_{1\prime} \lambda \mu\_1) \\ \frac{1}{M\_1} \approx (\frac{1}{\mu\_1}, \frac{1}{m\_1}, \frac{1}{l\_1}) \end{array}$$

#### *2.2. Single-Valued Neutrosophic Sets*

1 1 11 1 111 ( , ,) Neutrosophic set introduced by Smarandache [38] is an effective tool for solving the problems under complex environment. The definition of neutrosophic set is as follows.

1 1 11

(, , )

**Definition 1.** *X denote a space of points or objects, and each element of it is denoted as x. A neutrosophic set A in X is characterized by a truth-membership function TA, an indeterminacy-membership function IA and a falsity-membership function FA*. *TA*(*x*)*, IA*(*x*) *and FA*(*x*) *are real standard or non-standard subsets of* ]0 −*,* 1 +[*. That is:*

$$\begin{array}{c} \begin{array}{c} T\_A: X \mapsto [0^-, 1^+] \\ I\_A: X \mapsto [0^-, 1^+] \\ F\_A: X \mapsto [0^-, 1^+] \end{array} \end{array} \tag{1}$$
  $A \qquad \begin{array}{c} A \qquad \begin{array}{c} F\_A: X \mapsto [0^-, 1^+] \\ F\_A: X \mapsto [0^-, 1^+] \end{array} \end{array} \tag{1}$

]0 1 [ *For Formula (1)*, 0− ≤ *supTA*(*x*) + *supIA*(*x*) + *supFA*(*x*) ≤ 3 +.

: ]0 ,1 [ To facilitate the application of neutrosophic set in real scientific and engineering problems, the notion of SVNS was defined as follows.

: ]0 ,1 [ : ]0 ,1 [ **Definition 2.** *X denote a space of points or objects, and each element of it is denoted as x. A neutrosophic set A in X is characterized by a truth-membership function TA*(*x*)*, a indeterminacy-membership function IA*(*x*)

0 3

*and a falsity-membership function FA*(*x*)*. If TA*(*x*) : *X* → [0, 1]*, IA*(*x*) : *X* → [0, 1] *and FA*(*x*) : *X* → [0, 1] *satisfied:*

$$\begin{array}{l} \mathbf{x} \in \mathcal{X} \mapsto T\_A(\mathbf{x}) \in [0, 1] \\ \mathbf{x} \in \mathcal{X} \mapsto I\_A(\mathbf{x}) \in [0, 1] \\ \mathbf{x} \in \mathcal{X} \mapsto F\_A(\mathbf{x}) \in [0, 1] \\ 0 \le T\_A(\mathbf{x}) + I\_A(\mathbf{x}) + F\_A(\mathbf{x}) \le 3 \end{array} \tag{2}$$

*then an SVNS A in X can be denoted as:*

$$A = \left| \{ \mathbf{x}, \ T\_A(\mathbf{x}), I\_A(\mathbf{x}), F\_A(\mathbf{x}) \} \middle| \mathbf{x} \in \mathbf{X} \right| \tag{3}$$

*which is called an SVNS. Especially, if X includes only one element, N* = *TA*(*x*), *IA*(*x*), *FA*(*x*) *is called a single-valued neutrosophic number (SVN).*

For any two SVNSs (*A* = *TA*(*x*), *IA*(*x*), *FA*(*x*) , *B* = *TB*(*x*), *IB*(*x*), *FB*(*x*) operational relations are defined as follows:

$$\begin{aligned} (1) \ A + B &= \left\{ T\_A(\mathbf{x}) + T\_B(\mathbf{x}) - T\_A(\mathbf{x})T\_B(\mathbf{x}), \ I\_A(\mathbf{x}) + I\_B(\mathbf{x}) - I\_A(\mathbf{x})I\_B(\mathbf{x}) \right. \\ &F\_A(\mathbf{x}) + F\_B(\mathbf{x}) - F\_A(\mathbf{x})F\_B(\mathbf{x}) \right\} \\ (2) \ A \times \mathbf{B} &= \left\{ T\_A(\mathbf{x})T\_B(\mathbf{x}), I\_A(\mathbf{x})I\_B(\mathbf{x}), F\_A(\mathbf{x})F\_B(\mathbf{x}) \right\} \\ (3) \ \lambda A &= \left\{ 1 - (1 - T\_A(\mathbf{x}))^\lambda, 1 - (1 - I\_A(\mathbf{x}))^\lambda, 1 - (1 - F\_A(\mathbf{x}))^\lambda \right\}, \quad \lambda > 0 \\ (4) \ A^\lambda &= \left\{ T\_A(\mathbf{x})^\lambda, I\_A(\mathbf{x}), F\_A(\mathbf{x}) \right\}, \quad \lambda > 0 \end{aligned} \tag{4}$$

These are a series of common laws in operation for SVNSs.

Moreover, the assumption and operation requirements are as follows: Because ISVNS algorithm generates neutral set by triangular fuzzy number, the fault template data of each fault must be greater than or equal to 3.

#### **3. The Proposed Method**

#### *3.1. Correlation Coe*ffi*cient between Single-Valued Neutrosophic Sets*

For any two neutrosophic sets *A* = {*TA*, *FA*, *IA*} and *B* = {*TB*, *FB*, *IB*}, the improved correlation coefficient is defined as follows:

$$\begin{array}{lcl}\mathcal{W}(A,B) &=& \frac{2\cdot\mathbb{C}(A,\mathcal{B})}{\mathbb{C}(A,\mathcal{A}) + \mathbb{C}(B,\mathcal{B})}\\&=& \frac{2\cdot\sum\_{i=1}^{n}\left[T\_{A}(\mathbf{x}\_{i})\cdot T\_{B}(\mathbf{x}\_{i}) + F\_{A}(\mathbf{x}\_{i})\cdot F\_{B}(\mathbf{x}\_{i}) + I\_{A}(\mathbf{x}\_{i})\cdot I\_{B}(\mathbf{x}\_{i})\right]}{\sum\_{i=1}^{n}\left[T\_{A}^{\prime2}(\mathbf{x}\_{i}) + F\_{A}^{\prime2}(\mathbf{x}\_{i}) + I\_{A}^{\prime2}(\mathbf{x}\_{i})\right] + \sum\_{i=1}^{n}\left[T\_{B}^{\prime2}(\mathbf{x}\_{i}) + F\_{B}^{\prime2}(\mathbf{x}\_{i}) + I\_{B}^{\prime2}(\mathbf{x}\_{i})\right]}\end{array} \tag{5}$$

In addition, the correlation coefficient for any *A* = {*TA*, *FA*, *IA*} and *B* = {*TB*, *FB*, *IB*} must satisfy the following three mathematical rules10:

$$\begin{cases} \begin{array}{l} W(A,B) = W(B,A) \\ 0 \le W(A,B) \le 1 \end{array} \\\ if A = B, \; W(A,B) = 1 \end{cases} \tag{6} $$

For Formula (5), prove separately as follows:

(1) According to the structural symmetry of the Formula (5), the condition *W*(*A*, *B*) = *W*(*B*, *A*) is satisfied.

(2) For each element in the Formula (5), they are satisfied ≥ 0, so obviously *W*(*A*, *B*) ≥ 0; The proof of inequality *W*(*A*, *B*) ≤ 1 as follows:

$$\begin{array}{lcl} \mathcal{C}(A,B) &=& \sum\_{i=1}^{n} \left[ T\_A(\mathbf{x}\_i) \cdot T\_B(\mathbf{x}\_i) + F\_A(\mathbf{x}\_i) \cdot F\_B(\mathbf{x}\_i) + I\_A(\mathbf{x}\_i) \cdot I\_B(\mathbf{x}\_i) \right] \\ &=& T\_A(\mathbf{x}\_1) \cdot T\_B(\mathbf{x}\_1) + T\_A(\mathbf{x}\_2) \cdot T\_B(\mathbf{x}\_2) + \dots + T\_A(\mathbf{x}\_n) \cdot T\_B(\mathbf{x}\_n) \\ &+& F\_A(\mathbf{x}\_1) \cdot F\_B(\mathbf{x}\_1) + F\_A(\mathbf{x}\_2) \cdot F\_B(\mathbf{x}\_2) + \dots + F\_A(\mathbf{x}\_n) \cdot F\_B(\mathbf{x}\_n) \\ &+ I\_A(\mathbf{x}\_1) \cdot I\_B(\mathbf{x}\_1) + I\_A(\mathbf{x}\_2) \cdot I\_B(\mathbf{x}\_2) + \dots + I\_A(\mathbf{x}\_n) \cdot I\_B(\mathbf{x}\_n) \end{array}$$

And because of the inequality:

$$ab \le \frac{a^2 + b^2}{2}$$

Therefore, we can get:

*C*(*A*, *B*) = P*n i*=1 [*TA*(*xi*)·*TB*(*xi*) + *FA*(*xi*)·*FB*(*xi*) + *IA*(*xi*)·*IB*(*xi*)] = *TA*(*x*1)·*TB*(*x*1) + *TA*(*x*2)·*TB*(*x*2) + · · · + *TA*(*xn*)·*TB*(*xn*) +*FA*(*x*1)·*FB*(*x*1) + *FA*(*x*2)·*FB*(*x*2) + · · · + *FA*(*xn*)·*FB*(*xn*) +*IA*(*x*1)·*IB*(*x*1) + *IA*(*x*2)·*IB*(*x*2) + · · · + *IA*(*xn*)·*IB*(*xn*) ≤ *TA* 2 (*x*1 )+*TB* 2 (*x*1 ) 2 + *TA* 2 (*x*2)+*TB* 2 (*x*2) 2 + · · · + *TA* 2 (*xn*)+*TB* 2 (*xn*) 2 + *FA* 2 (*x*1 )+*FB* 2 (*x*1 ) 2 + *FA* 2 (*x*2)+*FB* 2 (*x*2) 2 + · · · + *FA* 2 (*xn*)+*FB* 2 (*xn*) 2 + *IA* 2 (*x*1 )+*IB* 2 (*x*1 ) 2 + *IA* 2 (*x*2)+*IB* 2 (*x*2) 2 + · · · + *IA* 2 (*xn*)+*IB* 2 (*xn*) 2 = 1 2 ( P*n i*=1 [*TA* 2 (*xi*) + *FA* 2 (*xi*) + *IA* 2 (*xi*)] + P*n i*=1 [*TB* 2 (*xi*) + *FB* 2 (*xi*) + *IB* 2 (*xi*)]) = 1 2 [*C*(*A*, *A*) + *C*(*B*, *B*)]

Therefore:

$$\mathcal{C}(A,B) \le \frac{1}{2} [\mathcal{C}(A,A) + \mathcal{C}(B,B)]^\top$$

There is:

$$2\cdot \mathbb{C}(A,B) \le \mathbb{C}(A,A) + \mathbb{C}(B,B)$$

Finally, contacting the previous types, there are:

$$W(A,B) = \frac{2 \cdot \mathbb{C}(A,B)}{\mathbb{C}(A,A) + \mathbb{C}(B,B)} \le 1$$

In summary, the condition 0 ≤ *W*(*A*, *B*) ≤ 1 is satisfied;

(3) If *A* = *B*, so for any *xi* ∈ *X*(*i* = 1, 2, · · · , *n*), all *TA*(*xi*) = *TB*(*xi*), *FA*(*xi*) = *FB*(*xi*), *IA*(*xi*) = *IB*(*xi*), we can see from the structure of Formula (5), *W*(*A*, *B*) = 1.

In practical application, it is usually necessary to consider the weight of neutrosophic sets, so the weighted correlation coefficient of neutrosophic sets is given:

$$W(A,B) = \frac{2 \cdot \sum\_{i=1}^{n} w\_i [T\_A(\mathbf{x}\_i) \cdot T\_B(\mathbf{x}\_i) + F\_A(\mathbf{x}\_i) \cdot F\_B(\mathbf{x}\_i) + I\_A(\mathbf{x}\_i) \cdot I\_B(\mathbf{x}\_i)]}{\sum\_{i=1}^{n} w\_i [T\_A^{-2}(\mathbf{x}\_i) + F\_A^{-2}(\mathbf{x}\_i) + I\_A^{-2}(\mathbf{x}\_i)] + \sum\_{i=1}^{n} w\_i [T\_B^{-2}(\mathbf{x}\_i) + F\_B^{-2}(\mathbf{x}\_i) + I\_B^{-2}(\mathbf{x}\_i)]} \tag{7}$$

Among them, *wj*(*j* = 1, 2, · · · , *n*) represents the weight of the *i* fault template, and P*n i*=1 *wi* = 1. In addition, the formula satisfies the three conditions of the formula and proves to be the same as the Formula (5), which is not repeated here.

#### *3.2. Fault Diagnosis Method*

( )

1

Based on the above analysis, properly handling the uncertain information in the fault diagnosis process is an important goal to be achieved, however, complicated and changeable environmental information, the mutual influence between the factors causing the failure are difficult to handle. Due to neutrosophic set's outstanding performance in handling uncertain information issues, this paper proposes a single-valued neutrosophic set ISVNS algorithm, which generates neutrosophic set by triangular fuzzy number and introduces the formula of the improved weighted correlation coefficient.

The objectives of the proposed algorithm are to rationally process the uncertain information in the diagnosis process, and obtain correct and reasonable fault diagnosis results from the fault data. Therefore, the laboratory simulation of the algorithm is carried out under the assumption that the actual fault is one of several known fault templates, the collected fault data is reasonable, and there is no major abnormality. The operating requirement is to collect fault data in a stable and equal time interval way. The detailed flow chart of the fault diagnosis method is shown in Figure 2:

**Figure 2.** The detailed flow chart of the fault diagnosis method.

1 2 1 2 **Step 1**: For fault template set *A*= {A1 , A2 , · · · , A*m*}, and test sample set *C*= {C1 , C2 , · · · ,*Cn*}. Firstly, three fuzzy numbers of fault template data and test sample data are generated, and the calculation method is as follows:

( )

In a group number, the largest value is the right end value of the triangle; the minimum value is the left end-point of the triangle; the average value is the upper-end value of the triangle; the height of the triangle is 1, as shown in Figure 1.

**Step 2**: By comparing the three fuzzy number of each attribute of the test sample and the three fuzzy number of the same attribute of the fault template, the degree of determinacy-membership *TAi Cj*), the degree of non-membership *FAi* (*Cj*), and the degree of indeterminacy-membership *IAi* (*Cj*) are obtained, as shown in Figure 3, and the calculation method is as follows: ( ) ( )

**Figure 3.** Each degree of membership and neutrsophic set generate a schematic diagram.

*TAi* (*Cj*): The ratio of the overlapping area of the analyzed sample triangular fuzzy number and the triangular fuzzy number of the fault template under a certain attribute, and the analyzed sample total area of the triangular fuzzy number;

*FAi* (*Cj*): The ratio of the area of the part not overlapped with the fault template in the analyzed sample triangular fuzzy number to the total area of the analyzed triangular fuzzy number. Moreover, for each attribute, the sum of the non-membership degree and the determined membership degree is 1;

*IAi* (*Cj*): The calculation of the indeterminacy-membership degree is as follows:

$$(R\_1 + R\_2)/2 \tag{8}$$

1 2 1 2

Among them, *R*1= 1−*S*1/*S*k, *R*2 = *S*k/max{*SA*, *SB*, *SC*}, *S*1 is the overlapping area of the analyzed sample triangular fuzzy number and the triangular fuzzy number of the fault template under the current attributes, {*S*k|*k* = *A*, *B*,*C*} is the area of the k fault triangular fuzzy number of the under the current attribute.

Finally, each degree of membership, the relationship of *Ai*(*i* = 1, 2, · · · *m*) and *Cj*(*j* = 1, 2, · · · *n*) is shown below:

$$A\_{i} = \left\{  \left| \mathbb{C}\_{j} \in \mathbb{C}\_{\prime} \right.\right\} = 1,2,\cdots,n\right\} \tag{9}$$

Among them *TAi* (*Cj*), *FAi* (*Cj*),*IAi* (*Cj*) ∈ [0, 1] respectively denote the degree of determinacy-membership, the degree of non-membership, and the degree of indeterminacy-membership between *Ai*(*i* = 1, 2, · · · *m*) and *Cj*(*j* = 1, 2, · · · *n*), also 0 ≤ *TAi* (*Cj*), *FAi* (*Cj*), *IAi* (*Cj*) ≤ 3.

**Step 3**: *TAi* (*Cj*), *FAi* (*Cj*),*IAi* (*Cj*) can be expressed as single-valued neutrosophic set *aij* =< *tij*, *fij*, *iij* >, at this point, a single-valued neutrsophic set decision matrix can be generated as follows:

$$\begin{array}{c} D \quad = \left( a\_{\mathrm{i}} \right)\_{\mathrm{m} \times \mathrm{n}} \\ \quad = \left[ \begin{array}{cccc} < t\_{11}, f\_{11}, i\_{11} > & < t\_{12}, f\_{12}, i\_{12} > & \cdots & < t\_{\mathrm{i}n}, f\_{\mathrm{i}\bar{j}}, i\_{\mathrm{i}\bar{j}} > \\ < t\_{21}, f\_{21}, i\_{21} > & < t\_{22}, f\_{22}, i\_{22} > & \cdots & < t\_{2n}, f\_{\mathrm{2n}}, i\_{2\bar{n}} > \\ & \vdots & & & \vdots \\ & & \vdots & & & \vdots \\ < t\_{m1}, f\_{m1}, i\_{m1} > & < t\_{m2}, f\_{m2}, i\_{m2} > & \cdots & < t\_{\mathrm{min}}, f\_{\mathrm{min}}, i\_{\mathrm{min}} > \end{array} \right] \end{array} \tag{10}$$

**Step 4**: After obtaining the single-valued neutrosophic set decision matrix *D*, the ideal single-valued neutrudophic number for attribute *j*(*j* = 1, 2, · · · , *n*) can be generated by column as follows:

$$a^\*\!\!\/ \_{\dot{\!\!\/}} =  = < \max\_{\dot{\!\!\/}} (t\_{\dot{\!\!\/}}) , \min\_{\dot{\!\!\/}} (f\_{\!\!\/}) , \min\_{\dot{\!\!\/}} (i\_{\dot{\!\!\/}}) > \tag{11}$$

Among them, max *i* (*tij*), min *i* (*fij*), min *i* (*iij*)respectively denote the maximum value of the *j*th column in *tij*, the minimum value of the *j*th column in *fij*, the minimum value of the *j*th column in *iij*.

