Open AccessArticle

On Single-Valued Neutrosophic Ideals in Šostak Sense

Yaser Saber

^1,2

Fahad Alsharari

and

Florentin Smarandache

^3,*

Department of Mathematics, College of Science and Human Studies, Hotat Sudair, Majmaah University, Majmaah 11952, Saudi Arabia

Department of Mathematics, Faculty of Science Al-Azhar University, Assiut 71524, Egypt

Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA

Author to whom correspondence should be addressed.

Symmetry 2020, 12(2), 193; https://doi.org/10.3390/sym12020193

Submission received: 15 December 2019 / Revised: 6 January 2020 / Accepted: 6 January 2020 / Published: 25 January 2020

(This article belongs to the Special Issue New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations)

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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 [3,4,5,6,7,8,9,10,11,12,13]

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 [34,35,36] 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

\begin{matrix} H = 〈 κ, T_{H}, I_{H}, F_{H} : κ \in S 〉, \end{matrix}

where

\begin{matrix} T_{H} : S \to ⌋^{-} 0, 1^{+} {⌊, I_{H} : S \to ⌋}^{-} 0, 1^{+} {⌊, F_{H} : S \to ⌋}^{-} 0, 1^{+} ⌊ \end{matrix}

and

\begin{matrix} ^{-} 0 \leq T_{H} (κ) + I_{H} (κ) + F_{H} (κ) \leq 3^{+}, \end{matrix}

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 \in 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 \in S

\begin{matrix} inf T_{H} (x) \leq inf T_{R} (κ), inf I_{H} (x) \geq inf I_{R} (κ), inf F_{H} (x) \geq inf F_{R} (κ) \end{matrix}

and

\begin{matrix} sup T_{H} (κ) \leq sup T_{R} (κ), sup I_{H} (κ) \geq sup I_{R} (κ), sup F_{H} (κ) \geq sup F_{R} (κ) . \end{matrix}

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)

S

H

has the form

H = 〈 T_{H}, I_{H}, F_{H} 〉

, where

T_{H}, I_{H}, F_{H} : S \to [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

S

SVNS (S)

Moreover, we will refer to the Null (empty)

SVNS

(or the absolute (universe)

SVNS

) in

S

0_{N}

(or

1_{N}

) and define by

0_{N} = (0, 1, 1)

(or

1_{N} = (1, 0, 0)

) for each

κ \in S

Definition 4

([47]). Let

H = 〈 T_{H}, I_{H}, F_{H} 〉

be an

SVNS

S

. The complement of the set

H

(

H^{c}, f o r s h o r t

) and is defined as follows: for every

κ \in S

\begin{matrix} T_{H^{c}} (κ) = F_{H} (κ), I_{H^{c}} (κ) = 1 - I_{H} (κ), F_{H^{c}} (κ) = T_{H} (κ) . \end{matrix}

Definition 5

([50]). Suppose that

H \in SVNS (S)

. Then,

(i): $H$ is said to be contained in $R$ , denoted by $H \subseteq R$ , if, for every $κ \in S$ ,

$\begin{matrix} T_{H} (κ) \leq T_{R} (κ), I_{H} (κ) \geq I_{R} (κ), F_{H} (κ) \geq F_{R} (κ); \end{matrix}$
(ii): $H$ is said to be equal to $R$ , denoted by $H = H$ , if $R \subseteq R$ and $H \supseteq R$ .

Definition 6

([51]). Suppose that

H, R \in SVNS (S)

. Then,

(i): the union of $H$ and $R$ ( $H \cup R, f o r s h o r t$ ) is an $SVNS$ in $S$ defined as

$\begin{matrix} H \cup R = (T_{H} \cup T_{R}, I_{H} \cap I_{R}, F_{H} \cap F_{R}), \end{matrix}$

where $(T_{H} \cup T_{R}) (κ) = T_{H} (κ) \cup T_{R} (κ)$ and $(F_{H} \cap F_{R}) (κ) = F_{H} (κ) \cap F_{R} (κ)$ , for each $κ \in S$ ;
(ii): the intersection of $H$ and $R$ , $(H \cap R, f o r s h o r t$ ), is an $SVNS$ in $S$ defined as

$\begin{matrix} H \cap R = (T_{H} \cap T_{R}, I_{H} \cup I_{R}, F_{H} \cup F_{R}) . \end{matrix}$

Definition 7

([45]). Let

H \in SVNS (S)

. Then,

(i): the union of ${H_{i}}_{i \in J}$ ( $⋃_{i \in J} H_{i}, f o r s h o r t)$ is an $SVNS$ in $S$ defined as follows: for every $κ \in S$ ,

$\begin{matrix} (⋃_{i \in J} H_{i}) (κ) = (⋃_{i \in J} T_{H_{i}} (κ), ⋂_{i \in J} I_{H_{i}} (κ), ⋂_{i \in J} F_{H_{i}} (κ); \end{matrix}$
(ii): the intersection of ${H_{i}}_{i \in J}$ ( $⋂_{i \in J} H_{i}, f o r s h o r t$ ) is an $SVNS$ in $S$ defined as follows: for every $κ \in S$ ,

$\begin{matrix} (⋂_{i \in J} H_{i}) (κ) = (⋂_{i \in J} T_{H_{i}} (κ), ⋃_{i \in J} I_{H_{i}} (κ), ⋃_{i \in J} F_{H_{i}} (κ) . \end{matrix}$

Definition 8

([52]). A single-valued neutrosophic topology on

S

is a map

(τ^{T}, τ^{I}, τ^{F}) : I^{S} \to I

satisfying the following three conditions:

(SVNT1): $τ^{T} (\underset{̲}{0}) = τ^{T} (\underset{̲}{1}) = 1$ and $τ^{I} (\underset{̲}{0}) = τ^{I} (\underset{̲}{1}) = τ^{F} (\underset{̲}{0}) = τ^{F} (\underset{̲}{1}) = 0$ ,
(SVNT2): $τ^{T} (H \cap R) \geq τ^{T} (H) \cap τ^{T} (R)$ , $τ^{I} (H \cap R) \leq τ^{I} (H) \cup τ^{I} (R)$ ,
$τ^{F} (H \cap R) \leq τ^{F} (H) \cup τ^{F} (R)$ , for any $H, R \in I^{S}$ ,
(SVNT3): $τ^{T} (\cup_{i \in j} H_{i}) \geq \cap_{i \in j} τ^{T} (H_{i})$ , $τ^{I} (\cup_{i \in j} H_{i}) \leq \cup_{i \in j} τ^{I} (H_{i})$ ,
$τ^{F} (\cup_{i \in j} H_{i}) \leq \cup_{i \in j} τ^{F} (H_{i})$ , for any ${H_{i}}_{i \in J} \in I^{S}$ .

The pair

(X, τ^{T}, τ^{I}, τ^{F})

is called single-valued neutrosophic topological spaces (

SVNTS, f o r s h o r t

). We will occasionally write

τ^{T I F}

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} \to 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

I_{o}

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} \times I_{0} \to I^{S}

is called a single-valued neutrosophic closure operator on

S

if, for every

H, R \in I^{S}

and

r, s \in I_{0}

, the following axioms are satisfied:

(

C_{1}

)

C ((0.1.1), s) = (0.1.1)

(

C_{2}

)

H \leq C (H, s)

(

C_{3}

)

C (H, s) \lor C (R, s) = C (H \lor R, s)

(

C_{4}

)

C (H, s) \leq C (H, r)

s \leq r

(

C_{5}

)

C (C (H, s), s) = C (H, s)

The pair

(X, C)

is a single-valued neutrosophic closure space (

SVNCS, f o r s h o r t

Suppose that

C_{1}

and

C_{2}

are single-valued neutrosophic closure operators on

S

. Then,

C_{1}

is finer than

C_{2}

, denoted by

C_{2} \leq C_{1}

iff

C_{1} (H, s) \leq C_{2} (H, s)

, for every

H \in I^{S}

and

s \in I_{0}

Theorem 1.

Let

(S, τ^{T I F})

be an

SVNTS

. Then, for any

H \in I^{S}

and

s \in I_{0}

, we define an operator

C_{τ^{T I F}} : I^{S} \times I_{0} \to I^{S}

as follows:

\begin{matrix} C_{τ^{T I F}} (H, s) = ⋀ {R \in I^{X} : H \leq R, τ^{T} (\underset{̲}{1} - R) \geq s, τ^{I} (\underset{̲}{1} - R) \leq 1 - s, τ^{F} (\underset{̲}{1} - R) \leq 1 - s} . \end{matrix}

Then,

(S, C_{τ^{T I F}})

is an

SVNCS

Proof.

Suppose that

(S, τ^{T I F})

is an

SVNTS

. Then,

C_{1}

(C_{2})

and (

C_{4}

) follows directly from the definition of

C_{τ^{T I F}}

(

C_{3}

) Since

R, H \leq H \cup R

C_{τ^{T I F}} (R, s) \leq C_{τ^{T I F}} (H \cup R, s)

and

C_{τ^{T I F}} (H, s) \leq C_{τ^{T I F}} (H \cup R, s)

, therefore,

\begin{matrix} C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (R, s) \leq C_{τ^{T I F}} (H \cup R, s) . \end{matrix}

Let

(X, τ^{T I F})

be an

SVNTS

. From (

C_{2}

), we have

\begin{matrix} H \leq C_{τ^{T I F}} (H, s), & τ^{T} (\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \geq s, τ^{I} (\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \leq 1 - s \\ a n d τ^{F} (\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \leq 1 - s, \end{matrix}

\begin{matrix} R \leq C_{τ^{T I F}} (R, s), & τ^{T} (\underset{̲}{1} - C_{τ^{T I F}} (R, s)) \geq s, τ^{I} (\underset{̲}{1} - C_{τ^{T I F}} (R, s)) \leq 1 - s \\ a n d τ^{F} (\underset{̲}{1} - C_{τ^{T I F}} (R, s)) \leq 1 - s . \end{matrix}

It implies that

H \cup R \leq C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (R, s)

\begin{matrix} τ^{T} (\underset{̲}{1} - (C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (R, s))) & = τ^{T} ((\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \cap (\underset{̲}{1} - C_{τ^{T I F}} (R, s))) \\ \geq τ^{T} (\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \cap τ^{T} (\underset{̲}{1} - C_{τ^{T I F}} (R, s)) \geq s, \end{matrix}

\begin{matrix} τ^{I} (\underset{̲}{1} - (C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (R, s))) & = τ^{I} ((\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \cap (\underset{̲}{1} - C_{τ^{T I F}} (R, s))) \\ \leq τ^{I} ((\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \cup τ^{I} (\underset{̲}{1} - C_{τ^{T I F}} (R, s)) \leq 1 - s, \end{matrix}

\begin{matrix} τ^{F} (\underset{̲}{1} - (C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (R, s))) & = τ^{F} ((\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \cap (\underset{̲}{1} - C_{τ^{T I F}} (R, s))) \\ \leq τ^{F} (\underset{̲}{1} - C_{τ^{T I F}} (H, s)) \cup τ^{F} (\underset{̲}{1} - C_{τ^{T I F}} (R, s)) \leq 1 - s . \end{matrix}

Hence,

C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (H \cup R, s) \geq C_{τ^{T I F}} (H \cup R, s)

. Therefore,

\begin{matrix} C_{τ^{T I F}} (H, s) \cup C_{τ^{T I F}} (H \cup R, s) = C_{τ^{T I F}} (H \cup R, s) . \end{matrix}

(

C_{5}

) Suppose that there exists

s \in I_{0}

H \in I^{S}

, and

κ \in S

such that

\begin{matrix} C_{τ^{T I F}} (C_{τ^{T I F}} (H, s), s) (κ) > C_{τ^{T I F}} (H, s) (κ) . \end{matrix}

By the definition of

C_{τ^{T I F}}

, there exists

D \in I^{S}

with

D \geq H

, and

τ^{T} (\underset{̲}{1} - D) \geq s

τ^{I} (\underset{̲}{1} - D) \leq 1 - s

and

τ^{F} (\underset{̲}{1} - D) \leq 1 - s

such that

\begin{matrix} C_{τ^{T I F}} (C_{τ^{T I F}} (H, s), s) (κ) > D (κ) \geq C_{τ^{T I F}} (H, s) (κ) . \end{matrix}

Since

C_{τ^{T I F}} (H, s) \leq D

and

τ^{T} (\underset{̲}{1} - D) \geq s

τ^{I} (\underset{̲}{1} - D) \leq 1 - s

, and

τ^{F} (\underset{̲}{1} - D) \leq 1 - s

, by the definition of

C_{τ^{T I F}} (C_{τ^{T I F}})

, we have

\begin{matrix} C_{τ^{T I F}} (C_{τ^{T I F}} (H, s), s) \leq D . \end{matrix}

It is a contradiction. Thus,

C_{τ^{T I F}} (C_{τ^{T I F}} (H, s), s) = C_{τ^{T I F}} (H, s)

. Hence,

C_{τ^{T I F}}

is a single-valued neutrosophic closure operator on

S

. □

Theorem 2.

Let

(S, C)

be an

SVNCS

and

H \in S

. Define the mapping

τ_{C}^{T I F} : I^{S} \to I

S

\begin{matrix} τ_{C}^{T} (H) = ⋃ {s \in I_{0} ∣ C (\bar{1} - H, s) = \bar{1} - H}, \end{matrix}

\begin{matrix} τ_{C}^{I} (H) = ⋂ {1 - s \in I_{0} ∣ C (\bar{1} - H, s) = \bar{1} - H}, \end{matrix}

\begin{matrix} τ_{C}^{F} (H) = ⋂ {1 - s \in I_{0} ∣ C (\bar{1} - H, s) = \bar{1} - H}, \end{matrix}

Then,

(1): $τ_{C}^{T I F}$ is an $SVNTS$ on $S$ ;
(2): $C_{τ_{C}^{T I F}}$ is finer than $C$ .

Proof.

(SVNT1) Let

(S, C)

be an

SVNCS

. Since

C ((0.1.1), r) = (0.1.1)

and

C (1, 0, 0), r) = (1, 0, 0)

for every

s \in I_{0}

, (SVNT1).

(SVNT2) Let

(S, C)

be an

SVNCS

. Suppose that there exists

H_{1}, H_{2} \in I^{S}

such that

\begin{matrix} τ_{C}^{T} (H_{1} \cap H_{2}) < τ_{C}^{T} (H_{1}) \cap τ_{C}^{T} (H_{2}), τ_{C}^{I} (H_{1} \cap H_{2}) > τ_{C}^{I} (H_{1}) \cup τ_{C}^{I} (H_{2}), \end{matrix}

\begin{matrix} τ_{C}^{F} (H_{1} \cap H_{2}) > τ_{C}^{F} (H_{1}) \cup τ_{C}^{F} (H_{2}) . \end{matrix}

There exists

s \in I_{0}

such that

\begin{matrix} τ_{C}^{T} (H_{1} \cap H_{2}) < s < τ_{C}^{T} (H_{1}) \cap τ_{C}^{T} (H_{2}), τ_{C}^{I} (H_{1} \cap H_{2}) > 1 - s > τ_{C}^{I} (H_{1}) \cup τ_{C}^{I} (H_{2}), \end{matrix}

\begin{matrix} τ_{C}^{F} (H_{1} \cap H_{2}) > 1 - s > τ_{C}^{F} (H_{1}) \cup τ_{C}^{F} (H_{2}) . \end{matrix}

For each

i \in {1, 2}

, there exists

s \in I_{0}

with

C (H_{i}, s_{i}) = \bar{1} - H_{i}

such that

\begin{matrix} s < s_{i} \leq τ_{C}^{T} (H_{i}), τ_{C}^{I} (H_{i}) \leq 1 - s_{i} < 1 - s, τ_{C}^{F} (H_{i}) \leq 1 - s_{i} < 1 - s . \end{matrix}

In addition, since

(\bar{1} - H_{i}, r) = \bar{1} - H_{i}

C_{2}

and

C_{4}

of Definition 9, for any

i \in {1, 2}

\begin{matrix} C ((\bar{1} - H_{1}) \cup (\bar{1} - H_{2}), s) = (\bar{1} - H_{1}) \cup (\bar{1} - H_{2}) . \end{matrix}

It follows that

τ_{C}^{T} (H_{1} \cap H_{2}) \geq s

τ_{C}^{I} (H_{1} \cap H_{2}) \leq 1 - s

, and

τ_{C}^{F} (H_{1} \cap H_{2}) \leq 1 - s

. It is a contradiction. Thus, for every

H, R \in I^{S}

τ_{C}^{T} (H \cap R) \geq τ_{C}^{T} (H) \cap τ_{C}^{T} (B)

τ_{C}^{I} (H \cap R) \leq τ_{C}^{I} (H) \cup τ_{C}^{I} (R)

, and

τ_{C}^{F} (H \cap R) \leq τ_{C}^{F} (H) \cup τ_{C}^{F} (R)

(SVNT3) Suppose that there exists

H = ⋃_{i \in I} H_{i} \in I^{S}

such that

\begin{matrix} τ_{C}^{T} (H) < ⋃_{i \in I} τ_{C}^{T} (H_{i}), τ_{C}^{I} (H) > ⋃_{i \in I} τ_{C}^{I} (H_{i}), τ_{C}^{F} (H) > ⋃_{i \in I} τ_{C}^{F} (H_{i}) . \end{matrix}

There exists

s_{0} \in I_{0}

such that

\begin{matrix} τ_{C}^{T} (H) < s_{0} < ⋃_{i \in I} τ_{C}^{T} (H_{i}), τ_{C}^{I} (H) > 1 - s_{0} > ⋃_{i \in I} τ_{C}^{I} (H_{i}), τ_{C}^{F} (H) > 1 - s_{0} > ⋃_{i \in I} τ_{C}^{F} (H_{i}) . \end{matrix}