**Step 5**: According to Formula (7), generated weighted correlation coefficient based on single-valued neutrsophic set decision matrix *D* and the ideal single-valued neutrudophic number *a* ∗ , the calculation formula is as follows:

$$\mathcal{W}(A\_i, B) = \frac{2 \cdot \sum\_{j=1}^n w\_j [t\_{ij} \cdot t^\* \ne f\_{ij} \cdot f^\* \ne i\_{ij} \cdot i^\* \ne j]}{\sum\_{j=1}^n w\_j [t\_{ij}^2 + f\_{ij}^2 + i\_{ij}^{-2}] + \sum\_{j=1}^n w\_j [t^\* \ne f^\* \ne j^2 + i^\* \ne j^2]} \tag{12}$$

Among them, *wj*(*j* = 1, 2, · · · , *n*) represents the weight of the *j* attribute, and P*n j*=1 *wj* = 1; *aij* =< *tij*, *fij*, *iij* > denote single-valued neutrosophic set for attribute *j* from decision matrix *D*, *a* ∗ *j* =< *t* ∗ *j* , *f* ∗ *j* , *i* ∗ *j* > denote the ideal single-valued neutrudophic number for attribute *j*.

**Step 6**: Finally, sorting the *W*(*Ai* , *B*) of each analyzed sample, the largest value indicates that the template data belongs to this kind of fault.

#### **4. Illustrative Example and Discussion**

In this section, in order to demonstrate the validity and accuracy rate of the proposed method, an example of a motor rotor is used. The data in this paper is originated from ref. [11], and the data analysis software is the LABVIEW environment12.

#### *4.1. Fault Diagnosis*

The specific steps for fault diagnosis using the ISVNS method proposed in this paper are as follows:

$$\begin{array}{lcl}B\_X = & \left[ <0.9612, 0.0388, 0.6747>, <0.7540, 0.2460, 0.5972>, \\ & < 0.9836, 0.0164, 0.6451>, < 0.9966, 0.0034, 0.5757> \right] \end{array} \tag{13}$$

(iv). The weights of attributes *j*(*j* = 1, 2, 3, 4) are all the same, that is the weight matrix *w* is as follows:

$$w = [0.25, 0.25, 0.25, 0.25] \tag{14}$$

Next, according to Table 3, Formula (7), (13), (14), for the fault template type *Ai* (*A*1= X11 − X45,*A*2= Y11 − Y45,*A*3= Z11 − Z45) and the ideal single-valued neutrosophic set *BX*, calculate the improved weight correlation coefficient as follows:

 *W*[*A*1, *BX*] = 0.8126 *W*[*A*2, *BX*] = 0.4133 *W*[*A*3, *BX*] = 0.5398 (15)

(v). Finally, according to Formula (15), *A*1 > *A*3 > *A*2, it can be seen that the analyzed samples X1–X4 belong to the first type of fault, namely, the X fault.

**Table 1.** Triangle fuzzy number of fault template.

**Table 2.** Triangular fuzzy numbers data of the analyzed sample.

**Table 3.** Single-valued neutrosophic set decision matrix under sample X.

#### *4.2. Fault Diagnosis Accuracy*

To verify the accuracy of fault diagnosis, separately extract arbitrary 40 faults data from the three faults template [18–42]; These 120 faults data are used as a diagnosis template. Diagnose the fault type to which it belongs. Finally, compare each test sample with its original fault template [43,44], Calculate the overall accuracy of fault diagnosis.

The SVNPWA algorithm in ref. [44] is used for verifying these 120 unknown fault samples, and the diagnosis accuracy is 98.33%. Moveover, the ISVNS algorithm proposed in this paper also applied for diagnosing the same 120 unknown fault samples, and the diagnosis accuracy is 99.16%. The diagnosis results are shown in Table 5.

**Table 5.** Diagnosis results of applying the SVNPWA and proposed algorithm.

It can be seen from Table 5: Compared with the SVNPWA algorithm, the fault diagnostic accuracy rate of the ISVNS algorithm, proposed in this paper, is improved by 0.83%. That is, the ISVNS algorithm can better satisfy the diagnostic needs than the basic SVNPWA algorithm.

#### **5. Conclusions**

This paper proposes an ISVNS algorithm, which introduces the improved weighted correlation coefficient formula, and more comprehensively considers both single-valued neutrosophic set and ideal neutrosophic set under various faults, effectively solved the problem of fault diagnosis. An example of a motor rotor is illustrated that the ISVNS algorithm could improve the diagnostic accuracy rate compared with the SVNPWA algorithm. In conclusion, the ISVNS algorithm can obtain better fault diagnosis accuracy, and satisfy the fault diagnosis needs in practice.

Since the collection and aggregation of data on technical faults is a laborious process. Moreover, the parameters in the diagnosis process are automatically generated in the laboratory simulation, without human intervention—therefore, the proposed method could be automating in practical application. Therefore, the next step of the work will focus on how to automating the proposed method in order to scale the use of the proposed algorithm.

**Author Contributions:** Conceptualization, W.J. and S.I.; methodology, W.J., S.I. and Y.Z.; validation, S.I.; formal analysis, W.J., S.I., Y.Z., X.D. and J.G.; writing—original draft preparation, S.I.; writing—review and editing, W.J., S.I., Y.Z., X.D. and J.G.; funding acquisition, W.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work is partially supported by National Natural Science Foundation of China (Program No. 61703338), Equipment Pre-Research Fund (Program No.<PHONE_NUMBER>9), National Science and Technology Major Project (2017-I-0001-0001).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

## **A New Multi-Sensor Fusion Target Recognition Method Based on Complementarity Analysis and Neutrosophic Set**

#### **Yuming Gong 1 , Zeyu Ma 1 , Meijuan Wang 1 , Xinyang Deng 1,\* and Wen Jiang 1,2,\***

Received: 1 August 2020; Accepted: 27 August 2020; Published: 31 August 2020

**Abstract:** To improve the efficiency, accuracy, and intelligence of target detection and recognition, multi-sensor information fusion technology has broad application prospects in many aspects. Compared with single sensor, multi-sensor data contains more target information and effective fusion of multi-source information can improve the accuracy of target recognition. However, the recognition capabilities of different sensors are different during target recognition, and the complementarity between sensors needs to be analyzed during information fusion. This paper proposes a multi-sensor fusion recognition method based on complementarity analysis and neutrosophic set. The proposed method mainly has two parts: complementarity analysis and data fusion. Complementarity analysis applies the trained multi-sensor to extract the features of the verification set into the sensor, and obtain the recognition result of the verification set. Based on recognition result, the multi-sensor complementarity vector is obtained. Then the sensor output the recognition probability and the complementarity vector are used to generate multiple neutrosophic sets. Next, the generated neutrosophic sets are merged within the group through the simplified neutrosophic weighted average (SNWA) operator. Finally, the neutrosophic set is converted into crisp number, and the maximum value is the recognition result. The practicality and effectiveness of the proposed method in this paper are demonstrated through examples.

**Keywords:** neutrosophic set; target recognition; complementarity analysis; data fusion

#### **1.  These applications also make target recognition technology more popular. The development of related technologies has greatly enriched the application scenarios of target recognition and tracking theories. Research on related theoretical methods has also received extensively attention. Target recognition involves image processing, calculation computer vision, pattern recognition and other subjects.

Generalized target recognition includes two stages, feature extraction, and classifier classification. Through feature extraction, image, video, and other target observation data are preprocessed to extract feature information, and then the classifier algorithm implements target classification based on the feature information [1]. Common image features can be divided into color gray statistical feature, texture edge feature, algebraic feature, and variation coefficient feature. The feature extraction methods corresponding to the above features are color histogram, gray-level co-occurrence matrix method, principal component analysis method, wavelet transform [2]. The classic target classification algorithms

include decision tree, support vector machine [3], neural network [4], logistic regression, and naive Bayes classification [5–8]. On the basis of a single classifier, the ensemble classifier integrates the classification results of a series of weak classifiers through an ensemble learning method, so as to obtain a better classification effect than a single classifier, mainly including Bagging and Boosting [9]. With the development of high-performance computing equipment and the enlargement in the amount of available data, deep learning related theories and methods have developed rapidly, and the application of deep learning in the direction of target recognition has also made it break through the limitations of traditional methods based on deep neural networks [10–12]. Classification network models include LeNet, AlexNet, VggNet.

In practical applications, multi-source sensor data is often processed in decision analysis [13–19] as the number of sensors increases. For the problem of uncertain information processing, there are many theoretical methods such as Dempster-Shafer evidence theory [20–26], fuzzy set theory [27,28], D number [29–32] and rough set theory [33].

Smarandache [34] firstly generalized the concepts of fuzzy sets [35], intuitionistic fuzzy sets (IFS) [36] and interval-valued intuitionistic fuzzy sets (IVIFS) [37], and proposed the neutrosophic set. It is very suitable to use the neutrosophic set to deal with uncertain and inconsistent information in the real world. However, its authenticity, uncertainty, and false membership function are defined in the real number standard or non-standard subset. Therefore, non-standard intervals are not suitable for scientific and engineering applications. Therefore, Ye [38] introduced a simplified neutrosophic set (SNS), which limits the true value, uncertainty, and false membership function to the actual standard interval [0, 1]. In addition, SNS also includes single value neutrosophic set (SVNS) [39–41] and interval neutrosophic set (INS) [42].

As a new kind of fuzzy set, neutrosophic set [43,44] have been used in many fields, such as decision-making [45–48], data analysis [49,50], fault diagnosis [51], the shortest path problem [52]. There is also a lot of progress in the related theoretical research of neutrosophic set. For example, score function of pentagonal neutrosophic set [53,54].

Existing multi-sensor fusion methods, such as evidence theory, have complex calculation and long calculation time [55], and there are a few methods that use neutrosophic set in multi-sensor fusion. Therefore, this paper proposes a multi-sensor fusion based on neutrosophic set. First, the complementarity vectors between multiple sensors are calculated. Then these complementarity vectors and the probability of sensor output are used to form a group of neutrosophic sets, and generated neutrosophic sets are fused through the SNWA operator. Finally, the neutrosophic set is converted to the crisp number, and the maximum value is the recognition result. The proposed method has simple calculation and fast operation, and can effectively improve the accuracy of target recognition.

The rest of this article is organized as follows: Section 2 introduces some necessary concepts, such as neutrosophic set and multi-category evaluation standard. The proposed multi-sensor fusion recognition method is listed step by step in Section 3. In Section 4, an example is used to illustrate and explain the effective of proposed method. Some results discussion are shown in Section 5.

#### **2. Preliminaries**

#### *2.1. Neutrosophic Set*

**Definition 1.** *The the simplified neutrosophic set (SNS) is defined as follows [38]:*

*X is a finite set, with a element of X denoted by x.  In particular, if X includes only one element, N* = h*x*, *TP*(*x*), *IP*(*x*), *FP*(*x*)i *is called a SNN (the simplified neutrosophic number) and is denoted by α* = h*µ*, *π*, *ν*i*. The numbers µ*, *π*, *ν denote the degree of membership, the degree of indeterminacy-membership, and the degree of non-membership.*

**Definition 2.** *The crisp number of each SNN is deneutrosophicated and calculated as follows [56]:*

$$S\_{\bar{i}} = \mu\_{\bar{i}} + (\pi\_{\bar{i}}) \times (\frac{\mu\_{\bar{i}}}{\mu\_{\bar{i}} + \nu\_{\bar{i}}}) \tag{6}$$

*Si can be regarded as the score of SNN, so SNN can be sorted according to crisp number Si .*

#### *2.2. Commonly Used Evaluation Indicators for Multi-Classification Problems*

The index for evaluating the performance of a classifier is generally the accuracy of the classifier, which is defined as the ratio of the number of samples correctly classified by the classifier to the total number of samples for a given test data set [57].

The commonly used evaluation indicators for classification problems are precision and recall. Usually, the category of interest is regarded as the positive category, and the other categories are regarded as the negative category. The prediction of the classifier on the test data set is correct or incorrect. There are four situations as follows:

Based on the above basic concepts, the commonly used evaluation indicators for multiclassification problems are as follows.

1. Precision or precision rate, also known as precision (P):

$$P = \frac{TP}{TP + FP} \tag{7}$$

2. Recall rate, also known as recall rate (R):

$$R = \frac{TP}{TP + FN} \tag{8}$$

3. F1 score is an index used to measure the accuracy of the classification model. It also takes into account the accuracy and recall of the classification model. The score can be regarded as a harmonic average of model accuracy and recall. Its maximum value is 1 and its minimum value is 0:

$$F\_1 = \frac{2 \times P \times R}{P + R} \tag{9}$$

#### *2.3. AdaBoost Algorithm*

AdaBoost is essentially an iterative algorithm [58], and its core idea is to train some weak classifier *hi* based on the initial sample using the decision tree algorithm. Use the classifier to detect the sample set. For each training sample point, adjust its weight according to whether the result of its classification is accurate: if *hi* makes it classified correctly, reduce the weight of the sample point; otherwise, increase the sample the weight of the point. The adjusted weight is calculated according to the accuracy of the detection result. The sample set after adjusting the weight constitutes the sample set to be trained at the next level, which is used to train the next level classifier. In this way, iterate step by step to obtain a new classifier until the classifier *hm* is obtained, and the sample detection error rate is 0.

Combine *h*1, *h*2, . . . , *hm* according to the error rate of the sample detection: make the weak classifier with the larger error account for the smaller weight in the combined classifier, and the weak classifier with the smaller error account for the larger weight to obtain a combined classifier.

The algorithm is essentially a comprehensive improvement of the weak classifier trained by the basic decision tree algorithm. Through continuous training of samples and weight adjustment, multiple classifiers are obtained, and the classifiers are combined by weight to obtain a comprehensive classifier that improves the ability of data classification. The whole process is as follows:

After *M* cycles, *M* weak classifiers are obtained, and the joint weight of the classifier is calculated according to the detection accuracy of each weak classifier, and finally a strong classifier is obtained.

#### *2.4. HOG Feature*

Histogram of oriented gradient (HOG), which is a feature descriptor for target detection. This technology counts the number of directional gradients that appear locally in the image. This method is similar to the histogram of edge orientation and scale-invariant feature transform, but the difference is hog calculate the density matrix based on the uniform space to improve accuracy. Navneet Dalal and Bill Triggs first proposed HOG in 2005 for pedestrian detection in static images or videos [59].

The core idea of HOG is that the shape of the detected local object can be described by the light intensity gradient or the distribution of the edge direction. By dividing the entire image into small connected areas (called cells). Each cell generates a directional gradient histogram or the edge direction of the pixel in the cell, and the descriptor is represented by combining the histogram. To improve the accuracy, the local histogram can be compared and standardized by calculating the light intensity of a larger area (called block) in the image as a measure, and then using this value (measure) to normalize all cells in the block. This normalization process completes better illumination/shadow invariance. Compared with other descriptors, the descriptors obtained by HOG maintain the invariance of geometric and optical transformations (unless the object orientation changes).

#### *2.5. Gabor Feature*

Gabor feature [60] is a feature that can be used to describe the texture information of an image. The frequency and direction of the Gabor filter are similar to the human visual system, and it is particularly suitable for texture representation and discrimination. The Gabor feature mainly relies on the Gabor kernel to window the signal in the frequency domain, so as to describe the local frequency information of the signal.

In terms of feature extraction, Gabor wavelet transform is compared with other methods: on the one hand, it processes less data and can meet the real-time requirements of the system; on the other hand, wavelet transform is insensitive to changes in illumination and can tolerate a certain degree of when image rotation and deformation are used for recognition based on Euclidean distance, the feature pattern and the feature to be measured do not need to correspond strictly, so the robustness of the system can be improved.

#### *2.6. D-AHP Theory*

Analytic Hierarchy Process (AHP) [61] is a systematic and hierarchical analysis method that combines qualitative and quantitative analysis. The characteristic of this method is that on the basis of in-depth research on the nature, influencing factors and internal relations of complex decision-making problems, it uses less quantitative information to mathematicize the thinking process of decision-making, thereby providing multi-objective, multi-criteria or complex decision-making problems with no structural characteristics provide simple decision-making methods.

The D-AHP method extends the traditional AHP method in theory. In the D-AHP method [62], the derived results about the ranking and priority weights of alternatives are impacted by the credibility of providing information. A parameter *λ* is used to express the credibility of information, and its value is associated with the cognitive ability of experts. If the comparison information used in the decision-making process is provided by an authoritative expert, *λ* will take a smaller value. If the comparison information comes from an expert whose judgment is with low belief, *λ* takes a higher value.