For every

i \in I

, there exists

C (H_{i}, s_{i}) = \bar{1} - H_{i}

and

s_{i} \in I_{0}

such that

\begin{matrix} s_{0} < s_{i} \leq τ_{C}^{T} (H_{i}), 1 - s_{0} > 1 - s_{i} \geq τ_{C}^{I} (H_{i}), 1 - s_{i} > 1 - s_{0} \geq τ_{C}^{F} (H_{i}) . \end{matrix}

In addition, since

C (\bar{1} - H_{i}, r_{0}) \leq C (\bar{1} - H_{i}, s_{i}) = \bar{1} - H_{i}

, by

C_{2}

of Definition 9,

\begin{matrix} C (\bar{1} - H_{i}, s_{0}) = \bar{1} - H_{i} . \end{matrix}

It implies, for all

i \in I

\begin{matrix} C (\bar{1} - H, s_{0}) \leq C (\bar{1} - H_{i}, s_{0}) = \bar{1} - H_{i} . \end{matrix}

It follows that

\begin{matrix} C (\bar{1} - H, r_{0}) \leq ⋂_{i \in J} (\bar{1} - H_{i}) = \bar{1} - H . \end{matrix}

Thus,

CI (\bar{1} - H, s_{0}) = \bar{1} - H

, that is,

τ_{C}^{T} (H) \geq s_{0}

τ_{C}^{I} (H) \leq 1 - s_{0}

, and

τ_{C}^{F} (H) \leq 1 - s_{0}

. It is a contradiction. Hence,

τ_{C}^{T I F}

is an

SVNTS

S

(2) Since

H \leq C (H, r)

\begin{matrix} τ_{C}^{T} (\bar{1} - C (H, s)) \geq s, τ_{C}^{I} (\bar{1} - C (H, s)) \leq 1 - s, τ_{C}^{F} (\bar{1} - C (H, s)) \leq 1 - s . \end{matrix}

From

C_{5}

of Definition 9, we have

C_{τ_{C}^{T I F}} (H, s) \leq C (H, s)

. Thus,

C_{τ_{C}^{T I F}}

is finer than

C

. □

Example 1.

Let

S = {a, b}

. Define

B, H, A \in I^{S}

as follows:

\begin{matrix} B = 〈 (0.2, 0.2), (0.3, 0.3), (0.3, 0.3) 〉; H = 〈 (0.5, 0.5), (0.1, 0.1), (0.1, 0.1) 〉 . \end{matrix}

We define the mapping

C : I^{S} \times I_{0} \to I^{S}

as follows:

\begin{matrix} C (A, s) = \{\begin{matrix} (0.1.1), i f A = (0.1.1), s \in I_{0}, \\ B \cap H, i f 0 \neq A \leq B \cap H, 0 < r < \frac{1}{2}, \\ B, i f A \leq B, A ≰ H, 0 < r < \frac{1}{2}, \\ o r 0 \neq A \leq B \frac{1}{2} < r < \frac{2}{3}, \\ H, i f A \leq H, A ≰ B, 0 < r < \frac{1}{2}, \\ B \cup H, i f 0 \neq A \leq B \cup H, 0 < r < \frac{1}{2}, \\ \bar{1}, otherwise . \end{matrix} \end{matrix}

Then,

C

is a single-valued neutrosophic closure operator.

From Theorem 2, we have a single-valued neutrosophic topology

(τ_{C}^{T}, τ_{C}^{I}, τ_{C}^{F})

S

as follows:

\begin{matrix} τ_{C}^{T} (A) = \{\begin{matrix} 1, i f A = (1, 0, 0) o r (0, 1, 1), \\ \frac{2}{3}, i f A = B^{c}, \\ \frac{1}{2}, i f A = H^{c}, \\ \frac{1}{2}, i f A = B^{c} \cup H^{c}, \\ \frac{1}{2}, i f A = B^{c} \cap H^{c}, \\ 0, otherwise . \end{matrix} \end{matrix}

\begin{matrix} τ_{C}^{I} (A) = \{\begin{matrix} 0, i f A = (1, 0, 0) o r (0, 1, 1), \\ \frac{1}{3}, i f A = B^{c}, \\ \frac{1}{2}, i f A = H^{c}, \\ \frac{1}{2}, i f A = B^{c} \cup H^{c}, \\ \frac{1}{2}, i f A = B^{c} \cap H^{c}, \\ 1, otherwise . \end{matrix} \end{matrix}

\begin{matrix} τ_{C}^{F} (A) = \{\begin{matrix} 0, i f A = (1, 0, 0) o r (0, 1, 1), \\ \frac{1}{3}, i f A = B^{c}, \\ \frac{1}{2}, i f A = H^{c}, \\ \frac{1}{2}, i f A = B^{c} \cup H^{c}, \\ \frac{1}{2}, i f A = B^{c} \cap H^{c}, \\ 1, otherwise . \end{matrix} \end{matrix}

Thus, the

τ_{C}^{T I F}

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} (\underset{̲}{0}) = 1

and

L^{I} (\underset{̲}{0}) = L^{F} (\underset{̲}{0}) = 0

(L_{2})

H \leq B,

then

L^{T} (R) \leq L^{T} (H)

L^{I} (R) \geq L^{I} (H)

, and

L^{F} (R) \geq L^{F} (H)

, for each single-valued neutrosophic set

R, H

I^{S}

(L_{3})

L^{T} (R \cup H) \geq L^{T} (R) \cap L^{T} (H)

L^{I} (R \cup H) \leq L^{I} (R) \cup L^{I} (H)

, and

L^{F} (R \cup H) \leq L^{F} (R) \cup L^{F} (H)

, for each single-valued neutrosophic set

R, H

I^{S}

L_{1}

and

L_{2}

are

SVNI

S

, we say that

L_{1}

is finer than

L_{2}

, denoted by

L_{1} \leq L_{2}

, iff

L_{1}^{T} (H) \leq L_{2}^{T} (H)

L_{1}^{I} (H) \geq L_{2}^{I} (A)

, and

L_{1}^{F} (H) \geq L_{2}^{F} (H)

, for

H \in 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 o r s h o r t

We will occasionally write

L^{T I F}

L_{i}^{T I F}

, and

L^{T I F} : I^{X} \to I

for

(L^{T}, L^{I}, L^{F})

(L_{i}^{T}, L_{i}^{I}, L_{i}^{F})

, and

L^{T}, L^{I}, L^{F} : I^{S} \to 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 \cup R) = L^{T} (H) \cap L^{T} (R)

L^{I} (H \cup R) \neq L^{I} (H) \cup L^{I} (R)

, and

L^{F} (H \cup R) \neq L^{F} (H) \cup L^{F} (R)

, for every

R, H \in 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} \to I

as follows:

\begin{matrix} R = 〈 (0.3, 0.5), (0.4, 0.5), (0.5, 0.5) 〉; C = 〈 (0.3, 0.4), (0.5, 0.5), (0.3, 0.4) 〉, \end{matrix}

\begin{matrix} H = 〈 (0.1, 0.2), (0.5, 0.5), (0.5, 0.5) 〉 . \end{matrix}

\begin{matrix} L^{T} (A) = \{\begin{matrix} 1, i f B = (0.1.1), \\ \frac{1}{2}, i f A = R, \\ \frac{2}{3}, i f (0.1.1) < A < R, \\ 0, otherwise . \end{matrix} \end{matrix}

\begin{matrix} L^{I} (A) = \{\begin{matrix} 0, i f A = (0.1.1), \\ \frac{1}{2}, i f A = C, \\ \frac{1}{4}, i f (0.1.1) < A < C, \\ 1, otherwise . \end{matrix} \end{matrix}

\begin{matrix} L^{T} (B) = \{\begin{matrix} 0, i f A = (0, 1, 1), \\ \frac{1}{2}, i f A = H, \\ \frac{1}{4}, i f (0.1.1) < A < H, \\ 1, otherwise . \end{matrix} \end{matrix}

Then,

L^{T I F}

is an

SVNI

S

Remark 2.

(i): If $L^{T} (\underset{̲}{1}) = 1$ , $L^{I} (\underset{̲}{1}) = 0$ , and $L^{F} (\underset{̲}{1}) = 0$ , then $L^{T I F}$ is called a single-valued neutrosophic proper ideal.
(ii): If $L^{T} (\underset{̲}{1}) = 0$ , $L^{I} (\underset{̲}{1}) = 1$ , and $L^{F} (\underset{̲}{1}) = 1$ , then $L^{T I F}$ is called a single-valued neutrosophic improper ideal.

Proposition 1.

Let

{L_{i}^{T I F}}_{i \in J}

be a family

o f SVNI o n S

. Then, their intersection

⋂_{i \in J} L_{i}^{T I F}

is also

SVNI

Proof.

Directly from Definition 7. □

Proposition 2.