#### **3. The Proposed Method**

In general sensor recognition, the training set is inputting to train the sensor by extracting feature, and then the test set is inputting to test and get the recognition result. To improve the accuracy of multi-sensor fusion recognition, this paper proposes a fusion recognition method based on neutrosophic set. The proposed method in this paper is mainly divided into two parts: complementarity analysis and data fusion. The main steps of the method proposed in this paper are shown in Figure 1.

**Figure 1.** Target fusion recognition based on sensor complementarity and neutrosophic set.

The essence of complementarity is to calculate the weight of the recognition ability of the base sensor in different categories. Based on this, the data set is divided into training set, validation set, and test set. First, the base sensor preference matrix is obtained from the recognition matrix of the

trained base sensor on the verification set, and then the sensor complementarity vector is calculated in each category.

Data fusion aims at different target types, based on the sensor complementarity vector, the recognition results of different sensors are generated by the neutrosophic set. Then the fusion of the neutrosophic set can be obtained in different categories, and finally the neutrosophic set is converted into crisp number, and the maximum value is taken as the recognition result.

#### *3.1. Complementarity*

The main steps of the multi-sensor complementarity analysis are proposed as follows:

#### 3.1.1. Sensor Recognition Matrix

Suppose there are *n* types of sensors *X*1, *X*2, *X*3, · · · , *Xn*, and *m* types of target types *Y*1,*Y*2,*Y*3, · · · ,*Ym*. For a certain data set *i*, the recognition results of all its samples can be represented by the following recognition matrix *Ri* .

$$R\_{l} = \begin{bmatrix} r\_{11}^{i} & r\_{12}^{i} & \cdots & r\_{1m}^{i} \\ r\_{21}^{i} & r\_{22}^{i} & \cdots & r\_{2m}^{i} \\ \vdots & \vdots & \ddots & \vdots \\ r\_{n1}^{i} & r\_{n2}^{i} & \cdots & r\_{nm}^{i} \end{bmatrix} \tag{10}$$

#### 3.1.2. Sensor Preference Matrix

To obtain the preference relationship matrix between sensors, the preference between sensors needs to be defined. For two sensors, if the recognition performance of the sensor *X*1 on the target *Yj* is better than the recognition performance of the sensor *X*2, then for the target *Yj* , the sensor *X*1 is better than the sensor *X*2. The recognition results of a certain sensor on the samples in the data set i can be organized in the form of a recognition matrix, but the rows and columns become the recognized category of the sample and the true category of the sample. Furthermore, for the target of category *Yj* , according to the recognition matrix, we can get Table 1:

**Table 1.** *R i j* : The recognition situation of a certain sensor to the target of category *Yj* .

Record the above matrix as *R i j* in Table 1, where *r i jj* is the number of correct recognition of the category *Yj* by the sensor, ∑ *k*6=*j r i kj* is the number of samples that the sensor misrecognizes non-targets. ∑ *k*6=*j r i jk* is the number of misrecognized category *Yj* samples into other categories. ∑ *l*6=*j* ∑ *k*6=*j r i kl* is the number of samples other than the above three cases number. If the optimal performance of sensor recognition is expressed as a matrix:

$$I\_{\vec{j}} = \begin{bmatrix} \sum\_{l=1}^{h} r\_{jl}^{i} & \mathbf{0} \\ \mathbf{0} & \sum\_{k \neq j} \sum\_{l=1}^{h} r\_{kl}^{i} \end{bmatrix} \tag{11}$$

Which means the category is fully recognized correctly. Then the recognition performance of this sensor to the category *Yj* is defined as 1 *R i j*−*Ij* . Therefore, if there are two sensors *Xk* and *Xl* , the preference value of the recognition accuracy rate of *Xk* versus *Xl* in the category (representing the priority of the recognition ability of *Xk* over *Xl* in the category of *Yj* ) is defined as:

$$p^{j}\_{kl} = \frac{\frac{1}{||\mathbb{R}^{k}\_{j} - I\_{j}||}}{\frac{1}{||\mathbb{R}^{k}\_{j} - I\_{j}||} + \frac{1}{||\mathbb{R}^{l}\_{j} - I\_{j}||}} = \frac{\left||\mathbb{R}^{l}\_{j} - I\_{j}||}{||\mathbb{R}^{k}\_{j} - I\_{j}|| + ||\mathbb{R}^{l}\_{j} - I\_{j}||}\tag{12}$$

In the same way, the recognition accuracy preference value of sensor *Xl* vs *Xk* on category *Yj* is defined as follows:

*p j lk* = *R k j* − *Ij Rk j* − *Ij* + *Rl j* − *Ij* (13)

It is easy to get from the above two formulas that *p j kl*+*p j lk* = 1, which is the sum of the two preference values is 1. If *l* = *k*, then *p j kl* = 0.5. For the case where multiple sensors recognize at the same time, the preference relationship matrix *P j* on the category *Yj* can be obtained. For each target category in the data set *i*, a corresponding preference relationship matrix can be obtained.

#### 3.1.3. Sensor Complementarity Vector

Next, using the method in D-AHP theory [62], the complementarity vector can be calculated by the preference relationship matrix.

According to the classifier preference relation matrix *Pj* of category *Yj* .

$$P^j = \begin{bmatrix} P^j{}\_{11} & P^j{}\_{12} & \cdots & P^j{}\_{1n} \\ P^j{}\_{21} & P^j{}\_{22} & \cdots & P^j{}\_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ P^j{}\_{n1} & P^j{}\_{n2} & \cdots & P^j{}\_{nn} \end{bmatrix} \tag{14}$$

It is calculated that for each sensor complementarity vector *C j* of category *Yj* , *C j* is a 1 × *n* dimensional vector. The flowchart of the *C j* calculation is presented as Figure 2, the calculation steps are as follows:

**Figure 2.** Complementary vector generation process [62].

#### *3.2. Data Fusion*

For a picture with unknown target type, the probability matrix is formed via the recognition result vectors of each sensor.

$$Q = \begin{bmatrix} q\_{11} & \cdots & q\_{1m} \\ \vdots & \ddots & \vdots \\ q\_{n1} & \cdots & q\_{nm} \end{bmatrix} \tag{15}$$

where *qij* is the probability that the sensor *Xi* considers the unknown target as the target type *Yj* .

At this time, the complementary vector is normalized to obtain the weight coefficient:

$$\mathbf{H}^{\mathbf{j}} = \mathbf{C}^{\mathbf{j}} / (\mathbf{C}^{\mathbf{j}}\_{\text{min}} + \mathbf{C}^{\mathbf{j}}\_{\text{max}}) \tag{16}$$

If the target type is *Yj* , the complementarity vector *Hj* and *Q* can be combined to obtain a neutrosophic set *α* = h*µ*, *π*, *ν*i. Since there are *n* sensors, *n* groups neutrosophic sets can be obtained according to *Q* and *Yj* as follows:

$$\mathfrak{a}\_{i} = \left[ \mathfrak{\mu} = q\_{\vert i \rangle} \times H\_{\vert \nu} \,\,\pi = (1 - q\_{\vert i \rangle}) \times H\_{\vert \nu} \,\,\nu = 1 - H\_{\vert} \right] \tag{17}$$

Combining *n* groups of neutrosophic sets, a group of fused neutrosophic set can be obtianed, and the target recognition neutrosophic set is calculated under the imaginary target type *Yj* . Since there are *m* types of target, we can finally use the SNWA operator [63] to get *m* neutrosophic sets:

$$\mathfrak{a}\_{\Pi} = \mathbb{W}\_1 \times \mathfrak{a}\_1 + \mathbb{W}\_2 \times \mathfrak{a}\_2 + \dots + \mathbb{W}\_{\mathbb{N}} \times \mathfrak{a}\_{\mathbb{N}} = \left[ 1 - \prod\_{i=1}^{n} (1 - \mu\_i)^{\mathbb{W}\_i}, \prod\_{i=1}^{n} (\pi\_i)^{\mathbb{W}\_i}, \prod\_{i=1}^{n} (\upsilon\_i)^{\mathbb{W}\_i} \right] \tag{18}$$

Finally, convert these *m* SNNs into crisp numbers, and take the maximum value as the recognition result.

$$\text{RIS} = \text{Max}(\text{S}) = \text{Max}[\text{S}\_1 = \mu\_{T\_1} + (\pi\_{T\_1}) \times (\frac{\mu\_{T\_1}}{\mu\_{T\_1} + \nu\_{T\_1}}), \dots, \text{S}\_{\text{ll}} = \mu\_{T\_m} + (\pi\_{T\_m}) \times (\frac{\mu\_{T\_m}}{\mu\_{T\_m} + \nu\_{T\_m}})] \tag{19}$$

#### **4. Simulation**

#### *4.1. Data Set*

The data source type of the experiment consists of two types: visible light image and infrared light image. There are four target types: sailboat (1), cargo ship (2), speed boat (3), and fishing boat (4). The structure of the experimental data is shown in Tables 2 and 3. The data consists of a train set, a verification set, and a test set. The verification set and the test set are the same pictures. Since the information of visible light and infrared sensors needs to be fused in the verification set and test set, two visible and infrared images taken at the same location are required for the same target, as shown in Figures 3–6. Due to the lack of data, K = 8 cross-validation is used for verificating and testing. This means that the validation set (test set) is divided into eight groups, and when a certain one group is tested, the remaining seven groups are used as the validation set. The image features use Gabor and HOG, and the classifier uses AdaBoost.

**Table 2.** Target recognition image of infrared data.

**Table 3.** Target recognition image of visible light data.

**Figure 3.** Visible light image and infrared light image for the same sailboat.

**Figure 4.** Visible light image and infrared light image for the same cargo ship.

**Figure 5.** Visible light image and infrared light image for the same speedboat.

**Figure 6.** Visible light image and infrared light image for the same fishing boat.

#### *4.2. Sensor*

Two data sources (visible light, infrared), two image features (HOG, Gabor), and the classification algorithm AdaBoost can be combined separately to obtain 4 classifiers, as shown in Table 4. Since the background of this research is target recognition, these classifiers are regarded as different sensors, so as to recognize the target and generate their respective recognition results. The specific process of recognition is shown in Figure 7.

**Figure 7.** The work process of sensor.

**Table 4.** Base sensor recognition confusion matrix.

#### *4.3. Base Sensor Recognition Confusion Matrix*

The verification set is input into the trained base sensor, and the recognition confusion matrix of the base sensor can be obtained according to the recognition result of the sensor. The Tables 5–8 are the identification confusion matrixs of the base sensors on the verification set.

**Table 5.** Recognition confusion matrix of sensor 1.

**Table 6.** Recognition confusion matrix of sensor 2.

**Table 7.** Recognition confusion matrix of sensor 3.

**Table 8.** Recognition confusion matrix of sensor 4.

#### *4.4. Preference Matrix*

Tables 9–12 show the preference comparison matrix of the four sensors for the four types of target.

**Table 9.** Preference matrix *P*(1, 1) of Sailboat.

**Table 10.** Preference matrix *P*(1, 2) of Cargo ship.

**Table 11.** Preference matrix *P*(1, 3) of Speedboat.

#### *4.5. Complementarity Vector*

According to the previous research, the complementarity vector can be obtained from the preference matrix, as shown in Table 13, which reflects the complementarity between the 4 sensors if the target to be identified is the first type of target sailboat. These information also reflect the importance of each sensor, so the complementary vector is used as the weight to generate the neutrosophic set when fusing the neutrosophic set.

**Table 13.** Complementarity vector: *C*.

#### *4.6. Data Fusion*

When identifying an unknown target, the four sensors can obtain the probability of the category through the trained classifier, as shown in Table 14.

**Table 14.** Probability of the 4 sensors to recognize the unknown target.

According to these probabilities, by Equation (16), the complementarity vectors are converted into weight vectors, as shown in Table 15.

**Table 15.** Weight vector: *H*.

Use Equation (17) to combine *P* and *W* and get four neutrosophic sets in one category. The current recognition framework has four categories, so four groups of neutrosophic set are obtained, as follows:

$$\begin{aligned} \alpha\_{1} &= \begin{bmatrix} \alpha\_{11} & [0.209, 0.607, 0.184] \\ \alpha\_{12} & [0.032, 0.205, 0.761] \\ \alpha\_{13} & [0.014, 0.168, 0.817] \\ \alpha\_{14} & [0.048, 0.439, 0.513] \end{bmatrix} \\ \alpha\_{2} &= \begin{bmatrix} \alpha\_{21} & [0.102, 0.731, 0.167] \\ \alpha\_{22} & [0.039, 0.127, 0.833] \\ \alpha\_{23} & [0.029, 0.242, 0.729] \\ \alpha\_{24} & [0.057, 0.297, 0.645] \end{bmatrix} \\ \alpha\_{3} &= \begin{bmatrix} \alpha\_{31} & [0.154, 0.699, 0.147] \\ \alpha\_{32} & [0.046, 0.101, 0.853] \\ \alpha\_{33} & [0.138, 0.255, 0.607] \\ \alpha\_{34} & [0.089, 0.222, 0.689] \end{bmatrix} \\ \alpha\_{4} &= \begin{bmatrix} \alpha\_{41} & [0.348, 0.442, 0.210] \\ \alpha\_{42} & [0.085, 0.189, 0.726] \\ \alpha\_{43} & [0.365, 0.425, 0.210] \\ \alpha\_{44} & [0.095, 0.116, 0.789] \end{bmatrix} \end{aligned}$$

Combine these four groups of neutrosophic sets according to Equation (18) to obtain 4 neutrosophic sets:

> *α*1 = 0.25 × *α*11 + 0.25 × *α*12 + 0.25 × *α*13 + 0.25 × *α*14 = [0.079, 0.492, 0.310] *α*2 = 0.25 × *α*21 + 0.25 × *α*22 + 0.25 × *α*23 + 0.25 × *α*24 = [0.058, 0.506, 0.286] *α*3 = 0.25 × *α*31 + 0.25 × *α*32 + 0.25 × *α*33 + 0.25 × *α*34 = [0.108, 0.479, 0.251] *α*4 = 0.25 × *α*41 + 0.25 × *α*42 + 0.25 × *α*43 + 0.25 × *α*44 = [0.234, 0.399, 0.253]

Furthermore, the four neutrosophic sets are transformed into crisp numbers by Equation (19).

$$\text{RS} = 4 = \text{Max}(\text{S}) = \text{Max}[\text{S}\_1 = 0.181, \text{S}\_2 = 0.142, \text{S}\_3 = 0.252, \text{S}\_4 = 0.427]$$

Therefore, the recognition result of the unknown target is fishing boat.

#### *4.7. Recognition Result*

After all the test sets are finally tested, the results of the method proposed in this paper are shown in Table 16. And the results of two other fusion methods such as simple fusion (Table 17) and D-S fusion (Table 18) are given to compare with the proposed method.

**Table 16.** The proposed method recognition result.

**Table 17.** Simple fusion result.

The recognition results of the 4 base sensors on the test set are shown in Tables 19–22:

**Table 19.** Sensor 1 recognition result.

**Table 20.** Sensor 2 recognition result.

**Table 21.** Sensor 3 recognition result.

**Table 22.** Sensor 4 recognition result.

The multi-category evaluation criteria is used in the previous article to evaluate the classification results, shown in Table 23. It can be seen that the method proposed in this paper has improved in the recall rate, accuracy rate, F1 score and other indicators compared with these four sensors. After multi-sensor fusion recognition, the accuracy rate of a single sensor is increased by 3.25% at the lowest and 35.77% at the highest, and the average performance of a single sensor is improved by 21.13%. At the same time, compared with the other two fusion methods, the method proposed in this parper also performs better.It can be seen that the accuracy of fusion recognition can be significantly improved.

**Table 23.** Recognition result analysis.

#### **5. Results**

Aiming at the problem of multi-sensor target recognition, this paper proposes a new method based on the complementary characteristics of sensors in the fusion of neutrosophic set, which improves the accuracy of target type recognition. Using the identification of the sea surface vessel type as the verification scenario, the category-oriented sensor complementarity vector is constructed through feature extraction, sensor training of the target's infrared and visible image training data. The multi-sensor neutrosophic set model is performed on the target to be recognized to realize the multi-sensor. Compared with other methods, the method proposed in this paper performs better in recognition accuracy, Compared with other fuzzy mathematics theories, the neutrosophic set theory is more helpful for us to deal with the complementary information between sensors. At the same time, the three sets of functions included in the neutrosophic set allow us to flexibly adjust the weight and other parameters, and the calculation of the neutrosophic set is simple, it takes less time to run the program. Further research will mostly concentrate on the the proposed method can be used to more complicated study to further demonstrate its efficiency.

**Author Contributions:** Y.G. wrote this paper, W.J., X.D. and M.W. reviewed and improved this article, Y.G. and Z.M. discussed and analyzed the numerical results. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work is partially supported by Equipment Pre-Research Fund (Program No.<PHONE_NUMBER>9), the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Program No. CX2020151).

**Conflicts of Interest:** The authors declare that they have no competing interest.

### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **On Single-Valued Neutrosophic Ideals in Šostak Sense**

### **Yaser Saber 1,2, Fahad Alsharari 1 and Florentin Smarandache 3,\***

Received: 15 December 2019; Accepted: 6 January 2020; Published: 25 January 2020

**Abstract:** Neutrosophy is a recent section of philosophy. It was initiated in 1980 by Smarandache. It was presented as the study of origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. In this paper, we introduce the notion of single-valued neutrosophic ideals sets in Šostak's sense, which is considered as a generalization of fuzzy ideals in Šostak's sense and intuitionistic fuzzy ideals. The concept of single-valued neutrosophic ideal open local function is also introduced for a single-valued neutrosophic topological space. The basic structure, especially a basis for such generated single-valued neutrosophic topologies and several relations between different single-valued neutrosophic ideals and single-valued neutrosophic topologies, are also studied here. Finally, for the purpose of symmetry, we also define the so-called single-valued neutrosophic relations.

**Keywords:** single-valued neutrosophic closure; single-valued neutrosophic ideal; single-valued neutrosophic ideal open local function; single-valued neutrosophic ideal closure; single-valued neutrosophic ideal interior; single-valued neutrosophic ideal open compatible

#### **1. Introduction**

The notion of fuzzy sets, employed as an ordinary set generalization, was introduced in 1965 by Zadeh [1]. Later on, using fuzzy sets through the fuzzy topology concept was initially introduced in 1968 by Chang [2]. Afterwards, many properties in fuzzy topological spaces have been explored by various researchers [3–13]

Paradoxically, it is to be emphasized that being fuzzy or what is termed as fuzzy topology in fuzzy openness concept is not highlighted and well-studied. Meanwhile, Samanta et al. [14,15] introduced what is called the graduation of openness of fuzzy sets. Later on, Ramadan [16] introduced smooth continuity, a number of their properties, and smooth topology. Demirci [17] investigated properties and systems of smooth *Q*-neighborhood and smooth neighborhood alike. It is worth mentioning that Chattopadhyay and Samanta [18] have initiated smooth connectedness and smooth compactness. On the other hand, Peters [19] tackled the notion of primary fuzzy smooth characteristics and structures together with smooth topology in Lowen sense. He [20] further evidenced that smooth topologies collection constitutes a complete lattice. Furthermore, Onassanya and Hošková-Mayerová [21] inspected certain features of subsets of *α*-level as an integral part of a fuzzy subset topology. Likewise, more specialists in the field like Çoker and Demirci [22], in addition to Samanta and Mondal [23,24], have provided definitions to the concept of graduation intuitionistic openness of fuzzy sets based on Šostak's sense [25] according to Atanassov's [26] intuitionistic fuzzy sets. Essentially, they focused on intuitionistic gradation of openness in light of Chang. On the other hand, Lim et al. [27] examined

Lowen's framework smooth intuitionistic topological spaces. In recent times, Kim et al. [28] considered systems of neighborhood and continuities within smooth intuitionistic topological spaces. Moreover, Choi et al. [29] scrutinized smooth interval-valued topology through graduation of the concept of interval-valued openness of fuzzy sets, as suggested by Gorzalczany [30] and Zadeh [31], respectively. Ying [32] put forward a topology notion termed as fuzzifying topology, taking into consideration the extent of ordinary subset of a set openness. General properties in ordinary smooth topological spaces were elaborated in 2012 by Lim et al. [33]. In addition, they [34–36] inspected compactness, interiors, and closures within normal smooth topological spaces. In 2014, Saber et al. [37] shaped the notion of fuzzy ideal and r-fuzzy open local function in fuzzy topological spaces in view of the definition of Šostak. In addition, they [38,39] inspected intuitionistic fuzzy ideals, fuzzy ideals and fuzzy open local function in fuzzy topological spaces in view of the definition of Chang.

Smarandache [40] determined the notion of a neutrosophic set as intuitionistic fuzzy set generalization. Meanwhile, Salama et al. [41,42] familiarized the concepts of neutrosophic crisp set and neutrosophic crisp relation neutrosophic set theory. Correspondingly, Hur et al. [43,44] initiated classifications NSet(H) and NCSet including neutrosophic crisp and neutrosophic sets, where they examined them in a universe topological position. Furthermore, Salama and Alblowi [45] presented neutrosophic topology as they claimed a number of its characteristics. Salama et al. [46] defined a neutrosophic crisp topology and studied some of its properties. Others, such as Wang et al. [47], defined the single-valued neutrosophic set concept. Currently, Kim et al. [48] has come to grips with a neutrosophic partition single-value, neutrosophic equivalence relation single-value, and neutrosophic relation single-value.

Preliminaries of single-value neutrosophic sets and single-valued neutrosophic topology are reviewed in Section 2. Section 3 is devoted to the concepts of single-valued neutrosophic closure space and single-valued neutrosophic ideal. Some of their characteristic properties are considered. Finally, the concepts of single-valued neutrosophic ideal open local function has been introduced and studied. Several preservation properties and some characterizations concerning single-valued neutrosophic ideal open compatible have been obtained.

#### **2. Preliminaries**

In this section, we attempt to cover enough of the fundamental concepts and definitions.

**Definition 1** ([49])**.** *A neutrosophic set* H *(***NS***, for short) on a nonempty set* S *is defined as*

$$\mathcal{H} = \langle \kappa\_\prime \, T\_{\mathcal{H}\prime} I\_{\mathcal{H}\prime} F\_{\mathcal{H}} : \kappa \in \mathcal{S} \rangle\_{\prime\prime}$$

*where*

$$T\_{\mathcal{H}} : \mathcal{S} \to \bigsqcup^{-} \mathbf{0}, \mathbf{1}^{+} \bigsqcup \bigsqcup \quad I\_{\mathcal{H}} : \mathcal{S} \to \bigsqcup^{-} \mathbf{0}, \mathbf{1}^{+} \bigsqcup \quad F\_{\mathcal{H}} : \mathcal{S} \to \bigsqcup^{-} \mathbf{0}, \mathbf{1}^{+} \bigsqcup$$

*and*

$$\mathbf{r}^-\mathbf{0} \le T\_{\mathcal{H}}(\kappa) + I\_{\mathcal{H}}(\kappa) + F\_{\mathcal{H}}(\kappa) \le \mathbf{3}^+\text{.}$$

*representing the degree of membership (namely, T*H(*κ*)*), the degree of indeterminacy (namely, I*H(*κ*)*), and the degree of nonmembership (namely, F*H(*κ*)*); for all κ* ∈ S *to the set* H*.*

**Definition 2** ([49])**.** *Let* H *and* R *be fuzzy neutrosophic sets in* S*. Then,* H *is a subset of* R *if, for each κ* ∈ S*,*

$$\inf T\_{\mathcal{H}}(\mathbf{x}) \le \inf T\_{\mathcal{R}}(\mathbf{x}), \quad \inf I\_{\mathcal{H}}(\mathbf{x}) \ge \inf I\_{\mathcal{R}}(\mathbf{x}), \quad \inf F\_{\mathcal{H}}(\mathbf{x}) \ge \inf F\_{\mathcal{R}}(\mathbf{x})$$

*and*

$$
\sup T\_{\mathcal{H}}(\kappa) \le \sup T\_{\mathcal{R}}(\kappa), \quad \sup I\_{\mathcal{H}}(\kappa) \ge \sup I\_{\mathcal{R}}(\kappa), \quad \sup F\_{\mathcal{H}}(\kappa) \ge \sup F\_{\mathcal{R}}(\kappa).
$$

**Definition 3** ([47])**.** *Let* H *be a space of points (objects) with a generic element in* S *denoted by κ. Then,* H *is called a single-valued neutrosophic set (in short,* **SVNS**) *in* S *if* H *has the form* H = h*T*H, *I*H, *F*Hi*, where T*H, *I*H, *F*H : S → [0, 1]*.*

*In this case, T*H, *I*H, *F*H *are called truth-membership function, indeterminacy-membership function, and falsity-membership function, respectively, and we will denote the set of all* **SVNS**′ **s** *in* S *as* **SVNS**(S)*.*

*Moreover, we will refer to the Null (empty)* **SVNS** *(or the absolute (universe)* **SVNS***) in* S *as* 0*N (or* 1*N) and define by* 0*N* = (0, 1, 1) *(or* 1*N* = (1, 0, 0)*) for each κ* ∈ S*.*

**Definition 4** ([47])**.** *Let* H = h*T*H, *I*H, *F*Hi *be an* **SVNS** *on* S*. The complement of the set* H *(*H*c* , *f or short) and is defined as follows: for every κ* ∈ S*,*

$$T\_{\mathcal{H}^\mathbb{C}}(\kappa) = F\_{\mathcal{H}}(\kappa), \quad I\_{\mathcal{H}^\mathbb{C}}(\kappa) = 1 - I\_{\mathcal{H}}(\kappa), \quad F\_{\mathcal{H}^\mathbb{C}}(\kappa) = T\_{\mathcal{H}}(\kappa).$$

**Definition 5** ([50])**.** *Suppose that* H ∈ **SVNS**(S)*. Then,*

*(i)* H *is said to be contained in* R*, denoted by* H ⊆ R*, if, for every κ* ∈ S*,*

$$T\_{\mathcal{H}}(\kappa) \le T\_{\mathcal{R}}(\kappa), \quad I\_{\mathcal{H}}(\kappa) \ge I\_{\mathcal{R}}(\kappa), \quad F\_{\mathcal{H}}(\kappa) \ge F\_{\mathcal{R}}(\kappa);$$

*(ii)* H *is said to be equal to* R*, denoted by* H = H*, if* R ⊆ R *and* H ⊇ R*.*

**Definition 6** ([51])**.** *Suppose that* H, R ∈ **SVNS**(S)*. Then,*

*(i) the union of* H *and* R *(*H ∪ R, *f or short) is an* **SVNS** *in* S *defined as*

$$\mathcal{H} \cup \mathcal{R} = (T\_{\mathcal{H}} \cup T\_{\mathcal{R}\prime} I\_{\mathcal{H}} \cap I\_{\mathcal{R}\prime} F\_{\mathcal{H}} \cap F\_{\mathcal{R}})\_{\prime}$$

*where* (*T*H ∪ *T*R)(*κ*) = *T*H(*κ*) ∪ *T*R(*κ*) *and* (*F*H ∩ *F*R)(*κ*) = *F*H(*κ*) ∩ *F*R(*κ*)*, for each κ* ∈ S*; (ii) the intersection of* H *and* R*,* (H ∩ R, *f or short), is an* **SVNS** *in* S *defined as*

$$
\mathcal{H} \cap \mathcal{R} = (T\_{\mathcal{H}} \cap T\_{\mathcal{R}\prime} I\_{\mathcal{H}} \cup I\_{\mathcal{R}\prime} F\_{\mathcal{H}} \cup F\_{\mathcal{R}}) .
$$

**Definition 7** ([45])**.** *Let* H ∈ **SVNS**(S)*. Then,*

*(i) the union of* {H*i*}*i*∈*J (* S *i*∈*J* H*i* , *f or short*) *is an* **SVNS** *in* S *defined as follows: for every κ* ∈ S*,*

$$(\bigcup\_{i \in I} \mathcal{H}\_i)(\kappa) = (\bigcup\_{i \in I} T\_{\mathcal{H}\_i}(\kappa), \bigcap\_{i \in I} I\_{\mathcal{H}\_i}(\kappa), \bigcap\_{i \in I} F\_{\mathcal{H}\_i}(\kappa);$$

*(ii) the intersection of* {H*i*}*i*∈*J (* T *i*∈*J* H*i* , *f or short) is an* **SVNS** *in* S *defined as follows: for every κ* ∈ S*,*

$$(\bigcap\_{i \in J} \mathcal{H}\_i)(\kappa) = (\bigcap\_{i \in J} T\_{\mathcal{H}\_i}(\kappa), \bigcup\_{i \in J} I\_{\mathcal{H}\_i}(\kappa), \bigcup\_{i \in J} F\_{\mathcal{H}\_i}(\kappa).$$

**Definition 8** ([52])**.** *A single-valued neutrosophic topology on* S *is a map* (*τ T* , *τ I* , *τ F* ) : *I* S → *I satisfying the following three conditions:*

*(SVNT1) τ T* (0) = *τ T* (1) = 1 *and τ I* (0) = *τ I* (1) = *τ F* (0) = *τ F* (1) = 0*, (SVNT2) τ T* (H ∩ R) ≥ *τ T* (H) ∩ *τ T* (R)*, τ I* (H ∩ R) ≤ *τ I* (H) ∪ *τ I* (R)*, τ F* (H ∩ R) ≤ *τ F* (H) ∪ *τ F* (R)*, for any* H, R ∈ *I* S *,*

$$\begin{array}{lll} (\mathsf{SVNT3}) & \tau^{T}(\cup\_{i\in j}\mathcal{H}\_{i}) \geq \cap\_{i\in j} \tau^{T}(\mathcal{H}\_{i}), & & \tau^{I}(\cup\_{i\in j}\mathcal{H}\_{i}) \leq \cup\_{i\in j} \tau^{I}(\mathcal{H}\_{i}),\\ & & \tau^{F}(\cup\_{i\in j}\mathcal{H}\_{i}) \leq \cup\_{i\in j} \tau^{F}(\mathcal{H}\_{i}), \text{for any } \{\mathcal{H}\_{i}\}\_{i\in I} \in I^{\mathcal{S}}. \end{array}$$

The pair (*X*, *τ T* , *τ I* , *τ F* ) is called single-valued neutrosophic topological spaces (**SVNTS**, *f or short*). We will occasionally write *τ T IF* for (*τ T* , *τ I* , *τ F* ) and it will cause no ambiguity.

#### **3. Single-Valued Neutrosophic Closure Space and Single-Valued Neutrosophic Ideal in Šostak Sense**

This section deals with the definition of single-valued neutrosophic closure space. The researchers examine the connection between single-valued neutrosophic closure space and **SVNTS** based in Šostak sense. Moreover, the researchers focused on the single-valued neutrosophic ideal notion where they obtained fundamental properties. Based on Šostak's sense, where a single-valued neutrosophic ideal takes the form (S, L *T* , L *I* , L *F* ) and the mappings L *T* , L *I* , L *F* : *I* S → *I*, where (L *T* , L *I* , L *F* ) are the degree of openness, the degree of indeterminacy, and the degree of non-openness, respectively.

In this paper, S is used to refer to nonempty sets, whereas *I* is used to refer to closed interval [0, 1] and *Io* is used to refer to the interval (0, 1]. Concepts and notations that are not described in this paper are standard, instead, S is usually used.

**Definition 9.** *A mapping* C : *I* S × *I*0 → *I* S *is called a single-valued neutrosophic closure operator on* S *if, for every* H, R ∈ *I* S *and r*,*s* ∈ *I*0*, the following axioms are satisfied:*

*(*C1*)* C((0.1.1),*s*) = (0.1.1)*, (*C2*)* H ≤ C(H,*s*)*, (*C3*)* C(H,*s*) ∨ C(R,*s*) = C(H ∨ R,*s*)*, (*C4*)* C(H,*s*) ≤ C(H,*r*) *if s* ≤ *r, (*C5*)* C(C(H,*s*),*s*) = C(H,*s*)*.*

The pair (*X*, C) is a single-valued neutrosophic closure space (SVN CS, *f or short*).

Suppose that C1 and C2 are single-valued neutrosophic closure operators on S. Then, C1 is finer than C2, denoted by C2 ≤ C1 iff C1(H,*s*) ≤ C2(H,*s*), for every H ∈ *I* S and *s* ∈ *I*0.

**Theorem 1.** *Let* (S, *τ T IF*) *be an* **SVNTS***. Then, for any* H ∈ *I* S *and s* ∈ *I*0*, we define an operator* C*τ T IF* : *I* S × *I*0 → *I* S *as follows:*

$$\mathbb{C}\_{\mathsf{T}^{\mathsf{T}}}(\mathsf{H},\mathsf{s}) = \bigwedge \{ \mathcal{R} \in I^{\mathsf{X}} : \mathcal{H} \le \mathcal{R}, \quad \mathsf{\tau}^{\mathsf{T}}(\underline{1} - \mathcal{R}) \ge \mathsf{s}, \quad \mathsf{\tau}^{\mathsf{I}}(\underline{1} - \mathcal{R}) \le 1 - \mathsf{s}, \quad \mathsf{\tau}^{\mathsf{F}}(\underline{1} - \mathcal{R}) \le 1 - \mathsf{s} \}.$$

*Then,* (S, C*τ T IF* ) *is an* SVN CS*.*

**Proof.** Suppose that (S, *τ T IF*) is an **SVNTS**. Then, C1, (C2) and (C4) follows directly from the definition of C*τ T IF* .