Let

{L_{i}^{T I F}}_{i \in J}

be a family

o f SVNI o n S

. Then, their union

⋃_{i \in J} L_{i}^{T I F}

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 \in I_{0}

and

s + t + p \leq 3

. A single-valued neutrosophic point

x_{s, t, r}

S

is the single-valued neutrosophic set in

I^{S}

for each

κ \in H

, defined by

\begin{matrix} x_{s, t, p} (κ) = \{\begin{matrix} (s, t, p), i f x = κ, \\ (0, 1, 1), i f x \neq κ . \end{matrix} \end{matrix}

A single-valued neutrosophic point

x_{s, t, p}

is said to belong to a single-valued neutrosophic set

H = 〈 T_{H}, I_{H}, F_{H} 〉 \in I^{S}

, denoted by

x_{s, t, p} \in H

iff

s < T_{H}

t \geq I_{H}

and

p \geq F_{H}

. 1. We indicate the set of all single-valued neutrosophic points in

S

SVNP (S)

For every

x_{s, t, p} \in SVNP (S)

and

H \in I^{S}

we shall write

x_{s, t, p}

quasi-coincident with

H

, denoted by

x_{s, t, p} q H

, if

\begin{matrix} s + T_{H} (κ) > 1, t + I_{H} (κ) \leq 1, p + F_{H} (κ) \leq 1 . \end{matrix}

For every

R, H \in S

we shall write

H \bar{q} R

to mean that

H

is quasi-coincident with

R

if there exists

κ \in S

such that

\begin{matrix} T_{H} (κ) + T_{R} (κ) > 1, I_{H} (κ) + I_{R} (κ) \leq 1, F_{H} (κ) + F_{R} (κ) \leq 1 . \end{matrix}

Definition 12.

Let

(S, τ^{T I F})

be an

SVNTS

. For each

r \in I_{0}

H \in I^{S}

x_{s, t, p} \in SVNP (S)

, a single-valued neutrosophic open

Q_{τ^{T I F}}

-neighborhood of

x_{s, t, p}

is defined as follows:

\begin{matrix} Q_{τ^{T I F}} (x_{s, t, p}, r) = {H | (x_{s, t, p}) q H, τ^{T} (H) \geq r, τ^{I} (H) \leq 1 - r, τ^{F} (H) \leq 1 - r} . \end{matrix}

Lemma 1.

A single-valued neutrosophic point

x_{s, t, p} \in C_{τ^{T I F}} (R, r)

iff every single-valued neutrosophic open

Q_{τ^{T I F}}

-neighborhood of

x_{s, t, p}

is quasi-coincident with

H

Definition 13.

Let

(S, τ^{T I F})

be an

SVNTS

for each

H \in I^{S}

. Then, the single-valued neutrosophic ideal open local function

H_{r}^{⭑} (τ^{T I F}, L^{T I F})

H

is the union of all single-valued neutrosophic points

x_{s, t, p}

such that if

R \in Q_{τ^{T I F}} (x_{s, t, p}, r)

and

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

, then there is at least one

κ \in S

for which

T_{R} (κ) + T_{H} (κ) - 1 > T_{C} (κ)

I_{R} (κ) + I_{H} (κ) - 1 \leq I_{C} (κ)

, and

F_{R} (κ) + F_{H} (κ) - 1 \leq F_{C} (κ)

Occasionally, we will write

H_{r}^{⭑}

for

H_{r}^{⭑} (τ^{T I F}, L^{T I F})

and it will have no ambiguity.

Example 3.

Let

(S, τ^{T I F}, L^{T I F})

be an

SVNITS

. The simplest single-valued neutrosophic ideal on

S

L_{0}^{T I F} : I^{S} \to I

, where

\begin{matrix} L_{0}^{T I F} (R) = \{\begin{matrix} 1, i f R = (1, 0, 0), \\ 0, otherwise . \end{matrix} \end{matrix}

If we take

L^{T I F} = L_{0}^{T I F}

, for each

H \in I^{S}

we have

H_{r}^{⭑} = C_{τ^{T I F}} (H, r)

Theorem 3.

Let

(S, τ^{T I F})

be an

SVNTS

and

L_{1}^{T I F}, L_{2}^{T I F} \in SVNI (S)

. Then, for any

H, R \in I^{S}

and

r \in I_{0}

, we have

(1): If $H \leq R,$ then $H_{r}^{⭑} \leq R_{r}^{⭑}$ ;
(2): If $L_{1}^{T} \leq L_{2}^{T}$ , $L_{1}^{I} \geq L_{2}^{I}$ and $L_{1}^{F} \geq L_{2}^{F}$ , then $H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \geq H_{r}^{⭑} ((L_{2}^{T I F}, τ^{T I F})$ ;
(3): $H_{r}^{⭑} = C_{τ^{T I F}} (A_{r}^{⭑}, r) \leq C_{τ^{T I F}} (H, r)$ ;
(4): ${(H_{r}^{⭑})}_{r}^{⭑} \leq H_{r}^{⭑}$ ;
(5): $(H_{r}^{⭑} \lor R_{r}^{⭑}) = {(H \lor R)}_{r}^{⭑}$ ;
(6): If $L^{T} (H) \geq r$ , $L^{I} (R) \leq 1 - r$ , and $L^{F} (R) \leq 1 - r$ then ${(H \lor R)}_{r}^{⭑} = A_{r}^{⭑} \lor R_{r}^{⭑} = H_{r}^{⭑}$ ;
(7): If $τ^{T} (R) \geq r$ , $τ^{I} (R) \leq 1 - r$ , and $τ^{F} (R) \leq 1 - r$ , then $(R \land H_{r}^{⭑}) \leq {(R \land H)}_{r}^{⭑}$ ;
(8): $(H_{r}^{⭑} \land R_{r}^{⭑}) \geq {(H \land R)}_{r}^{⭑}$ .

Proof.

(1) Suppose that

H \in I^{S}

and

H_{r}^{⭑} \leq R_{r}^{⭑}

. Then, there exists

κ \in S

and

s, t, p \in I_{0}

such that

\begin{matrix} T_{H_{r}^{⭑}} (κ) \geq s > T_{R_{r}^{⭑}} (κ), I_{H_{r}^{⭑}} (κ) < t \leq I_{R_{r}^{⭑}} (κ), F_{H_{r}^{⭑}} (κ) < p \leq F_{R_{r}^{⭑}} (κ) . \end{matrix}

(1)

Since

T_{R_{r}^{⭑}} (κ) < s

I_{R_{r}^{⭑}} (κ) \geq t

, and

F_{R_{r}^{⭑}} (κ) \geq p

. Then, there exists

D \in Q_{(τ^{T I F})} (x_{s, t, p}, r)

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

, and

L^{F} (C) \leq 1 - r

such that for any

κ_{1} \in S

\begin{matrix} T_{D} (κ_{1}) + T_{R} (κ_{1}) - 1 \leq T_{C} (κ_{1}), I_{D} (κ_{1}) + I_{R} (κ_{1}) - 1 > I_{C} (κ_{1}), F_{D} (κ_{1}) + F_{R} (κ_{1}) - 1 > F_{C} (κ_{1}) . \end{matrix}

Since

H \leq R

\begin{matrix} T_{D} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C} (κ_{1}), I_{D} (κ_{1}) + I_{H} (κ_{1}) - 1 > I_{C} (κ_{1}), F_{D} (κ_{1}) + F_{H} (κ_{1}) - 1 > F_{C} (κ_{1}) . \end{matrix}

So,

T_{H_{r}^{⭑}} (κ) < s

I_{H_{r}^{⭑}} (κ) \geq t

, and

F_{H_{r}^{⭑}} (κ) \geq p

and we arrive at a contradiction for Equation (1). Hence,

H_{r}^{⭑} \leq R_{r}^{⭑} .