(C3) Since R, H ≤ H ∪ R, C*τ T IF* (R,*s*) ≤ C*τ T IF* (H ∪ R,*s*) and C*τ T IF* (H,*s*) ≤ C*τ T IF* (H ∪ R,*s*), therefore,

$$
\mathbb{C}\_{\mathsf{T}^{TIF}}(\mathcal{H}\_{\mathsf{r}}\mathbf{s}) \cup \mathbb{C}\_{\mathsf{T}^{TIF}}(\mathcal{R}\_{\mathsf{r}}\mathbf{s}) \le \mathbb{C}\_{\mathsf{T}^{TIF}}(\mathcal{H}\cup\mathcal{R}\_{\mathsf{r}}\mathbf{s}).
$$

Let (*X*, *τ T IF*) be an **SVNTS**. From (C2), we have

$$\begin{aligned} \mathcal{H} \le \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{H}, \mathsf{s}), \quad \mathsf{r}^{T}(\underline{\mathsf{1}} - \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{H}, \mathsf{s})) \ge \mathsf{s}, \; \mathsf{r}^{I}(\underline{\mathsf{1}} - \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{H}, \mathsf{s})) \le \mathsf{1} - \mathsf{s}, \\\ \text{and } \mathsf{r}^{\mathrm{F}}(\underline{\mathsf{1}} - \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{H}, \mathsf{s})) \le \mathsf{1} - \mathsf{s}, \end{aligned}$$

*Symmetry* **2020**, *12*, 193

$$\begin{aligned} \mathcal{R} \le \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{R}, s), \quad \tau^{\mathrm{T}}(\underline{1} - \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{R}, s)) \ge s, \ \tau^{\mathrm{I}}(\underline{1} - \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{R}, s)) \le 1 - s, \\\ \text{and } \tau^{\mathrm{F}}(\underline{1} - \mathbb{C}\_{\tau^{\mathrm{TIF}}}(\mathcal{R}, s)) \le 1 - s. \end{aligned}$$

It implies that H ∪ R ≤ C*τ T IF* (H,*s*) ∪ C*τ T IF* (R,*s*),

$$\begin{aligned} \tau^T(\underline{\mathfrak{T}} - (\mathbb{C}\_{\tau^{\scriptscriptstyle{T}IF}}(\mathcal{H}, s) \cup \mathbb{C}\_{\tau^{\scriptscriptstyle{T}IF}}(\mathcal{R}, s))) &= \tau^T((\underline{\mathfrak{1}} - \mathbb{C}\_{\tau^{\scriptscriptstyle{T}IF}}(\mathcal{H}, s)) \cap (\underline{\mathfrak{1}} - \mathbb{C}\_{\tau^{\scriptscriptstyle{T}IF}}(\mathcal{R}, s))) \\ &\ge \tau^T(\underline{\mathfrak{1}} - \mathbb{C}\_{\tau^{\scriptscriptstyle{T}IF}}(\mathcal{H}, s)) \cap \tau^T(\underline{\mathfrak{1}} - \mathbb{C}\_{\tau^{\scriptscriptstyle{T}IF}}(\mathcal{R}, s)) \ge s\_r \end{aligned}$$

$$\begin{split} \tau^{I}(\underline{\mathfrak{l}} - (\mathbb{C}\_{\tau^{\rm{TF}}}(\mathcal{H}\_{\prime}s) \cup \mathbb{C}\_{\tau^{\rm{TF}}}(\mathcal{R}\_{\prime}s))) &= \tau^{I}((\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\rm{TF}}}(\mathcal{H}\_{\prime}s)) \cap (\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\rm{TF}}}(\mathcal{R}\_{\prime}s))) \\ &\leq \tau^{I}((\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\rm{TF}}}(\mathcal{H}\_{\prime}s)) \cup \tau^{I}(\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\rm{TF}}}(\mathcal{R}\_{\prime}s)) \leq 1 - s\_{\star} \end{split}$$

$$\begin{split} \tau^{F}(\underline{\mathfrak{l}} - (\mathbb{C}\_{\tau^{\mathrm{TF}}}(\mathcal{H}, \mathsf{s}) \cup \mathbb{C}\_{\tau^{\mathrm{TF}}}(\mathcal{R}, \mathsf{s}))) &= \tau^{F}((\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\mathrm{TF}}}(\mathcal{H}, \mathsf{s})) \cap (\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\mathrm{TF}}}(\mathcal{R}, \mathsf{s}))) \\ &\leq \tau^{F}(\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\mathrm{TF}}}(\mathcal{H}, \mathsf{s})) \cup \tau^{F}(\underline{\mathfrak{l}} - \mathbb{C}\_{\tau^{\mathrm{TF}}}(\mathcal{R}, \mathsf{s})) \leq 1 - s. \end{split}$$

Hence, C*τ T IF* (H,*s*) ∪ C*τ T IF* (H ∪ R,*s*) ≥ C*τ T IF* (H ∪ R,*s*). Therefore,

$$
\mathbb{C}\_{\tau^{TIF}}(\mathcal{H}, \mathbf{s}) \cup \mathbb{C}\_{\tau^{TIF}}(\mathcal{H} \cup \mathcal{R}, \mathbf{s}) = \mathbb{C}\_{\tau^{TIF}}(\mathcal{H} \cup \mathcal{R}, \mathbf{s}) .
$$

(C5) Suppose that there exists *s* ∈ *I*0, H ∈ *I* S , and *κ* ∈ S such that

$$
\mathbb{C}\_{\tau^{TIF}}(\mathbb{C}\_{\tau^{TIF}}(\mathcal{H}\_{\mathsf{t}}s), \mathsf{s})(\mathsf{x}) > \mathbb{C}\_{\tau^{TIF}}(\mathcal{H}\_{\mathsf{t}}s)(\mathsf{x}).
$$

By the definition of C*τ T IF* , there exists D ∈ *I* S with D ≥ H, and *τ T* (1 − D) ≥ *s*, *τ I* (1 − D) ≤ 1 − *s* and *τ F* (1 − D) ≤ 1 − *s* such that

$$
\mathbb{C}\_{\tau^{TIF}}(\mathbb{C}\_{\tau^{TIF}}(\mathcal{H}\_{\prime}s),s)(\kappa) > \mathcal{D}(\kappa) \ge \mathbb{C}\_{\tau^{TIF}}(\mathcal{H}\_{\prime}s)(\kappa).
$$

Since C*τ T IF* (H,*s*) ≤ D and *τ T* (1 − D) ≥ *s*, *τ I* (1 − D) ≤ 1 − *s*, and *τ F* (1 − D) ≤ 1 − *s*, by the definition of C*τ T IF* (C*τ T IF* ), we have

$$\mathbb{C}\_{\tau^{TIF}}(\mathbb{C}\_{\tau^{TIF}}(\mathcal{H}, s), s) \le \mathcal{D}.$$

It is a contradiction. Thus, C*τ T IF* (C*τ T IF* (H,*s*),*s*) = C*τ T IF* (H,*s*). Hence, C*τ T IF* is a single-valued neutrosophic closure operator on S.

**Theorem 2.** *Let* (S, C) *be an* SVN CS *and* H ∈ S *. Define the mapping τ T IF C* : *I* S → *I on* S *by*

$$\pi\_{\mathbb{C}}^{\mathrm{T}}(\mathcal{H}) = \bigcup \{ s \in I\_{0} \mid \mathbb{C}(\overline{1} - \mathcal{H}, s) = \overline{1} - \mathcal{H} \},$$

$$\pi\_{\mathbb{C}}^{I}(\mathcal{H}) = \bigcap \{ 1 - s \in I\_{0} \mid \mathbb{C}(\overline{1} - \mathcal{H}, s) = \overline{1} - \mathcal{H} \},$$

$$\pi\_{\mathbb{C}}^{\mathbf{F}}(\mathcal{H}) = \bigcap \{ 1 - s \in I\_{0} \mid \mathbb{C}(\overline{1} - \mathcal{H}, s) = \overline{1} - \mathcal{H} \} \dots$$

*Then,*

**Proof.** (SVNT1) Let (S, C) be an SVN CS. Since C((0.1.1),*r*) = (0.1.1) and C(1, 0, 0),*r*) = (1, 0, 0) for every *s* ∈ *I*0, (SVNT1).

(SVNT2) Let (S, C) be an SVN CS. Suppose that there exists H1, H2 ∈ *I* S such that

*τ T* C (H1 ∩ H2) < *τ T* C (H1) ∩ *τ T* C (H2), *τ I* C (H1 ∩ H2) > *τ I* C (H1) ∪ *τ I* C (H2),

$$
\pi^{\mathsf{F}}\_{\mathsf{C}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) > \pi^{\mathsf{F}}\_{\mathsf{C}}(\mathcal{H}\_{1}) \cup \pi^{\mathsf{F}}\_{\mathsf{C}}(\mathcal{H}\_{2}).
$$

There exists *s* ∈ *I*0 such that

$$\pi\_{\mathsf{C}}^{\mathsf{T}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) < s < \pi\_{\mathsf{C}}^{\mathsf{T}}(\mathcal{H}\_{1})\cap\pi\_{\mathsf{C}}^{\mathsf{T}}(\mathcal{H}\_{2}), \qquad \pi\_{\mathsf{C}}^{\mathsf{I}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) > 1 - s > \pi\_{\mathsf{C}}^{\mathsf{I}}(\mathcal{H}\_{1})\cup\pi\_{\mathsf{C}}^{\mathsf{I}}(\mathcal{H}\_{2}).$$

*τ F* C (H1 ∩ H2) > 1 − *s* > *τ F* C (H1) ∪ *τ F* C (H2).

For each *i* ∈ {1, 2}, there exists *s* ∈ *I*0 with C(H*i* ,*si*) = 1 − H*i* such that

$$s < s\_i \le \pi\_\mathbb{C}^T(\mathcal{H}\_i), \quad \pi\_\mathbb{C}^I(\mathcal{H}\_i) \le 1 - s\_i < 1 - s, \quad \pi\_\mathbb{C}^F(\mathcal{H}\_i) \le 1 - s\_i < 1 - s.$$

In addition, since (1 − H*i* ,*r*) = 1 − H*i* by C2 and C4 of Definition 9, for any *i* ∈ {1, 2},

$$
\mathbb{C}((\overline{1} - \mathcal{H}\_1) \cup (\overline{1} - \mathcal{H}\_2), s) = (\overline{1} - \mathcal{H}\_1) \cup (\overline{1} - \mathcal{H}\_2).
$$

It follows that *τ T* C (H1 ∩ H2) ≥ *s*, *τ I* C (H1 ∩ H2) ≤ 1 − *s*, and *τ F* C (H1 ∩ H2) ≤ 1 − *s*. It is a contradiction. Thus, for every H, R ∈ *I* S , *τ T* C (H ∩ R) ≥ *τ T* C (H) ∩ *τ T* C (B), *τ I* C (H ∩ R) ≤ *τ I* C (H) ∪ *τ I* C (R), and *τ F* C (H ∩ R) ≤ *τ F* C (H) ∪ *τ F* C (R).

(SVNT3) Suppose that there exists H = S *i*∈*I* H*i* ∈ *I* S such that

$$\tau\_{\mathbb{C}}^{\mathrm{T}}(\mathcal{H}) < \bigcup\_{i \in I} \tau\_{\mathbb{C}}^{\mathrm{T}}(\mathcal{H}\_{i}), \quad \tau\_{\mathbb{C}}^{\mathrm{I}}(\mathcal{H}) > \bigcup\_{i \in I} \tau\_{\mathbb{C}}^{\mathrm{I}}(\mathcal{H}\_{i}), \quad \tau\_{\mathbb{C}}^{\mathrm{F}}(\mathcal{H}) > \bigcup\_{i \in I} \tau\_{\mathbb{C}}^{\mathrm{F}}(\mathcal{H}\_{i}).$$

There exists *s*0 ∈ *I*0 such that

$$\tau\_{\mathbb{C}}^{\mathrm{T}}(\mathcal{H}) < s\_{0} < \bigcup\_{i \in I} \tau\_{\mathbb{C}}^{\mathrm{T}}(\mathcal{H}\_{i}), \quad \tau\_{\mathbb{C}}^{\mathrm{I}}(\mathcal{H}) > 1 - s\_{0} > \bigcup\_{i \in I} \tau\_{\mathbb{C}}^{\mathrm{I}}(\mathcal{H}\_{i}), \quad \tau\_{\mathbb{C}}^{\mathrm{F}}(\mathcal{H}) > 1 - s\_{0} > \bigcup\_{i \in I} \tau\_{\mathbb{C}}^{\mathrm{F}}(\mathcal{H}\_{i}).$$

For every *i* ∈ *I*, there exists C(H*i* ,*si*) = 1 − H*i* and *si* ∈ *I*0 such that

$$s\_0 < s\_i \le \tau\_\mathbb{C}^T(\mathcal{H}\_i), \quad 1 - s\_0 > 1 - s\_i \ge \tau\_\mathbb{C}^I(\mathcal{H}\_i), \quad 1 - s\_i > 1 - s\_0 \ge \tau\_\mathbb{C}^F(\mathcal{H}\_i).$$

In addition, since C(1 − H*i* ,*r*0) ≤ C(1 − H*i* ,*si*) = 1 − H*i* , by C2 of Definition 9,

$$
\mathbb{C}(\overline{1} - \mathcal{H}\_{i\prime}s\_0) = \overline{1} - \mathcal{H}\_{i\prime}.
$$

It implies, for all *i* ∈ *I*,

$$\mathbb{C}(\overline{1} - \mathcal{H}\_{\prime}s\_0) \le \mathbb{C}(\overline{1} - \mathcal{H}\_{i\prime}s\_0) = \overline{1} - \mathcal{H}\_{i\prime}.$$

It follows that

$$\mathbb{C}(\overline{1} - \mathcal{H}, r\_0) \le \bigcap\_{i \in I} (\overline{1} - \mathcal{H}\_i) = \overline{1} - \mathcal{H}.$$

Thus, CI(1 − H,*s*0) = 1 − H, that is, *τ T* C (H) ≥ *s*0, *τ I* C (H) ≤ 1 − *s*0, and *τ F* C (H) ≤ 1 − *s*0. It is a contradiction. Hence, *τ T IF* C is an **SVNTS** on S.

(2) Since H ≤ C(H,*r*),

$$\pi\_{\mathbb{C}}^{T}(\overline{1} - \mathbb{C}(\mathcal{H}, s)) \ge s,\\ \pi\_{\mathbb{C}}^{I}(\overline{1} - \mathbb{C}(\mathcal{H}, s)) \le 1 - s,\\ \pi\_{\mathbb{C}}^{F}(\overline{1} - \mathbb{C}(\mathcal{H}, s)) \le 1 - s.$$

From C5 of Definition 9, we have C*τ T IF* C (H,*s*) ≤ C(H,*s*). Thus, C*τ T IF* C is finer than C.

**Example 1.** *Let* S = {*a*, *b*}*.  From Theorem 2, we have a single-valued neutrosophic topology* (*τ T* C , *τ I* C , *τ F* C ) *on* S *as follows:*

$$
\pi\_{\mathbb{C}}^{T}(\mathcal{A}) = \begin{cases}
1, & \text{if } \mathcal{A} = (1,0,0) \text{ or } (0,1,1), \\
\frac{2}{3}, & \text{if } \mathcal{A} = \mathcal{B}^{c}, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{H}^{c}, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{B}^{c} \cup \mathcal{H}^{c}, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{B}^{c} \cap \mathcal{H}^{c}, \\
0, & \text{otherwise.}
\end{cases}
$$

$$
\pi^I\_\mathbb{C}(\mathcal{A}) = \begin{cases}
0, & \text{if } \mathcal{A} = (1,0,0) \text{ or } (0,1,1), \\
\frac{1}{3}, & \text{if } \mathcal{A} = \mathcal{B}^c, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{H}^c, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{B}^c \cup \mathcal{H}^c, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{B}^c \cap \mathcal{H}^c, \\
1, & \text{otherwise.}
\end{cases}
$$

$$
\pi^{F}\_{\mathbb{C}}(\mathcal{A}) = \begin{cases}
0, & \text{if } \mathcal{A} = (1,0,0) \text{ or } (0,1,1), \\
\frac{1}{3}, & \text{if } \mathcal{A} = \mathcal{B}^{c}, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{H}^{c}, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{B}^{c} \cup \mathcal{H}^{c}, \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{B}^{c} \cap \mathcal{H}^{c}, \\
1, & \text{otherwise.}
\end{cases}
$$

*Thus, the τ T IF* C *is a single-valued neutrosophic topology on* S*.*

**Definition 10.** *A single-valued neutrosophic ideal (***SVNI***) on* S *in Šostak's sense on a nonempty set* S *is a family* L *T* ,L *I* ,L *F of single-valued neutrosophic sets in* S *satisfying the following axioms:*

(*L*1) L *T* (0) = 1 *and* L *I* (0) = L *F* (0) = 0*.*

(*L*2) *If* H ≤ B, *then* L *T* (R) ≤ L*T* (H)*,* L *I* (R) ≥ L*I* (H)*, and* L *F* (R) ≥ L*F* (H)*, for each single-valued neutrosophic set* R, H *in I*S *.*

(*L*3) L *T* (R ∪ H) ≥ L*T* (R) ∩ L*T* (H)*,* L *I* (R ∪ H) ≤ L*I* (R) ∪ L*I* (H)*, and* L *F* (R ∪ H) ≤ L*F* (R) ∪ L *F* (H)*, for each single-valued neutrosophic set* R, H *in I*S *.*

*If* L1 *and* L2 *are* **SVNI** *on* S*, we say that* L1 *is finer than* L2*, denoted by* L1 ≤ L2*, iff* L *T* 1 (H) ≤ L*T* 2 (H)*,* L *I* 1 (H) ≥ L*I* 2 (A)*, and* L *F* 1 (H) ≥ L*F* 2 (H)*, for* H ∈ *I* S .

*The triable* (*X*,(*τ T* , *τ I* , *τ F* ),(L *T* ,L *I* ,L *F* ) *is called a single-valued neutrosophic ideal topological space in Šostak sense (***SVNITS**, *f or short).*

*We will occasionally write* L *T IF ,* L *T IF i , and* L *T IF* : *I X* → *I for* (L *T* ,L *I* ,L *F* )*,* (L *T i* ,L *I i* ,L *F i* )*, and* L *T* ,L *I* ,L *F* : *I* S → *I, respectively.*

**Remark 1.** *The conditions* (*L*2) *and* (*L*3)*, which are given in Definition 10, are equivalent to the following axioms:* L *T* (H ∪ R) = L *T* (H) ∩ L*T* (R)*,* L *I* (H ∪ R) 6= L *I* (H) ∪ L*I* (R)*, and* L *F* (H ∪ R) 6= L *F* (H) ∪ L *F* (R)*, for every* R, H ∈ *I* S *.*

**Example 2.** *Let* S = {*a*, *b*}*. Define the single-valued neutrosophic sets* R, C, H, A *and* (L *T* , L *T* , L *T* ) : *I* S → *I as follows:*

R = h(0.3, 0.5),(0.4, 0.5),(0.5, 0.5)i; C = h(0.3, 0.4),(0.5, 0.5),(0.3, 0.4)i,

$$\mathcal{H} = \langle (0.1, 0.2), (0.5, 0.5), (0.5, 0.5) \rangle.$$

$$\mathcal{L}^{T}(\mathcal{A}) = \begin{cases} 1, & \text{if } \mathcal{B} = (0.1.1), \\ \frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{R}, \\ \frac{2}{3}, & \text{if } (0.1.1) < \mathcal{A} < \mathcal{R}, \\ 0, & \text{otherwise}. \end{cases}$$

$$\mathcal{L}^I(\mathcal{A}) = \begin{cases} \begin{array}{ll} 0, & \text{if } \mathcal{A} = (0.1.1), \\ \frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{C}, \\ \frac{1}{4}, & \text{if } (0.1.1) < \mathcal{A} < \mathcal{C}, \\ 1, & \text{otherwise}. \end{array} \end{cases}$$

$$
\mathcal{L}^T(\mathcal{B}) = \begin{cases}
0, & \text{if } \mathcal{A} = (0, 1, 1), \\
\frac{1}{2}, & \text{if } \mathcal{A} = \mathcal{H}, \\
\frac{1}{4}, & \text{if } (0.1.1) < \mathcal{A} < \mathcal{H}, \\
1, & \text{otherwise}.
\end{cases}
$$

*Then,* L *T IF is an* **SVNI** *on* S*.*

**Proposition 1.** *Let* {L*T IF i* }*i*∈*J be a family o f* **SVNI** *on* S*. Then, their intersection* T *i*∈*J* L *T IF i is also* **SVNI***.*

**Proof.** Directly from Definition 7.

**Proposition 2.** *Let* {L*T IF i* }*i*∈*J be a family o f* **SVNI** *on* S*. Then, their union* S *i*∈*J* L *T IF i is also an* **SVNI***.*

**Proof.** Directly from Definition 7.

#### **4. Single-Valued Neutrosophic Ideal Open Local Function in Šostak Sense**

In this section, we study the single-valued neutrosophic ideal open local function in Šostak's sense and present some of their properties. Additionally, properties preserved by single-valued neutrosophic ideal open compatible are examined.

**Definition 11.** *Let s*, *t*, *p* ∈ *I*0 *and s* + *t* + *p* ≤ 3*. A single-valued neutrosophic point xs*,*t*,*r of* S *is the single-valued neutrosophic set in I*S *for each κ* ∈ H*, defined by*

$$\mathfrak{x}\_{\mathsf{s},t,p}(\kappa) = \begin{cases} (\mathsf{s},t,p)\_{\mathsf{s}} & \text{if } \mathsf{x} = \mathsf{x}\_{\mathsf{s}}\\ (\mathsf{0},1,1)\_{\mathsf{s}} & \text{if } \mathsf{x} \neq \mathsf{x}\_{\mathsf{s}} \end{cases}$$

*A single-valued neutrosophic point xs*,*t*,*p is said to belong to a single-valued neutrosophic set* H = h*T*H, *I*H, *F*Hi ∈ *I* S *, denoted by xs*,*t*,*p* ∈ H *iff s* < *T*H*, t* ≥ *I*H *and p* ≥ *F*H*. 1. We indicate the set of all single-valued neutrosophic points in* S *as* **SVNP**(S)*.*

For every *xs*,*t*,*p* ∈ **SVNP**(S) and H ∈ *I* S we shall write *xs*,*t*,*p* quasi-coincident with H, denoted by *xs*,*t*,*pq*H, if

$$s + T\_{\mathcal{H}}(\kappa) > 1, \qquad t + I\_{\mathcal{H}}(\kappa) \le 1, \qquad p + F\_{\mathcal{H}}(\kappa) \le 1.$$

For every R, H ∈ S we shall write H*q*R to mean that H is quasi-coincident with R if there exists *κ* ∈ S such that

$$T\_{\mathcal{H}}(\kappa) + T\_{\mathcal{R}}(\kappa) > 1, \quad I\_{\mathcal{H}}(\kappa) + I\_{\mathcal{R}}(\kappa) \le 1, \quad F\_{\mathcal{H}}(\kappa) + F\_{\mathcal{R}}(\kappa) \le 1.$$

**Definition 12.** *Let* (S, *τ T IF*) *be an* **SVNTS***. For each r* ∈ *I*0*,* H ∈ *I* S *, xs*,*t*,*p* ∈ **SVNP**(S)*, a single-valued neutrosophic open Qτ T IF -neighborhood of xs*,*t*,*p is defined as follows:*

$$Q\_{\mathsf{T}^{\mathsf{T}\mathsf{H}}}(\mathbf{x}\_{\mathsf{s},t,p\prime}r) = \{\mathsf{H}|(\mathbf{x}\_{\mathsf{s},t,p})q\mathcal{H}, \qquad \mathsf{\tau}^{\mathsf{T}}(\mathsf{H}) \ge r, \qquad \mathsf{\tau}^{\mathsf{I}}(\mathsf{H}) \le 1-r, \qquad \mathsf{\tau}^{\mathsf{F}}(\mathsf{H}) \le 1-r\}.$$

**Lemma 1.** *A single-valued neutrosophic point xs*,*t*,*p* ∈ C*τ T IF* (R,*r*) *iff every single-valued neutrosophic open Qτ T IF -neighborhood of xs*,*t*,*p is quasi-coincident with* H*.*

**Definition 13.** *Let* (S, *τ T IF*) *be an* **SVNTS** *for each* H ∈ *I* S *. Then, the single-valued neutrosophic ideal open local function* H⋆ *r* (*τ T IF* , L *T IF*) *of* H *is the union of all single-valued neutrosophic points xs*,*t*,*p such that if* R ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) *and* L *T* (C) ≥ *r,* L *I* (C) ≤ 1 − *r,* L *F* (C) ≤ 1 − *r, then there is at least one κ* ∈ S *for which T*R(*κ*) + *T*H(*κ*) − 1 > *T*C (*κ*)*, I*R(*κ*) + *I*H(*κ*) − 1 ≤ *I*C (*κ*)*, and F*R(*κ*) + *F*H(*κ*) − 1 ≤ *F*C (*κ*)*.*

Occasionally, we will write H⋆ *r* for H⋆ *r* (*τ T IF* ,L *T IF*) and it will have no ambiguity.

**Example 3.** *Let* (S, *τ T IF* ,L *T IF*) *be an* **SVNITS***. The simplest single-valued neutrosophic ideal on* S *is* L *T IF* 0 : *I* S → *I, where*

$$\mathcal{L}\_0^{TIF}(\mathcal{R}) = \begin{cases} \ 1, & \text{if } \mathcal{R} = (1, 0, 0), \\ 0, & \text{otherwise.} \end{cases}$$

*If we take* L *T IF* = L *T IF* 0 *, for each* H ∈ *I* S *we have* H⋆ *r* = C*τ T IF* (H,*r*)*.* **Theorem 3.** *Let* (S, *τ T IF*) *be an* **SVNTS** *and* L *T IF* 1 , L *T IF* 2 ∈ **SVNI**(S)*. Then, for any* H, R ∈ *I* S *and r* ∈ *I*0*, we have*

*(1) If* H ≤ R, *then* H⋆ *r* ≤ R⋆ *r ; (2) If* L *T* 1 ≤ L*T* 2 *,* L *I* 1 ≥ L*I* 2 *and* L *F* 1 ≥ L*F* 2 *, then* H⋆ *r* (L *T IF* 1 , *τ T IF*) ≥ H⋆ *r* ((L *T IF* 2 , *τ T IF*)*; (3)* H⋆ *r* = *Cτ T IF* (A⋆ *r* ,*r*) ≤ C*τ T IF* (H,*r*)*; (4)* (H⋆ *r* ) ⋆ *r* ≤ H⋆ *r ; (5)* (H⋆ *r* ∨ R⋆ *r* ) = (H ∨ R) ⋆ *r ; (6) If* L *T* (H) ≥ *r,* L *I* (R) ≤ 1 − *r, and* L *F* (R) ≤ 1 − *r then* (H ∨ R) ⋆ *r* = A⋆ *r* ∨ R⋆ *r* = H⋆ *(7) If τ T* (R) ≥ *r, τ I* (R) ≤ 1 − *r, and τ F* (R) ≤ 1 − *r, then* (R ∧ H⋆ *r* ) ≤ (R ∧ H) ⋆ *r ; (8)* (H⋆ *r* ∧ R⋆ *r* ) ≥ (H ∧ R) ⋆ *r .*

**Proof.** (1) Suppose that H ∈ *I* S and H⋆ *r* 6≤ R⋆ *r* .  Then, there exists D ∈ *Q*(*τ T IF*) (*xs*,*t*,*p*,*r*), L *T* (C) ≥ *r*, L *I* (C) ≤ 1 − *r*, and L *F* (C) ≤ 1 − *r* such that for any *κ*1 ∈ S,

$$T\_{\mathcal{D}}(\mathbf{x}\_{1}) + T\_{\mathcal{R}}(\mathbf{x}\_{1}) - 1 \leq T\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad I\_{\mathcal{D}}(\mathbf{x}\_{1}) + I\_{\mathcal{R}}(\mathbf{x}\_{1}) - 1 > I\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad F\_{\mathcal{D}}(\mathbf{x}\_{1}) + F\_{\mathcal{R}}(\mathbf{x}\_{1}) - 1 > F\_{\mathcal{C}}(\mathbf{x}\_{1}).$$

Since H ≤ R,

$$T\_{\mathcal{D}}(\mathbf{x}\_{1}) + T\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 \le T\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad I\_{\mathcal{D}}(\mathbf{x}\_{1}) + I\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 > I\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad F\_{\mathcal{D}}(\mathbf{x}\_{1}) + F\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 > F\_{\mathcal{C}}(\mathbf{x}\_{1}).$$

So, *T*H⋆ *r* (*κ*) < *s*, *I*H⋆ *r* (*κ*) ≥ *t*, and *F*H⋆ *r* (*κ*) ≥ *p* and we arrive at a contradiction for Equation (1). Hence, H⋆ *r* ≤ R⋆ *r* .

(2) Suppose H⋆ *r* (L *T IF* 1 , *τ T IF*) 6≥ H⋆ *r* (L *T IF* 2 , *τ T IF*). Then, there exists *s*, *t*, *p* ∈ *I*0 and *κ* ∈ S such that

$$\begin{aligned} \mathcal{T}\_{\mathcal{H}\_{\mathbf{r}}^{\star}(\mathcal{L}\_{1}^{\mathrm{TIF}}, \boldsymbol{\tau}^{\mathrm{TIF}})}(\kappa) &< s \le \mathcal{T}\_{\mathcal{H}\_{\mathbf{r}}^{\star}(\mathcal{L}\_{2}^{\mathrm{TIF}}, \boldsymbol{\tau}^{\mathrm{TIF}})}(\kappa), \\\\ \mathcal{I}\_{\mathcal{H}\_{\mathbf{r}}^{\star}(\mathcal{L}\_{1}^{\mathrm{TIF}}, \boldsymbol{\tau}^{\mathrm{TIF}})}(\kappa) &\ge t > \mathcal{I}\_{\mathcal{H}\_{\mathbf{r}}^{\star}(\mathcal{L}\_{2}^{\mathrm{TIF}}, \boldsymbol{\tau}^{\mathrm{TIF}})}(\kappa), \end{aligned} \tag{2}$$

$$F\_{\mathcal{H}\_r^\star(\mathcal{L}\_1^{TF}, \mathfrak{r}^{TF})}(\kappa) \ge p > F\_{\mathcal{H}\_r^\star(\mathcal{L}\_2^{TF}, \mathfrak{r}^{TF})}(\kappa).$$

Since *T*H⋆ *r* (L*T IF* 1 ,*τ T IF*) (*κ*) < *s*, *I*H⋆ *r* (L*T IF* 1 ,*τ T IF*) (*κ*) ≥ *t*, and *F*H⋆ *r* (L*T IF* 1 ,*τ T IF*) (*κ*) ≥ *p*, D ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) with L *T* 1 (C) ≥ *r*, L *I* 1 (C) ≤ 1 − *r* and L *F* 1 (C) ≤ 1 − *r*. Thus, for every *κ*1 ∈ S,

$$T\_{\mathcal{D}}(\kappa\_1) + T\_{\mathcal{H}}(\kappa\_1) - 1 \le T\_{\mathcal{C}}(\kappa\_1), \quad I\_{\mathcal{D}}(\kappa\_1) + I\_{\mathcal{H}}(\kappa\_1) - 1 > I\_{\mathcal{C}}(\kappa\_1), \quad F\_{\mathcal{D}}(\kappa\_1) + F\_{\mathcal{H}}(\kappa\_1) - 1 > F\_{\mathcal{C}}(\kappa\_1).$$

$$\text{Since } \mathcal{L}\_2^{\mathrm{T}}(\mathcal{C}) \ge \mathcal{L}\_1^{\mathrm{T}}(\mathcal{C})) \ge r, \; \mathcal{L}\_2^{\mathrm{I}}(\mathcal{C}) \le \mathcal{L}\_1^{\mathrm{I}}(\mathcal{C})) \le 1 - r, \text{and } \mathcal{L}\_2^{\mathrm{F}}(\mathcal{C}) \le \mathcal{L}\_1^{\mathrm{F}}(\mathcal{C})) \le 1 - r.$$

$$T\_{\mathcal{D}}(\mathbf{x}\_{1}) + T\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 \leq T\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad I\_{\mathcal{D}}(\mathbf{x}\_{1}) + I\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 > I\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad F\_{\mathcal{D}}(\mathbf{x}\_{1}) + F\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 > F\_{\mathcal{C}}(\mathbf{x}\_{1}).$$

Thus, *T*H⋆ *r* (L*T IF* 2 ,*τ T IF*) (*κ*) < *s*, *I*H⋆ *r* (L*T IF* 2 ,*τ T IF*) (*κ*) ≥ *t*, and *F*H⋆ *r* (L*T IF* 2 ,*τ T IF*) (*κ*) ≥ *p*. This is a contradiction for Equation (2). Hence, H⋆ *r* ((L *T IF* 1 , *τ T IF*)) ≥ H⋆ *r* ((L *T IF* 2 , *τ T IF*)).

(3)(⇒) Suppose H⋆ *r* 6≤ C*τ T IF* (H,*r*). Then, there exists *s*, *t*, *p* ∈ *I*0 and *κ* ∈ S such that

$$T\_{\mathsf{H}\_{\mathsf{r}}^{\star}}(\mathbf{x}) \ge \mathbf{s} > T\_{\mathbb{C}\_{\tau \to \mathsf{I}\mathcal{F}}(\mathsf{H}\_{\mathsf{r}}\mathcal{I})}(\mathbf{x}), \quad I\_{\mathsf{H}\_{\mathsf{r}}^{\star}}(\mathbf{x}) < t \le I\_{\mathbb{C}\_{\tau \to \mathsf{I}\mathcal{F}}(\mathsf{H}\_{\mathsf{r}}\mathcal{I})}(\mathbf{x}), \quad F\_{\mathsf{H}\_{\mathsf{r}}^{\star}}(\mathbf{x}) < p \le F\_{\mathbb{C}\_{\tau \to \mathsf{I}\mathcal{F}}(\mathsf{H}\_{\mathsf{r}}\mathcal{I})}(\mathbf{x}). \tag{3}$$

Since *T*H⋆ *r* (*κ*) ≥ *s*, *I*H⋆ *r* (*κ*) < *t* and *F*H⋆ *r* (*κ*) < *p*, *xs*,*t*,*p* ∈ H⋆ *r* . So there is at least one *κ*1 ∈ S for every D ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) with L *T* 1 (C) ≥ *r*, L *I* 1 (C) ≤ 1 − *r*, L *F* 1 (C) ≤ 1 − *r* such that

$$T\_{\mathcal{D}}(\mathbf{x}\_{1}) + T\_{\mathcal{H}}(\mathbf{x}\_{1}) > T\_{\mathcal{C}}(\mathbf{x}\_{1}) + 1, \quad I\_{\mathcal{D}}(\mathbf{x}\_{1}) + I\_{\mathcal{H}}(\mathbf{x}\_{1}) \leq I\_{\mathcal{C}}(\mathbf{x}\_{1}) + 1, \quad F\_{\mathcal{D}}(\mathbf{x}\_{1}) + F\_{\mathcal{H}}(\mathbf{x}\_{1}) \leq F\_{\mathcal{C}}(\mathbf{x}\_{1}) + 1.$$

Therefore, by Lemma 1, *xs*,*t*,*p* ∈ C*τ T IF* (H,*r*) which is a contradiction for Equation (3). Hence, H⋆ *r* ≤ C*τ T IF* (H,*r*).