(2) Suppose

H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) ≱ H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})

. Then, there exists

s, t, p \in I_{0}

and

κ \in S

such that

\begin{matrix} T_{H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F})} (κ) < s \leq T_{H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})} (κ), \end{matrix}

\begin{matrix} I_{H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F})} (κ) \geq t > I_{H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})} (κ), \end{matrix}

(2)

\begin{matrix} F_{H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F})} (κ) \geq p > F_{H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})} (κ) . \end{matrix}

Since

T_{H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F})} (κ) < s

I_{H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F})} (κ) \geq t

, and

F_{H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F})} (κ) \geq p

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

with

L_{1}^{T} (C) \geq r

L_{1}^{I} (C) \leq 1 - r

and

L_{1}^{F} (C) \leq 1 - r

. Thus, for every

κ_{1} \in S

\begin{matrix} T_{D} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C} (κ_{1}), I_{D} (κ_{1}) + I_{H} (κ_{1}) - 1 > I_{C} (κ_{1}), F_{D} (κ_{1}) + F_{H} (κ_{1}) - 1 > F_{C} (κ_{1}) . \end{matrix}

Since

L_{2}^{T} (C) \geq L_{1}^{T} (C)) \geq r

L_{2}^{I} (C) \leq L_{1}^{I} (C)) \leq 1 - r

, and

L_{2}^{F} (C) \leq L_{1}^{F} (C)) \leq 1 - r

\begin{matrix} T_{D} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C} (κ_{1}), I_{D} (κ_{1}) + I_{H} (κ_{1}) - 1 > I_{C} (κ_{1}), F_{D} (κ_{1}) + F_{H} (κ_{1}) - 1 > F_{C} (κ_{1}) . \end{matrix}

Thus,

T_{H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})} (κ) < s

I_{H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})} (κ) \geq t

, and

F_{H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})} (κ) \geq p

. This is a contradiction for Equation (2). Hence,

H_{r}^{⭑} ((L_{1}^{T I F}, τ^{T I F})) \geq H_{r}^{⭑} ((L_{2}^{T I F}, τ^{T I F}))

(3)

(\Rightarrow)

Suppose

H_{r}^{⭑} \leq C_{τ^{T I F}} (H, r)

. Then, there exists

s, t, p \in I_{0}

and

κ \in S

such that

\begin{matrix} T_{H_{r}^{⭑}} (κ) \geq s > T_{C_{τ^{T I F}} (H, r)} (κ), I_{H_{r}^{⭑}} (κ) < t \leq I_{C_{τ^{T I F}} (H, r)} (κ), F_{H_{r}^{⭑}} (κ) < p \leq F_{C_{τ^{T I F}} (H, r)} (κ) . \end{matrix}

(3)

Since

T_{H_{r}^{⭑}} (κ) \geq s

I_{H_{r}^{⭑}} (κ) < t

and

F_{H_{r}^{⭑}} (κ) < p

x_{s, t, p} \in H_{r}^{⭑}

. So there is at least one

κ_{1} \in S

for every

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

with

L_{1}^{T} (C) \geq r

L_{1}^{I} (C) \leq 1 - r

L_{1}^{F} (C) \leq 1 - r

such that

\begin{matrix} T_{D} (κ_{1}) + T_{H} (κ_{1}) > T_{C} (κ_{1}) + 1, I_{D} (κ_{1}) + I_{H} (κ_{1}) \leq I_{C} (κ_{1}) + 1, F_{D} (κ_{1}) + F_{H} (κ_{1}) \leq F_{C} (κ_{1}) + 1 . \end{matrix}

Therefore, by Lemma 1,

x_{s, t, p} \in C_{τ^{T I F}} (H, r)

which is a contradiction for Equation (3). Hence,

H_{r}^{⭑} \leq C_{τ^{T I F}} (H, r)

(\Leftarrow)

Suppose

H_{r}^{⭑} \geq C_{τ^{T I F}} (H_{r}^{⭑}, r)

. Then, there exists

s, t, p \in I_{0}

and

κ \in S

such that

\begin{matrix} T_{H_{r}^{⭑}} (κ) < s \leq T_{C_{τ^{T I F}} (H_{r}^{⭑}, r)} (κ), I_{H_{r}^{⭑}} (κ) \geq t > I_{C_{τ^{T I F}} (H_{r}^{⭑}, r)} (κ), F_{H_{r}^{⭑}} (κ) \geq p > F_{C_{τ^{T I F}} (H_{r}^{⭑}, r)} (κ) . \end{matrix}

(4)

Since

T_{C_{τ^{T I F}} (H_{r}^{⭑}, r)} (κ) \geq t

I_{C_{τ^{T I F}} (H_{r}^{⭑}, r)} (κ) < s

C_{τ^{T I F}} (H_{r}^{⭑}, r) (κ) < p

we have

x_{s, t, p} \in C_{τ^{T I F}} (H_{r}^{⭑}, r) .

So, there is at least one

κ_{1} \in S

with

R \in Q_{τ^{T I F}} (x_{s, t, p}, r)

such that

\begin{matrix} T_{R} (κ_{1}) + T_{H_{r}^{⭑}} (κ_{1}) > 1, I_{R} (κ_{1}) + I_{H_{r}^{⭑}} (κ_{1}) \leq 1, F_{R} (κ_{1}) + F_{H_{r}^{⭑}} (κ_{1}) \leq 1 . \end{matrix}

Therefore,

H_{r}^{⭑} (κ_{1}) \neq 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}} \in H_{r}^{*}

and

s_{1} + T_{R} (κ_{1}) > 1

t_{1} + I_{R} (κ_{1}) \leq 1

, and

p_{1} + F_{R} (κ_{1}) \leq 1

so that

R \in Q_{τ^{T I F}} ({(κ_{1})}_{s_{1}, t_{1}, p_{1}}, r)

. Now,

{(κ_{1})}_{s_{1}, t_{1}, p_{1}} \in H_{r}^{⭑}

implies there is at least one

κ^{^{'}} \in S

such that

T_{D} (κ^{^{'}}) + T_{H} (κ^{^{'}}) - 1 > T_{C} (κ^{^{'}})

I_{D} (κ^{^{'}}) + I_{H} (κ^{^{'}}) - 1 \leq I_{C} (κ^{^{'}})

, and

F_{D} (κ^{^{'}}) + F_{H} (κ^{^{'}}) - 1 \leq F_{C} (κ^{^{'}})

, for all

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

, and

D \in Q_{τ^{T I F}} ({(κ_{1})}_{s_{1}, t_{1}, p_{1}}, r)

. That is also true for

R

. So there is at least one

κ^{^{' ″}} \in S

such that

T_{R} (κ^{^{″}}) + T_{H} (κ^{^{″}}) - 1 > T_{C} (κ^{^{″}})

I_{R} (κ^{^{″}}) + I_{H} (κ^{^{″}}) - 1 \leq I_{C} (κ^{^{″}})

, and

F_{R} (x^{^{″}}) + F_{H} (κ^{^{″}}) - 1 \leq F_{C} (κ^{^{″}})

. Since

R \in Q_{τ^{T I F}} (κ_{s, t, p}, r)

and

R

is arbitrary; then

T_{H_{r}^{⭑}} (κ) > s

I_{H_{r}^{⭑}} (κ) \leq t

and

T_{H_{r}^{⭑}} (κ) \leq p

. It is a contradiction for (4). Thus,

H_{r}^{⭑} \geq C_{τ^{T I F}} (H_{r}^{⭑}, r)

(4)

(\Rightarrow)

Can be easily established using standard technique.

(5)

(\Rightarrow)

Since

H, R \leq H \cup R .

By (1),

H_{r}^{⭑} \leq {(H \cup R)}_{r}^{⭑}

and

R_{r}^{⭑} \leq {(H \cup R)}_{r}^{⭑}

. Hence,

H_{r}^{⭑} \cup B_{r}^{⭑} \leq {(H \cup R)}_{r}^{⭑}

(\Leftarrow)

Suppose

(H_{r}^{⭑} \cup R_{r}^{⭑}) ≱ {(H \cup R)}_{r}^{⭑}

. Then, there exists

s, t, p \in I_{0}

and

κ \in S

such that

\begin{matrix} T_{(H_{r}^{⭑} \cup R_{r}^{⭑})} (κ) < s \leq T_{{(H \cup R)}_{r}^{⭑}} (κ), I_{(H_{r}^{⭑} \cup R_{r}^{⭑})} (κ) \geq t > I_{{(H \cup R)}_{r}^{⭑}} (κ), F_{(H_{r}^{⭑} \cup R_{r}^{⭑})} (κ) \geq p > F_{{(H \cup R)}_{r}^{⭑}} (κ) . \end{matrix}

(5)

Since

T_{(H_{r}^{⭑} \cup R_{r}^{⭑})} (κ) < s

I_{(H_{r}^{⭑} \cup R_{r}^{⭑})} (κ) \geq t

, and

F_{(H_{r}^{⭑} \cup R_{r}^{⭑})} (κ) \geq p

, we have

T_{H_{r}^{⭑}} (κ) < s

I_{H_{r}^{⭑}} (κ) \geq t

F_{H_{r}^{⭑}} (κ) \geq p

T_{R_{r}^{⭑}} (κ) < t

I_{R_{r}^{⭑}} (κ) \geq t

F_{R_{r}^{⭑}} (κ) \geq t

. So, there exists

D_{1} \in Q_{τ^{T I F}} (x_{s, t, p}, r)

such that for every

κ_{1} \in S

and for some

L^{T} (C_{1}) \geq r

L^{I} (C_{1}) \leq 1 - r

L^{F} (C_{1}) \leq 1 - r

, we have

\begin{matrix} T_{D_{1}} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C_{1}} (κ_{1}), I_{D_{1}} (κ_{1}) + I_{H} (κ_{1}) - 1 > I_{C_{1}} (κ_{1}), F_{D_{1}} (κ_{1}) + F_{H} (κ_{1}) - 1 > F_{C_{1}} (κ_{1}) . \end{matrix}