(⇐) Suppose H⋆ *r* 6≥ C*τ T IF* (H⋆ *r* ,*r*). Then, there exists *s*, *t*, *p* ∈ *I*0 and *κ* ∈ S such that

$$T\_{\mathsf{H}\_{\mathsf{T}}^{\star}}(\mathbf{x}) < s \le T\_{\mathbb{C}\_{\mathsf{T}\to\mathsf{T}}(\mathsf{H}\_{\mathsf{T}}^{\star}\mathsf{I})}(\mathbf{x}), \quad I\_{\mathsf{H}\_{\mathsf{T}}^{\star}}(\mathbf{x}) \ge t > I\_{\mathbb{C}\_{\mathsf{T}\to\mathsf{T}}(\mathsf{H}\_{\mathsf{T}}^{\star}\mathsf{I})}(\mathbf{x}), \quad F\_{\mathsf{H}\_{\mathsf{T}}^{\star}}(\mathbf{x}) \ge p > F\_{\mathbb{C}\_{\mathsf{T}\to\mathsf{T}}(\mathsf{H}\_{\mathsf{T}}^{\star}\mathsf{I})}(\mathbf{x}). \tag{4}$$

Since *T*C*τT IF* (H⋆ *r* ,*r*) (*κ*) ≥ *t*, *I*C*τT IF* (H⋆ *r* ,*r*) (*κ*) < *s*, C*τ T IF* (H⋆ *r* ,*r*)(*κ*) < *p* we have *xs*,*t*,*p* ∈ C*τ T IF* (H⋆ *r* ,*r*). So, there is at least one *κ*1 ∈ S with R ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) such that

$$T\_{\mathcal{R}}(\kappa\_1) + T\_{\mathcal{H}\_r^\pi}(\kappa\_1) > 1, \quad I\_{\mathcal{R}}(\kappa\_1) + I\_{\mathcal{H}\_r^\pi}(\kappa\_1) \le 1, \quad F\_{\mathcal{R}}(\kappa\_1) + F\_{\mathcal{H}\_r^\pi}(\kappa\_1) \le 1.$$

Therefore, H⋆ *r* (*κ*1) 6= 0. Let *s*1 = *T*H⋆ *r* (*κ*1), *t*1 = *I*H⋆ *r* (*κ*1), and *p*1 = *F*H⋆ *r* (*κ*1). Then, (*κ*1)*s*1 ,*t*1 ,*p*1 ∈ H∗ *r* and *s*1 + *T*R(*κ*1) > 1, *t*1 + *I*R(*κ*1) ≤ 1, and *p*1 + *F*R(*κ*1) ≤ 1 so that R ∈ *Qτ T IF* ((*κ*1)*s*1 ,*t*1 ,*p*1 ,*r*). Now, (*κ*1)*s*1 ,*t*1 ,*p*1 ∈ H⋆ *r* implies there is at least one *κ* ′ ∈ S such that *T*D(*κ* ′ ) + *T*H(*κ* ′ ) − 1 > *T*C (*κ* ′ ), *I*D(*κ* ′ ) + *I*H(*κ* ′ ) − 1 ≤ *I*C (*κ* ′ ), and *F*D(*κ* ′ ) + *F*H(*κ* ′ ) − 1 ≤ *F*C (*κ* ′ ), for all L *T* (C) ≥ *r*, L *I* (C) ≤ 1 − *r*, L *F* (C) ≤ 1 − *r*, and D ∈ *Qτ T IF* ((*κ*1)*s*1 ,*t*1 ,*p*1 ,*r*). That is also true for R. So there is at least one *κ* ′′ ∈ S such that *T*R(*κ* ′′) + *T*H(*κ* ′′) − 1 > *T*C (*κ* ′′), *I*R(*κ* ′′) + *I*H(*κ* ′′) − 1 ≤ *I*C (*κ* ′′), and *F*R(*x* ′′) + *F*H(*κ* ′′) − 1 ≤ *F*C (*κ* ′′). Since R ∈ *Qτ T IF* (*κs*,*t*,*p*,*r*) and R is arbitrary; then *T*H⋆ *r* (*κ*) > *s*, *I*H⋆ *r* (*κ*) ≤ *t* and *T*H⋆ *r* (*κ*) ≤ *p*. It is a contradiction for (4). Thus, H⋆ *r* ≥ C*τ T IF* (H⋆ *r* ,*r*).

(4) (⇒) Can be easily established using standard technique.

(5) (⇒) Since H, R ≤ H ∪ R. By (1), H⋆ *r* ≤ (H ∪ R) ⋆ *r* and R⋆ *r* ≤ (H ∪ R) ⋆ *r* . Hence, H⋆ *r* ∪ B⋆ *r* ≤ (H ∪ R) ⋆ *r* .

(⇐) Suppose (H⋆ *r* ∪ R⋆ *r* ) 6≥ (H ∪ R) ⋆ *r* . Then, there exists *s*, *t*, *p* ∈ *I*0 and *κ* ∈ S such that

$$T\_{\left(\mathsf{H}\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{u}}\cup\mathsf{R}\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}\right)}(\mathsf{x}) < s \le T\_{\left(\mathsf{H}\cup\mathsf{R}\right)\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}}(\mathsf{x}),\ I\_{\left(\mathsf{H}\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}\cup\mathsf{R}\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}\right)}(\mathsf{x}) \ge t > I\_{\left(\mathsf{H}\cup\mathsf{R}\right)\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}}(\mathsf{x}),\ F\_{\left(\mathsf{H}\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}\cup\mathsf{R}\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}\right)}(\mathsf{x}) \ge p > F\_{\left(\mathsf{H}\cup\mathsf{R}\right)\_{\mathsf{r}}^{\scriptscriptstyle\mathsf{s}}}(\mathsf{x}).\tag{5}$$

Since *T*(H⋆ *r* ∪R⋆ *r* ) (*κ*) < *s*, *I* (H⋆ *r* ∪R⋆ *r* ) (*κ*) ≥ *t*, and *F*(H⋆ *r* ∪R⋆ *r* ) (*κ*) ≥ *p*, we have *T*H⋆ *r* (*κ*) < *s*, *I*H⋆ *r* (*κ*) ≥ *t*, *F*H⋆ *r* (*κ*) ≥ *p* or *T*R⋆ *r* (*κ*) < *t*, *I*R⋆ *r* (*κ*) ≥ *t*, *F*R⋆ *r* (*κ*) ≥ *t*. So, there exists D1 ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) such that for every *κ*1 ∈ S and for some L *T* (C1) ≥ *r*, L *I* (C1) ≤ 1 − *r*, L *F* (C1) ≤ 1 − *r*, we have

$$T\_{\mathcal{D}\_1}(\mathbf{x}\_1) + T\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \le T\_{\mathcal{C}\_1}(\mathbf{x}\_1), \quad I\_{\mathcal{D}\_1}(\mathbf{x}\_1) + I\_{\mathcal{H}}(\mathbf{x}\_1) - 1 > I\_{\mathcal{C}\_1}(\mathbf{x}\_1), \quad F\_{\mathcal{D}\_1}(\mathbf{x}\_1) + F\_{\mathcal{H}}(\mathbf{x}\_1) - 1 > F\_{\mathcal{C}\_1}(\mathbf{x}\_1).$$

Similarly, there exists D2 ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) such that for every *κ*1 ∈ S and for some L *T* (C2) ≥ *r*, L *I* (C2) ≤ 1 − *r*, L *F* (C2) ≤ 1 − *r*, we have

$$T\_{\mathcal{D}\_2}(\mathbf{x}\_1) + T\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \le T\_{\mathcal{C}\_2}(\mathbf{x}\_1), \quad I\_{\mathcal{D}\_2}(\mathbf{x}\_1) + I\_{\mathcal{H}}(\mathbf{x}\_1) - 1 > I\_{\mathcal{C}\_2}(\mathbf{x}\_1), \quad F\_{\mathcal{D}\_2}(\mathbf{x}\_1) + F\_{\mathcal{H}}(\mathbf{x}\_1) - 1 > F\_{\mathcal{C}\_2}(\mathbf{x}\_1).$$

Since D = D1 ∧ D2 ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) and by (*L*3), L *T* (C1 ∪ C2) ≥ L*T* (C1) ∩ L*T* (C2) ≥ *r*, L *I* (C1 ∪ C2) ≤ L *I* (C1) ∪ L*I* (C2) ≤ 1 − *r*, and L *F* (C1 ∪ C2) ≤ L*T* (C1) ∪ L*T* (C2) ≤ 1 − *r*. Thus, for every *κ*1 ∈ S,

$$\begin{aligned} T\_{\mathcal{D}}(\kappa\_1) + T\_{\mathcal{R}\cup\mathcal{H}}(\kappa\_1) - 1 &\leq T\_{\mathcal{C}\_1 \cup \mathcal{C}\_2}(\kappa\_1), \\ I\_{\mathcal{D}}(\kappa\_1) + I\_{\mathcal{R}\cup\mathcal{H}}(\kappa\_1) - 1 &\geq I\_{\mathcal{C}\_1 \cup \mathcal{C}\_2}(\kappa\_1), \\ F\_{\mathcal{D}}(\kappa\_1) + F\_{\mathcal{R}\cup\mathcal{H}}(\kappa\_1) &\geq F\_{\mathcal{C}\_1 \cup \mathcal{C}\_2}(\kappa\_1). \end{aligned}$$

Therefore, *T*(H∪R) ⋆ *r* (*κ*) < *s*, *I* (H∪R) ⋆ *r* (*κ*) ≥ *t*, and *F*(H∪R) ⋆ *r* (*κ*) ≥ *p*. So, we arrive at a contradiction for (5). Hence, (H⋆ *r* ∪ R⋆ *r* ) ≥ (H ∪ R) ⋆ *r* .

(6), (7), and (8) can be easily established using the standard technique.

**Example 4.** *Let* S = {*a*, *b*}*. Define* R, C, H ∈ S *as follows:* R1 = h(0.5, 0.5, 0.5),(0.5, 0.5, 0.5),(0.5, 0.5, 0.5)i; R2 = h(0.4, 0.4, 0.4),(0.1, 0.1, 0.1),(0.1, 0.1, 0.1)i; R3 = h(0.3, 0.3, 0.3),(0.1, 0.1, 0.1),(0.1, 0.1, 0.1); C1 = h(0.3, 0.3, 0.3),(0.3, 0.3, 0.3),(0.1, 0.1, 0.1)i;

C2 = h(0.2, 0.2, 0.2),(0.2, 0.2, 0.2),(0.1, 0.1, 0.1); C3 = h(0.1, 0.1, 0.1),(0.1, 0.1, 0.1),(0.1, 0.1, 0.1)i. *Define τ T IF* ,L *T IF* : *I X* → *I as follows:*

*τ T* (H) = 1, *if* H = (0, 1, 1), 1, *if* H = (1, 0, 0), 1 2 , *if* H = R1; L *T* (H) = 1, *if* H = (0, 1, 1), 1 2 , *if* H = C1, 2 3 , *if* 0 < H < C1; *τ I* (H) = 0, *if* H = (0, 1, 1), 0, *if* H = (1, 0, 0), 1 2 , *if* H = R2; L *I* (R) = 0, *if* H = (0, 1, 1), 1 2 , *if* H = C2, 1 4 , *if* 0 < H < C2; *τ F* (H) = 0, *if* H = (0, 1, 1), 0, *if* H = (1, 0, 0), 1 2 , *if* H = R3; L *F* (H) = 0, *if* H = (0, 1, 1), 1 2 , *if* H = C3, 1 4 , *if* 0 < H < C3.

*Let* G = h(0.4, 0.4, 0.4),(0.4, 0.4, 0.4),(0.4, 0.4, 0.4)i*. Then,* G ⋆ 1 2 = R1*.*

**Theorem 4.** *Let* {H*i*}*i*∈*J* ⊂ *I* S *be a family of single-valued neutrosophic sets on* S *and* (S, *τ T IF* , L *T IF*) *be an* **SVNITS***. Then,*

*(1)* ( S (H*i*) ⋆ *r* : *i* ∈ *J*) ≤ ( S H*i* : *i* ∈ *J*) ⋆ *r ; (2)* ( T (H*i*) ⋆ *r* : *i* ∈ *J*) ≥ ( T H*i* : *i* ∈ *J*) ⋆ *r .*

**Proof.** (1) Since H*i* ≤ S H*i* for all *i* ∈ *J*, and by Theorem 3 (1), we obtain ( S (H*i*) ⋆ *r* , *i* ∈ *J*) ≤ ( S H*i* , *i* ∈ *J*) ⋆ *r* . Then, (1) holds. (2) Easy, so omitted.

**Remark 3.** *Let* (S, *τ T IF* ,L *T IF*) *be an* **SVNITS** *and* H ∈ *I* S *, we can define*

$$\mathbb{C}\_{\tau^{TIF}}^{\star}(\mathcal{H},r) = \mathcal{H} \cup \mathcal{H}\_{r}^{\star}, \qquad \operatorname{int}\_{\tau^{TIF}}^{\star}(\mathcal{H},r) = \mathcal{H} \wedge [\underline{1} - (\underline{1} - \mathcal{H})\_{r}^{\star}].$$

*It is clear,* C⋆ *τ T IF is a single-valued neutrosophic closure operator and* (*τ T*⋆ (L *T* ), *τ I*⋆ (L *I* ), *τ F*⋆ (L *F* ) *is the single-valued neutrosophic topology generated by* C⋆ *τ T IF , i.e.,*

$$\tau^\star(\mathcal{T})(\mathcal{H}) = \bigcup \{ r \mid \mathbb{C}^\star\_{\tau^{TIF}}(\underline{1} - \mathcal{H}, r) = \underline{1} - \mathcal{H} \}.$$

*Now, if* L *T IF* = L *T IF* 0 *, then,* C⋆ *τ T IF* (H,*r*) = H∗ *r* ∪ H = C⋆ *τ T IF* (H,*r*) ∪ H = C*τ T IF* (H,*r*), *for* H ∈ *I* S . *So, τ T IF*⋆ (L *T IF* 0 ) = *τ T IF .*

**Proposition 3.** *Let* (S, *τ T IF* ,L *T IF*) *be an* **SVNITS***, r* ∈ *I*0*, and* H ∈ *I* S *. Then,*

*(1)* C⋆ *τ T IF* (1,*r*) = 1; *Symmetry* **2020**, *12*, 193

**Proof.** Follows directly from definitions of C⋆ *τ T IF* , *int*⋆ *τ T IF* , C*τ T IF* , and Theorem 3 (5).

**Theorem 5.** *Let* (S, *τ T IF* 1 ,L *T IF*) *and* (S, *τ T IF* 2 ,L *T IF*) *be* **SVNTS**′ **s** *and τ T IF* 1 ≤ *τ T IF* 2 *. Then,* H⋆ *r* (*τ T IF* 2 ,L *T IF*) ≤ H⋆ *r* (*τ T IF* 1 ,L *T IF*)*.*

**Proof.** Suppose H⋆ *r* (*τ T IF* 2 ,L *T IF*) 6≤ H⋆ *r* (*τ T IF* 1 ,L *T IF*). Then, there exists *s*, *t*, *p* ∈ *I*0, *κ* ∈ S such that

$$\begin{aligned} \mathcal{I}\_{\mathcal{H}^s\_r(\mathbf{z}\_1^{\mathrm{TF}}, \mathcal{L}^{\mathrm{TF}})}(\kappa) &\geq s > \mathcal{I}\_{\mathcal{H}^s\_r(\mathbf{z}\_1^{\mathrm{TF}}, \mathcal{L}^{\mathrm{TF}})}(\kappa), \\\\ \mathcal{I}\_{\mathcal{H}^s\_r(\mathbf{z}\_2^{\mathrm{TF}}, \mathcal{L}^{\mathrm{TF}})}(\kappa) &< t \leq \mathcal{I}\_{\mathcal{H}^s\_r(\mathbf{z}\_1^{\mathrm{TF}}, \mathcal{L}^{\mathrm{TF}})}(\kappa), \\\\ \mathcal{F}\_{\mathcal{H}^s\_r(\mathbf{z}\_2^{\mathrm{TF}}, \mathcal{L}^{\mathrm{TF}})}(\kappa) &< t \leq \mathcal{F}\_{\mathcal{H}^s\_r(\mathbf{z}\_1^{\mathrm{TF}}, \mathcal{L}^{\mathrm{TF}})}(\kappa). \end{aligned} \tag{6}$$

Since *T*H⋆ *r* (*τ T IF* 1 ,L*T IF*) (*κ*) < *s*, *I*H⋆ *r* (*τ T IF* 1 ,L*T IF*) (*κ*) ≥ *t*, *F*H⋆ *r* (*τ T IF* 1 ,L*T IF*) (*κ*) ≥ *p*, there exists D ∈ *Qτ T IF* 1 (*xs*,*t*,*p*,*r*) with L *T* (C1) ≥ *r*, L *I* (C1) ≤ 1 − *r* and L *F* (C1) ≤ 1 − *r*, such that for any *κ*1 ∈ S,

$$T\_{\mathcal{D}}(\mathbf{x}\_{1}) + T\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 \le T\_{\mathcal{C}}\mathbf{x}\_{1}), \quad I\_{\mathcal{D}}(\mathbf{x}\_{1}) + I\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 > I\_{\mathcal{C}}(\mathbf{x}\_{1}), \quad F\_{\mathcal{D}}(\mathbf{x}\_{1}) + F\_{\mathcal{H}}(\mathbf{x}\_{1}) - 1 > F\_{\mathcal{C}}(\mathbf{x}\_{1}).$$

Since *τ T IF* 1 ≤ *τ T IF* 2 , D ∈ *Qτ T IF* 2 (*xs*,*t*,*p*,*r*). Thus, *T*H⋆ *r* (*τ T IF* 2 ,L*T IF*) (*κ*) < *s*, *I*H⋆ *r* (*τ T IF* 2 ,L*T IF*))(*κ*) ≥ *t*, *F*H⋆ *r* (*τ T IF* 2 ,L*T IF*) (*κ*) ≥ *p*. It is a contradiction for Equation (6).