Similarly, there exists

D_{2} \in Q_{τ^{T I F}} (x_{s, t, p}, r)

such that for every

κ_{1} \in S

and for some

L^{T} (C_{2}) \geq r

L^{I} (C_{2}) \leq 1 - r

L^{F} (C_{2}) \leq 1 - r

, we have

\begin{matrix} T_{D_{2}} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C_{2}} (κ_{1}), I_{D_{2}} (κ_{1}) + I_{H} (κ_{1}) - 1 > I_{C_{2}} (κ_{1}), F_{D_{2}} (κ_{1}) + F_{H} (κ_{1}) - 1 > F_{C_{2}} (κ_{1}) . \end{matrix}

Since

D = D_{1} \land D_{2} \in Q_{τ^{T I F}} (x_{s, t, p}, r)

and by (

L_{3}

L^{T} (C_{1} \cup C_{2}) \geq L^{T} (C_{1}) \cap L^{T} (C_{2}) \geq r

L^{I} (C_{1} \cup C_{2}) \leq L^{I} (C_{1}) \cup L^{I} (C_{2}) \leq 1 - r

, and

L^{F} (C_{1} \cup C_{2}) \leq L^{T} (C_{1}) \cup L^{T} (C_{2}) \leq 1 - r

. Thus, for every

κ_{1} \in S

\begin{matrix} T_{D} (κ_{1}) + T_{R \cup H} (κ_{1}) - 1 \leq T_{C_{1} \cup C_{2}} (κ_{1}), \\ I_{D} (κ_{1}) + I_{R \cup H} (κ_{1}) - 1 \geq I_{C_{1} \cup C_{2}} (κ_{1}), \\ F_{D} (κ_{1}) + F_{R \cup H} (κ_{1}) \geq F_{C_{1} \cup C_{2}} (κ_{1}) . \end{matrix}

Therefore,

T_{{(H \cup R)}_{r}^{⭑}} (κ) < s

I_{{(H \cup R)}_{r}^{⭑}} (κ) \geq t

, and

F_{{(H \cup R)}_{r}^{⭑}} (κ) \geq p

. So, we arrive at a contradiction for (5). Hence,

(H_{r}^{⭑} \cup R_{r}^{⭑}) \geq {(H \cup R)}_{r}^{⭑}

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

Example 4.

Let

S = {a, b}

. Define

R, C, H \in S

as follows:

\begin{matrix} R_{1} = 〈 (0.5, 0.5, 0.5), (0.5, 0.5, 0.5), (0.5, 0.5, 0.5) 〉; R_{2} = 〈 (0.4, 0.4, 0.4), (0.1, 0.1, 0.1), (0.1, 0.1, 0.1) 〉; \end{matrix}

\begin{matrix} R_{3} = 〈 (0.3, 0.3, 0.3), (0.1, 0.1, 0.1), (0.1, 0.1, 0.1); C_{1} = 〈 (0.3, 0.3, 0.3), (0.3, 0.3, 0.3), (0.1, 0.1, 0.1) 〉; \end{matrix}

\begin{matrix} C_{2} = 〈 (0.2, 0.2, 0.2), (0.2, 0.2, 0.2), (0.1, 0.1, 0.1); C_{3} = 〈 (0.1, 0.1, 0.1), (0.1, 0.1, 0.1), (0.1, 0.1, 0.1) 〉 . \end{matrix}

Define

τ^{T I F}, L^{T I F} : I^{X} \to I

as follows:

\begin{matrix} τ^{T} (H) = \{\begin{matrix} 1, i f H = (0, 1, 1), \\ 1, i f H = (1, 0, 0), \\ \frac{1}{2}, i f H = R_{1}; \end{matrix} L^{T} (H) = \{\begin{matrix} 1, i f H = (0, 1, 1), \\ \frac{1}{2}, i f H = C_{1}, \\ \frac{2}{3}, i f \underset{̲}{0} < H < C_{1}; \end{matrix} \end{matrix}

\begin{matrix} τ^{I} (H) = \{\begin{matrix} 0, i f H = (0, 1, 1), \\ 0, i f H = (1, 0, 0), \\ \frac{1}{2}, i f H = R_{2}; \end{matrix} L^{I} (R) = \{\begin{matrix} 0, i f H = (0, 1, 1), \\ \frac{1}{2}, i f H = C_{2}, \\ \frac{1}{4}, i f \underset{̲}{0} < H < C_{2}; \end{matrix} \end{matrix}

\begin{matrix} τ^{F} (H) = \{\begin{matrix} 0, i f H = (0, 1, 1), \\ 0, i f H = (1, 0, 0), \\ \frac{1}{2}, i f H = R_{3}; \end{matrix} L^{F} (H) = \{\begin{matrix} 0, i f H = (0, 1, 1), \\ \frac{1}{2}, i f H = C_{3}, \\ \frac{1}{4}, i f \underset{̲}{0} < H < C_{3} . \end{matrix} \end{matrix}

Let

G = 〈 (0.4, 0.4, 0.4), (0.4, 0.4, 0.4), (0.4, 0.4, 0.4) 〉

. Then,

G_{\frac{1}{2}}^{⭑} = R_{1}

Theorem 4.

Let

{H_{i}}_{i \in J} \subset I^{S}

be a family of single-valued neutrosophic sets on

S

and

(S, τ^{T I F}, L^{T I F})

be an

SVNITS

. Then,

(1): $(⋃ {(H_{i})}_{r}^{⭑} : i \in J) \leq {(⋃ H_{i} : i \in J)}_{r}^{⭑}$ ;
(2): $(⋂ {(H_{i})}_{r}^{⭑} : i \in J) \geq {(⋂ H_{i} : i \in J)}_{r}^{⭑}$ .

Proof.

(1) Since

H_{i} \leq ⋃ H_{i}

for all

i \in J

, and by Theorem 3 (1), we obtain

(⋃ {(H_{i})}_{r}^{⭑}, i \in J) \leq {(⋃ H_{i}, i \in J)}_{r}^{⭑}

. Then, (1) holds.

(2) Easy, so omitted. □

Remark 3.

Let

(S, τ^{T I F}, L^{T I F})

be an

SVNITS

and

H \in I^{S}

, we can define

\begin{matrix} C_{τ^{T I F}}^{⭑} (H, r) = H \cup H_{r}^{⭑}, i n t_{τ^{T I F}}^{⭑} (H, r) = H \land [\underset{̲}{1} - {(\underset{̲}{1} - H)}_{r}^{⭑}] . \end{matrix}

It is clear,

C_{τ^{T I F}}^{⭑}

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 I F}}^{⭑}

, i.e.,

\begin{matrix} τ^{⭑} (I) (H) = ⋃ {r | C_{τ^{T I F}}^{⭑} (\underset{̲}{1} - H, r) = \underset{̲}{1} - H} . \end{matrix}

Now, if

L^{T I F} = L_{0}^{T I F}

, then,

C_{τ^{T I F}}^{⭑} (H, r) = H_{r}^{*} \cup H = C_{τ^{T I F}}^{⭑} (H, r) \cup H = C_{τ^{T I F}} (H, r),

for

H \in I^{S} .

So,

τ^{T I F ⭑} (L_{0}^{T I F}) = τ^{T I F}

Proposition 3.

Let

(S, τ^{T I F}, L^{T I F})

be an

SVNITS

r \in I_{0}

, and

H \in I^{S}

. Then,

(1): $C_{τ^{T I F}}^{⭑} (\underset{̲}{1}, r) = \underset{̲}{1};$
(2): $C_{τ^{T I F}}^{⭑} (\underset{̲}{0}, r) = \underset{̲}{0};$
(3): $i n t_{τ^{T I F}}^{⭑} (H \cup R, r) \leq i n t_{τ^{T I F}}^{⭑} (H, r) \cup i n t_{τ^{T I F}}^{⭑} (R, r);$
(4): $i n t_{τ^{T I F}}^{⭑} (H, r) \leq H \leq C_{τ^{T I F}}^{⭑} (H, r) \leq C_{τ^{T I F}} (H, r);$
(5): $C_{τ^{T I F}}^{⭑} (\underset{̲}{1} - H, r) = \underset{̲}{1} - i n t_{τ^{T I F}}^{⭑} (H, r)$ and $\underset{̲}{1} - C_{τ^{T I F}}^{⭑} ⭑ (H, r) = i n t_{τ^{T I F}}^{⭑} (\underset{̲}{1} - H, r);$
(6): $i n t_{τ^{T I F}}^{⭑} (H \cap R, r) = i n t_{τ^{T I F}}^{⭑} (H, r) \cap i n t_{τ^{T I F}}^{⭑} (R, r) .$

Proof.

Follows directly from definitions of

C_{τ^{T I F}}^{⭑}, i n t_{τ^{T I F}}^{⭑}

C_{τ^{T I F}}

, and Theorem 3 (5). □

Theorem 5.

Let

(S, τ_{1}^{T I F}, L^{T I F})

and

(S, τ_{2}^{T I F}, L^{T I F})

{SVNTS}^{'} s

and

τ_{1}^{T I F} \leq τ_{2}^{T I F}

. Then,

H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F}) \leq H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})

Proof.

Suppose

H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F}) \leq H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})

. Then, there exists

s, t, p \in I_{0}

κ \in S

such that

\begin{matrix} T_{H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F})} (κ) \geq s > T_{H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})} (κ), \end{matrix}

\begin{matrix} I_{H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F})} (κ) < t \leq I_{H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})} (κ), \end{matrix}

(6)

\begin{matrix} F_{H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F})} (κ) < t \leq F_{H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})} (κ) . \end{matrix}

Since

T_{H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})} (κ) < s

I_{H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})} (κ) \geq t

F_{H_{r}^{⭑} (τ_{1}^{T I F}, L^{T I F})} (κ) \geq p

, there exists

D \in Q_{τ_{1}^{T I F}} (x_{s, t, p}, r)

with

L^{T} (C_{1}) \geq r

L^{I} (C_{1}) \leq 1 - r

and

L^{F} (C_{1}) \leq 1 - r

, such that for any

κ_{1} \in S

\begin{matrix} T_{D} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C} κ_{1}), I_{D} (κ_{1}) + I_{H} (κ_{1}) - 1 > I_{C} (κ_{1}), F_{D} (κ_{1}) + F_{H} (κ_{1}) - 1 > F_{C} (κ_{1}) . \end{matrix}

Since

τ_{1}^{T I F} \leq τ_{2}^{T I F}

D \in Q_{τ_{2}^{T I F}} (x_{s, t, p}, r)

. Thus,

T_{H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F})} (κ) < s

I_{H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F}))} (κ) \geq t

F_{H_{r}^{⭑} (τ_{2}^{T I F}, L^{T I F})} (κ) \geq p

. It is a contradiction for Equation (6). □

Theorem 6.

Let

(S, τ^{T I F}, L_{1}^{T I F})

and

(S, τ^{T I F}, L_{2}^{T I F})

{SVNTS}^{'} s

and

L_{1}^{T I F} \leq L_{2}^{T I F}

. Then,

H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \geq H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})

Proof.

Clear. □

Definition 14.

Let Θ be a subset of

I^{S}

, and

\underset{̲}{0} \notin Θ

. A mapping

β^{T}, β^{I}, β^{F} : Θ \to I

is called a single-valued neutrosophic base on

S

if it satisfies the following conditions:

(1): $β^{T} (\underset{̲}{1}) = 1$ and $β^{I} (\underset{̲}{1}) = β^{F} (\underset{̲}{1}) = 0$ ;
(2): For all $H$ , $R \in Θ$ ,

$\begin{matrix} β^{T} (H \cap R) \geq β^{T} (H) \cap β^{T} (R), β^{I} (H \cap R) \leq β^{I} (H) \cup β^{I} (R), β^{F} (H \cap R) \leq β^{F} (H) \cup β^{F} (R) . \end{matrix}$

Theorem 7.

Define a mapping

β : Θ \to I

S

\begin{matrix} β^{I} (H) = ⋃ {τ^{T} (R) \cap I^{T} (C) | H = R \cap (\underset{̲}{1} - C)}, \end{matrix}

\begin{matrix} β^{I} (H) = ⋂ {τ^{I} (R) \cup I^{I} (C) | H = R \cap (\underset{̲}{1} - C)}, \end{matrix}

\begin{matrix} β^{F} (H) = ⋂ {τ^{F} (R) \cup I^{F} (C) | H = R \cap (\underset{̲}{1} - C)} . \end{matrix}

Then,

β^{T I F}

is a base for the single-valued neutrosophic topology

τ^{T I F ⭑}

Proof.

(1): Since $L^{T} (\underset{̲}{0}) = 1$ and $L^{I} (\underset{̲}{0}) = L^{F} (\underset{̲}{0}) = 0$ , we have $β^{T} (\underset{̲}{1}) = 1$ and $β^{I} (\underset{̲}{1}) = β^{F} (\underset{̲}{1}) = 0$ ;
(2): Suppose that there exists $H_{1}, H_{2} \in Θ$ such that

$\begin{matrix} β^{T} (H_{1} \cap H_{2}) ≱ β^{T} (H_{1}) \cap β^{T} (H_{2}), \\ β^{I} (H_{1} \cap H_{2}) ≰ β^{I} (H_{1}) \cup β^{I} (H_{2}), \\ β^{F} (H_{1} \cap H_{2}) ≰ β^{F} (H_{1}) \cup β^{F} (H_{2}) . \end{matrix}$

There exists

s, t, p \in I_{0}

and

κ \in S

such that

\begin{matrix} β^{T} (H_{1} \cap H_{2}) (κ) < s \leq β^{T} (H_{1}) (x) \cap β^{T} (H_{2}) (κ), \end{matrix}

\begin{matrix} β^{I} (H_{1} \cap H_{2}) (κ) \geq t > β^{I} (H_{1}) (κ) \cap β^{I} (H_{2}) (κ), \end{matrix}

(7)

\begin{matrix} β^{F} (H_{1} \cap H_{2}) (κ) \geq p > β^{F} (H_{1}) (κ) \cup β^{F} (H_{2}) (κ) . \end{matrix}

Since

β^{T} (H_{1}) (κ) \geq s

β^{I} (H_{1}) (κ) < t

β^{F} (H_{1}) (κ) < p

, and

β^{T} (H_{2}) (κ) \geq s

β^{I} (H_{2}) (κ) < t

β^{F} (H_{2}) (κ) < p

, then there exists

R_{1}, R_{1}, C_{1}, C_{2} \in Θ

with

H_{1} = R_{1} \cap (\underset{̲}{1} - C_{1})

and

H_{2} = R_{2} \cap (\underset{̲}{1} - C_{2})

, such that

β^{T} (H_{1}) \geq τ^{T} (R_{1}) \cap L^{T} (C_{1}) \geq s

β^{I} (H_{1}) \leq τ^{I} (R_{1}) \cup L^{I} (C_{1}) < t

β^{F} (H_{1}) \leq τ^{F} (R_{1}) \cup L^{F} (C_{1}) < p

, and

β^{T} (H_{2}) \geq τ^{T} (R_{2}) \cap L^{T} (C_{2}) \geq s

β^{I} (H_{2}) \leq τ^{I} (R_{2}) \cup L^{I} (C_{2}) < t

β^{F} (H_{2}) \leq τ^{F} (R_{2}) \cup L^{F} (C_{2}) < p

. Therefore,

\begin{matrix} H_{1} \cap H_{2} & = (R_{1} \cap (\underset{̲}{1} - C_{1})) \cap (R_{2} \cap (\underset{̲}{1} - C_{2})) \\ = (R_{1} \cap R_{2}) \cap ((\underset{̲}{1} - C_{1}) \cap (\underset{̲}{1} - C_{2})) \\ = (R_{1} \cap R_{2}) \cap (\underset{̲}{1} - (C_{1} \cup C_{2})) . \end{matrix}

Hence, from Definition 14, we have

\begin{matrix} β^{T} (H_{1} \cap H_{2}) & \geq τ^{T} (R_{1} \cap R_{2}) \cap L^{T} (C_{1} \cup C_{2}) \\ \geq τ^{T} (R_{1}) \cap τ^{T} (R_{2}) \cap L^{T} (C_{1}) \cap L^{T} (C_{2}) \\ = (τ^{T} (R_{1}) \cap L^{T} (C_{1})) \cap (τ^{T} (R_{2}) \cap L^{T} (C_{2})) \geq s, \end{matrix}

\begin{matrix} β^{I} (H_{1} \cap H_{2}) & \leq τ^{I} (R_{1} \cap R_{2}) \cup L^{I} (C_{1} \cup C_{2}) \\ \leq τ^{I} (R_{1}) \cup τ^{I} (R_{2}) \cup L^{I} (C_{1}) \cup L^{I} (C_{2}) \\ = (τ^{I} (R_{1}) \cup L^{F} (C_{1})) \cup (τ^{I} (R_{2}) \cup L^{I} (C_{2})) < t, \end{matrix}

\begin{matrix} β^{F} (H_{1} \cap H_{2}) & \leq τ^{F} (R_{1} \cap R_{2}) \cup L^{F} (C_{1} \cup C_{2}) \\ \leq τ^{F} (R_{1}) \cup τ^{F} (R_{2}) \cup L^{F} (C_{1}) \cup L^{F} (C_{2}) \\ = (τ^{F} (R_{1}) \cup L^{F} (C_{1})) \cup (τ^{F} (R_{2}) \cup L^{F} (C_{2})) < p . \end{matrix}

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

\begin{matrix} β^{T} (H_{1} \cap H_{2}) \geq β^{T} (H_{1}) \cap β^{T} (H_{2}), β^{I} (H_{1} \cap H_{2}) \leq β^{I} (H_{1}) \cup β^{I} (H_{2}), β^{F} (H_{1} \cap H_{2}) \leq β^{F} (H_{1}) \cup β^{F} (H_{2}) . \end{matrix}

□

Theorem 8.