**Theorem 6.** *Let* (S, *τ T IF* ,L *T IF* 1 ) *and* (S, *τ T IF* ,L *T IF* 2 ) *be* **SVNTS**′ **s** *and* L *T IF* 1 ≤ L*T IF* 2 *. Then,* H⋆ *r* (L *T IF* 1 , *τ T IF*) ≥ H⋆ *r* (L *T IF* 2 , *τ T IF*)*.*

**Proof.** Clear.

**Definition 14.** *Let* Θ *be a subset of I* S *, and* 0 6∈ Θ*. A mapping β T* , *β I* , *β F* : Θ → *I is called a single-valued neutrosophic base on* S *if it satisfies the following conditions:*

*(1) β T* (1) = 1 *and β I* (1) = *β F* (1) = 0*;*

*(2) For all* H*,*R ∈ Θ*,*

$$
\mathfrak{G}^{\mathsf{T}}(\mathsf{H}\cap\mathcal{R})\geq\mathfrak{G}^{\mathsf{T}}(\mathsf{H})\cap\mathfrak{G}^{\mathsf{T}}(\mathsf{R}), \quad \mathfrak{G}^{\mathsf{I}}(\mathsf{H}\cap\mathcal{R})\leq\mathfrak{G}^{\mathsf{I}}(\mathsf{H})\cup\mathfrak{G}^{\mathsf{I}}(\mathsf{R}), \quad \mathfrak{G}^{\mathsf{F}}(\mathsf{H}\cap\mathcal{R})\leq\mathfrak{G}^{\mathsf{F}}(\mathsf{H})\cup\mathfrak{G}^{\mathsf{F}}(\mathsf{R}).
$$

**Theorem 7.** *Define a mapping β* : Θ → *I on* S *by*

$$\begin{aligned} \beta^{I}(\mathcal{H}) &= \bigcup \{ \tau^{T}(\mathcal{R}) \cap \mathcal{T}^{T}(\mathcal{C}) \,|\,\mathcal{H} = \mathcal{R} \cap (\underline{1} - \mathcal{C}) \}, \\\\ \beta^{I}(\mathcal{H}) &= \bigcap \{ \tau^{I}(\mathcal{R}) \cup \mathcal{T}^{I}(\mathcal{C}) \,|\,\mathcal{H} = \mathcal{R} \cap (\underline{1} - \mathcal{C}) \}, \\\\ \beta^{F}(\mathcal{H}) &= \bigcap \{ \tau^{F}(\mathcal{R}) \cup \mathcal{T}^{F}(\mathcal{C}) \,|\,\mathcal{H} = \mathcal{R} \cap (\underline{1} - \mathcal{C}) \}. \end{aligned}$$

*Then, β T IF is a base for the single-valued neutrosophic topology τ T IF*⋆ *.*

**Proof.** (1) Since L *T* (0) = 1 and L *I* (0) = L *F* (0) = 0, we have *β T* (1) = 1 and *β I* (1) = *β F* (1) = 0; (2) Suppose that there exists H1, H2 ∈ Θ such that

$$
\begin{split}
\boldsymbol{\beta}^{\mathrm{T}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) &\mathop{\rm g}\nolimits^{\mathrm{T}}(\mathcal{H}\_{1})\cap\boldsymbol{\beta}^{\mathrm{T}}(\mathcal{H}\_{2}), \\
\boldsymbol{\beta}^{\mathrm{I}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) &\mathop{\rm g}\nolimits^{\mathrm{I}}(\mathcal{H}\_{1})\cup\boldsymbol{\beta}^{\mathrm{I}}(\mathcal{H}\_{2}), \\
\boldsymbol{\beta}^{\mathrm{F}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) &\mathop{\rm g}\nolimits^{\mathrm{F}}(\mathcal{H}\_{1})\cup\boldsymbol{\beta}^{\mathrm{F}}(\mathcal{H}\_{2}).
\end{split}
$$

There exists *s*, *t*, *p* ∈ *I*0 and *κ* ∈ S such that

$$
\boldsymbol{\beta}^{T}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2})(\kappa) < s \le \boldsymbol{\beta}^{T}(\mathcal{H}\_{1})(\kappa) \cap \boldsymbol{\beta}^{T}(\mathcal{H}\_{2})(\kappa),
$$

$$
\boldsymbol{\beta}^{I}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2})(\kappa) \ge t > \boldsymbol{\beta}^{I}(\mathcal{H}\_{1})(\kappa) \cap \boldsymbol{\beta}^{I}(\mathcal{H}\_{2})(\kappa),
\tag{7}
$$

$$
\beta^{\mathcal{F}}(\mathcal{H}\_1 \cap \mathcal{H}\_2)(\kappa) \ge p > \beta^{\mathcal{F}}(\mathcal{H}\_1)(\kappa) \cup \beta^{\mathcal{F}}(\mathcal{H}\_2)(\kappa).
$$

Since *β T* (H1)(*κ*) ≥ *s*, *β I* (H1)(*κ*) < *t*, *β F* (H1)(*κ*) < *p*, and *β T* (H2)(*κ*) ≥ *s*, *β I* (H2)(*κ*) < *t*, *β F* (H2)(*κ*) < *p*, then there exists R1, R1, C1, C2 ∈ Θ with H1 = R1 ∩ (1 − C1) and H2 = R2 ∩ (1 − C2), such that *β T* (H1) ≥ *τ T* (R1) ∩ L*T* (C1) ≥ *s*, *β I* (H1) ≤ *τ I* (R1) ∪ L*I* (C1) < *t*, *β F* (H1) ≤ *τ F* (R1) ∪ L*F* (C1) < *p*, and *β T* (H2) ≥ *τ T* (R2) ∩ L*T* (C2) ≥ *s*, *β I* (H2) ≤ *τ I* (R2) ∪ L*I* (C2) < *t*, *β F* (H2) ≤ *τ F* (R2) ∪ L*F* (C2) < *p*. Therefore,

$$\begin{aligned} \mathcal{H}\_1 \cap \mathcal{H}\_2 &= (\mathcal{R}\_1 \cap (\underline{1} - \mathcal{C}\_1)) \cap (\mathcal{R}\_2 \cap (\underline{1} - \mathcal{C}\_2)) \\ &= (\mathcal{R}\_1 \cap \mathcal{R}\_2) \cap ((\underline{1} - \mathcal{C}\_1) \cap (\underline{1} - \mathcal{C}\_2)) \\ &= (\mathcal{R}\_1 \cap \mathcal{R}\_2) \cap (\underline{1} - (\mathcal{C}\_1 \cup \mathcal{C}\_2)). \end{aligned}$$

Hence, from Definition 14, we have

$$\begin{aligned} \mathfrak{F}^{\mathrm{T}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) & \quad \geq \mathfrak{r}^{\mathrm{T}}(\mathcal{R}\_{1}\cap\mathcal{R}\_{2})\cap\mathcal{L}^{\mathrm{T}}(\mathcal{C}\_{1}\cup\mathcal{C}\_{2})\\ & \geq \mathfrak{r}^{\mathrm{T}}(\mathcal{R}\_{1})\cap\mathfrak{r}^{\mathrm{T}}(\mathcal{R}\_{2})\cap\mathcal{L}^{\mathrm{T}}(\mathcal{C}\_{1})\cap\mathcal{L}^{\mathrm{T}}(\mathcal{C}\_{2})\\ &= (\mathfrak{r}^{\mathrm{T}}(\mathcal{R}\_{1})\cap\mathcal{L}^{\mathrm{T}}(\mathcal{C}\_{1}))\cap(\mathfrak{r}^{\mathrm{T}}(\mathcal{R}\_{2})\cap\mathcal{L}^{\mathrm{T}}(\mathcal{C}\_{2})) \geq s. \end{aligned}$$

$$\begin{aligned} \beta^{I}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) &\leq \tau^{I}(\mathcal{R}\_{1}\cap\mathcal{R}\_{2})\cup\mathcal{L}^{I}(\mathcal{C}\_{1}\cup\mathcal{C}\_{2})\\ &\leq \tau^{I}(\mathcal{R}\_{1})\cup\tau^{I}(\mathcal{R}\_{2})\cup\mathcal{L}^{I}(\mathcal{C}\_{1})\cup\mathcal{L}^{I}(\mathcal{C}\_{2})\\ &= (\tau^{I}(\mathcal{R}\_{1})\cup\mathcal{L}^{F}(\mathcal{C}\_{1}))\cup(\tau^{I}(\mathcal{R}\_{2})\cup\mathcal{L}^{I}(\mathcal{C}\_{2}))$$

$$\begin{split} \beta^{F}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) &\quad \leq \tau^{F}(\mathcal{R}\_{1}\cap\mathcal{R}\_{2})\cup\mathcal{L}^{F}(\mathcal{C}\_{1}\cup\mathcal{C}\_{2})\\ &\leq \tau^{F}(\mathcal{R}\_{1})\cup\tau^{F}(\mathcal{R}\_{2})\cup\mathcal{L}^{F}(\mathcal{C}\_{1})\cup\mathcal{L}^{F}(\mathcal{C}\_{2})\\ &= (\tau^{F}(\mathcal{R}\_{1})\cup\mathcal{L}^{F}(\mathcal{C}\_{1}))\cup(\tau^{F}(\mathcal{R}\_{2})\cup\mathcal{L}^{F}(\mathcal{C}\_{2}))$$

It is a contradiction for Equation (7). Thus,

$$
\boldsymbol{\beta}^{\mathrm{T}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) \geq \boldsymbol{\beta}^{\mathrm{T}}(\mathcal{H}\_{1}) \cap \boldsymbol{\beta}^{\mathrm{T}}(\mathcal{H}\_{2}),
\boldsymbol{\beta}^{\mathrm{I}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) \leq \boldsymbol{\beta}^{\mathrm{I}}(\mathcal{H}\_{1}) \cup \boldsymbol{\beta}^{\mathrm{I}}(\mathcal{H}\_{2}),
\boldsymbol{\beta}^{\mathrm{F}}(\mathcal{H}\_{1}\cap\mathcal{H}\_{2}) \leq \boldsymbol{\beta}^{\mathrm{F}}(\mathcal{H}\_{1}) \cup \boldsymbol{\beta}^{\mathrm{F}}(\mathcal{H}\_{2}).
$$

$$
\Box
$$

**Theorem 8.** *Let* (S, *τ T IF*) *be an* **SVNTS***, and* L *T IF* 1 *and* L *T IF* 1 *be two single-valued neutrosophic ideals on* S*. Then, for every r* ∈ *I*0 *and* H ∈ *I* S *,*

**Proof.**

Now, *T*H⋆ *r* (L*T* 1 ,*τ T*) (*κ*) < *s*, *I*H⋆ *r* (L*I* 1 ,*τ I*) (*κ*) ≥ *t*, *F*H⋆ *r* (L*F* 1 ,*τ F*) (*κ*) ≥ *p* implies that there exists D1 ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) and for some L *T* 1 (C1) ≥ *r*, L *I* 1 (C1) ≤ 1 − *r* and L *F* 1 (C1) ≤ 1 − *r* such that for every *κ*1 ∈ S,

$$T\_{\mathcal{D}\_1}(\mathbf{x}\_1) + T\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \le T\_{\mathcal{C}\_1}(\mathbf{x}\_1), \quad I\_{\mathcal{D}\_1}(\mathbf{x}\_1) + I\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \ge I\_{\mathcal{C}\_1}(\mathbf{x}\_1), \quad F\_{\mathcal{D}\_1}(\mathbf{x}\_1) + F\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \ge F\_{\mathcal{C}\_1}(\mathbf{x}\_1).$$

Once again, *T*H⋆ *r* (L*T* 2 ,*τ T*) (*κ*) < *s*, *I*H⋆ *r* (L*I* 2 ,*τ I*) (*κ*) ≥ *t*, *F*H⋆ *r* (L*F* 2 ,*τ F*) (*κ*) ≥ *p*, implies there exists D2 ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) and for some L *T* 2 (C2) ≥ *r*, L *I* 2 (C2) ≤ 1 − *r* and L *F* 2 (C2) ≤ 1 − *r*, such that for *κ*1 ∈ S,

$$T\_{\mathcal{D}\_2}(\mathbf{x}\_1) + T\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \le T\_{\mathcal{C}\_2}(\mathbf{x}\_1), \quad I\_{\mathcal{D}\_2}(\mathbf{x}\_1) + I\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \ge I\_{\mathcal{C}\_2}(\mathbf{x}), \quad F\_{\mathcal{D}\_2}(\mathbf{x}\_1) + F\_{\mathcal{H}}(\mathbf{x}\_1) - 1 \ge F\_{\mathcal{C}\_2}(\mathbf{x}\_1),$$

Therefore, for every *κ*1 ∈ S, we have

$$T\_{\mathcal{D}\_1 \cap \mathcal{D}\_2}(\kappa\_1) + T\_{\mathcal{H}}(\kappa\_1) - 1 \le T\_{\mathcal{C}\_1 \cap \mathcal{C}\_2}(\kappa\_1), \quad I\_{\mathcal{D}\_1 \cup \mathcal{D}\_2}(\kappa\_1) + I\_{\mathcal{H}}(\kappa\_1) - 1 \ge I\_{\mathcal{C}\_1 \cup \mathcal{C}\_2}(\kappa\_1).$$

$$F\_{\mathcal{D}\_1 \cup \mathcal{D}\_2}(\kappa\_1) + F\_{\mathcal{H}}(\kappa\_1) - 1 \ge F\_{\mathcal{C}\_1 \cup \mathcal{C}\_2}(\kappa\_1).$$

Since (D1 ∧ D2) ∈ *Qτ T IF* (*xs*,*t*,*p*,*r*) and (L *T* 1 ∩ L*T* 2 )(C1 ∩ C2) ≥ *r*, (L *I* 1 ∩ L*I* 2 )(C1 ∪ C2) ≤ 1 − *r*, and (L *F* 1 ∩ L *F* 2 )(C1 ∪ C2) ≥ 1 − *r* we have *T*H⋆ *r* (L*T* 1 ∩L*T* 2 ,*τ T*) (*κ*) ≤ *s*, *I*H⋆ *r* (L*I* 1 ∩L*I* 2 ,*τ I*) (*κ*) > *t*, and *F*H⋆ *r* (L*F* 1 ∩L*F* 2 ,*τ F*) (*κ*) > *t* and this is a contradiction for Equation (8). So that

$$
\mathcal{H}\_r^\star(\mathcal{L}\_1^{\rm TIF} \cap \mathcal{L}\_2^{\rm TIF}, \mathfrak{r}^{\rm TIF}) \le \mathcal{H}\_r^\star(\mathcal{L}\_1^{\rm TIF}, \mathfrak{r}^{\rm TIF}) \cup \mathcal{H}\_r^\star(\mathcal{L}\_2^{\rm TIF}, \mathfrak{r}^{\rm TIF}).
$$

On the opposite direction, L *T IF* 1 ≥ L*T IF* 1 ∩ L*T IF* 2 and L *T IF* 2 ≥ L*T IF* 1 ∩ L*T IF* 2 , so by Theorem 3 (2),

$$
\mathcal{H}\_r^\star(\mathcal{L}\_1^{\rm TIF} \cap \mathcal{L}\_2^{\rm TIF}, \mathfrak{r}^T) \ge \mathcal{H}\_r^\star(\mathcal{L}\_1^{\rm TIF}, \mathfrak{r}^{\rm TIF}) \cup \mathcal{H}\_r^\star(\mathcal{L}\_2^{\rm TIF}, \mathfrak{r}^{\rm TIF}).
$$

Then,

$$\mathcal{H}\_r^\star(\mathcal{L}\_1^{\rm TIF} \cap \mathcal{L}\_2^{\rm TIF}, \mathfrak{r}^{\rm TIF}) = \mathcal{H}\_r^\star(\mathcal{L}\_1^{\rm TIF}, \mathfrak{r}^{\rm TIF}) \cup \mathcal{H}\_r^\star(\mathcal{L}\_2^{\rm TIF}, \mathfrak{r}^{\rm TIF}).$$

(2) Straightforward.

The above theorem results in an important consequence. *τ T IF*⋆ (L *T IF*) and [*τ T IF*⋆ (L *T IF*)]⋆ (L *T IF*) (in short *τ* ⋆⋆) are equal for any single-valued neutrosophic ideal on S.

**Corollary 1.** *Let* (S, *τ T IF* ,L *T IF*) *be an* **SVNITS***. For every r* ∈ *I*0 *and* H ∈ *I X,* H⋆ *r* (L *T IF*) = H⋆ *r* (L *T IF* , *τ T IF*⋆ ) *and τ T IF*⋆ (L *T IF*) = *τ T IF*⋆⋆ *.*

**Proof.** Putting L *T IF* 1 = L *T IF* 2 in Theorem 8 (2), we have the required result.

**Corollary 2.** *Let* (S, *τ T IF*) *be an* **SVNTS***, and* L *T IF* 1 *and* L *T IF* 1 *be two single-valued neutrosophic ideals on* S*. Then, for any* H ∈ *I* S *and r* ∈ *I*0*,*