Let

(S, τ^{T I F})

be an

SVNTS

, and

L_{1}^{T I F}

and

L_{1}^{T I F}

be two single-valued neutrosophic ideals on

S

. Then, for every

r \in I_{0}

and

H \in I^{S}

(1): $H_{r}^{⭑} (L_{1}^{T I F} \cap L_{2}^{T I F}, τ^{T I F}) = H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \cup H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})$ ,
(2): $H_{r}^{⭑} (L_{1}^{T I F} \cup L_{2}^{T I F}, τ) = H_{r}^{⭑} (L_{1}^{T I F}, τ^{T ⭑} (L_{2}^{T I F})) \cap H^{⭑} (L_{2}^{T I F}, τ^{T ⭑} (L_{1}^{T I F}))$ .

Proof.

(1) Suppose that

H_{r}^{⭑} (L_{1}^{T I F} \in L_{2}^{T I F}, τ^{T I F}) ≰ H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \cup H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F})

, there exists

κ \in S

and

s, t, p \in I_{0}

such that

\begin{matrix} T_{H_{r}^{⭑} (L_{1}^{T} \cap L_{2}^{T}, τ^{T})} (κ) \geq s > T_{H_{r}^{⭑} (L_{1}^{T}, τ^{T})} (κ) \cup T_{H_{r}^{⭑} (L_{2}^{T}, τ^{T})} (κ), \\ I_{H_{r}^{⭑} (L_{1}^{I} \cap L_{2}^{I}, τ^{I})} (κ) < t \leq I_{H_{r}^{⭑} (L_{1}^{I}, τ^{I})} (κ) \cup I_{H_{r}^{⭑} (L_{2}^{I}, τ^{I})} (κ), \end{matrix}

(8)

\begin{matrix} F_{H_{r}^{⭑} (L_{1}^{F} \cap L_{2}^{F}, τ^{F})} (κ) < p \leq F_{H_{r}^{⭑} (L_{1}^{F}, τ^{F})} (κ) \cap F_{H_{r}^{⭑} (L_{2}^{F}, τ^{F})} (κ) . \end{matrix}

Since

T_{H_{r}^{⭑} (L_{1}^{T}, τ^{T})} (κ) \cup T_{H_{r}^{⭑} (L_{2}^{T}, τ^{T})} (κ) < s

I_{H_{r}^{⭑} (L_{1}^{I}, τ^{I})} (κ) \cap I_{H_{r}^{⭑} (L_{2}^{I}, τ^{I})} (κ) \geq t

F_{H_{r}^{⭑} (L_{1}^{F}, τ^{F})} (κ) \cap F_{H_{r}^{⭑} (L_{2}^{F}, τ^{F})} (κ) \geq p

, we have,

T_{H_{r}^{⭑} (L_{1}^{T}, τ^{T})} (κ) < s

I_{H_{r}^{⭑} (L_{1}^{I}, τ^{I})} (κ) \geq t

F_{H_{r}^{⭑} (L_{1}^{F}, τ^{F})} (κ) \geq p

, and

I_{H_{r}^{⭑} (L_{2}^{I}, τ^{I})} (κ) < s

I_{H_{r}^{⭑} (L_{2}^{I}, τ^{I})} (κ) \geq t

F_{H_{r}^{⭑} (L_{2}^{F}, τ^{F})} (κ) \geq p

Now,

T_{H_{r}^{⭑} (L_{1}^{T}, τ^{T})} (κ) < s

I_{H_{r}^{⭑} (L_{1}^{I}, τ^{I})} (κ) \geq t

F_{H_{r}^{⭑} (L_{1}^{F}, τ^{F})} (κ) \geq p

implies that there exists

D_{1} \in Q_{τ^{T I F}} (x_{s, t, p}, r)

and for some

L_{1}^{T} (C_{1}) \geq r

L_{1}^{I} (C_{1}) \leq 1 - r

and

L_{1}^{F} (C_{1}) \leq 1 - r

such that for every

κ_{1} \in S,

\begin{matrix} T_{D_{1}} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C_{1}} (κ_{1}), I_{D_{1}} (κ_{1}) + I_{H} (κ_{1}) - 1 \geq I_{C_{1}} (κ_{1}), F_{D_{1}} (κ_{1}) + F_{H} (κ_{1}) - 1 \geq F_{C_{1}} (κ_{1}) . \end{matrix}

Once again,

T_{H_{r}^{⭑} (L_{2}^{T}, τ^{T})} (κ) < s

I_{H_{r}^{⭑} (L_{2}^{I}, τ^{I})} (κ) \geq t

F_{H_{r}^{⭑} (L_{2}^{F}, τ^{F})} (κ) \geq p

, implies there exists

D_{2} \in Q_{τ^{T I F}} (x_{s, t, p}, r)

and for some

L_{2}^{T} (C_{2}) \geq r

L_{2}^{I} (C_{2}) \leq 1 - r

and

L_{2}^{F} (C_{2}) \leq 1 - r

, such that for

κ_{1} \in S

\begin{matrix} T_{D_{2}} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C_{2}} (κ_{1}), I_{D_{2}} (κ_{1}) + I_{H} (κ_{1}) - 1 \geq I_{C_{2}} (κ), F_{D_{2}} (κ_{1}) + F_{H} (κ_{1}) - 1 \geq F_{C_{2}} (κ_{1}), \end{matrix}

Therefore, for every

κ_{1} \in S

, we have

\begin{matrix} T_{D_{1} \cap D_{2}} (κ_{1}) + T_{H} (κ_{1}) - 1 \leq T_{C_{1} \cap C_{2}} (κ_{1}), I_{D_{1} \cup D_{2}} (κ_{1}) + I_{H} (κ_{1}) - 1 \geq I_{C_{1} \cup C_{2}} (κ_{1}), \end{matrix}

\begin{matrix} F_{D_{1} \cup D_{2}} (κ_{1}) + F_{H} (κ_{1}) - 1 \geq F_{C_{1} \cup C_{2}} (κ_{1}) . \end{matrix}

Since

(D_{1} \land D_{2}) \in Q_{τ^{T I F}} (x_{s, t, p}, r)

and

(L_{1}^{T} \cap L_{2}^{T}) (C_{1} \cap C_{2}) \geq r

(L_{1}^{I} \cap L_{2}^{I}) (C_{1} \cup C_{2}) \leq 1 - r

, and

(L_{1}^{F} \cap L_{2}^{F}) (C_{1} \cup C_{2}) \geq 1 - r

we have

T_{H_{r}^{⭑} (L_{1}^{T} \cap L_{2}^{T}, τ^{T})} (κ) \leq s

I_{H_{r}^{⭑} (L_{1}^{I} \cap L_{2}^{I}, τ^{I})} (κ) > t

, and

F_{H_{r}^{⭑} (L_{1}^{F} \cap L_{2}^{F}, τ^{F})} (κ) > t

and this is a contradiction for Equation (8). So that

\begin{matrix} H_{r}^{⭑} (L_{1}^{T I F} \cap L_{2}^{T I F}, τ^{T I F}) \leq H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \cup H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F}) . \end{matrix}

On the opposite direction,

L_{1}^{T I F} \geq L_{1}^{T I F} \cap L_{2}^{T I F}

and

L_{2}^{T I F} \geq L_{1}^{T I F} \cap L_{2}^{T I F}

, so by Theorem 3 (2),

\begin{matrix} H_{r}^{⭑} (L_{1}^{T I F} \cap L_{2}^{T I F}, τ^{T}) \geq H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \cup H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F}) . \end{matrix}

Then,

\begin{matrix} H_{r}^{⭑} (L_{1}^{T I F} \cap L_{2}^{T I F}, τ^{T I F}) = H_{r}^{⭑} (L_{1}^{T I F}, τ^{T I F}) \cup H_{r}^{⭑} (L_{2}^{T I F}, τ^{T I F}) . \end{matrix}

(2) Straightforward. □

The above theorem results in an important consequence.

τ^{T I F ⭑} (L^{T I F})

and

{[τ^{T I F ⭑} (L^{T I F})]}^{⭑} (L^{T I F})

(in short

τ^{⭑ ⭑}

) are equal for any single-valued neutrosophic ideal on

S

Corollary 1.

Let

(S, τ^{T I F}, L^{T I F})

be an

SVNITS

. For every

r \in I_{0}

and

H \in I^{X}

H_{r}^{⭑} (L^{T I F}) = H_{r}^{⭑} (L^{T I F}, τ^{T I F ⭑})

and

τ^{T I F ⭑} (L^{T I F}) = τ^{T I F ⭑ ⭑}

Proof.

Putting

L_{1}^{T I F} = L_{2}^{T I F}

in Theorem 8 (2), we have the required result. □

Corollary 2.

Let

(S, τ^{T I F})

be an

SVNTS

, and

L_{1}^{T I F}

and

L_{1}^{T I F}

be two single-valued neutrosophic ideals on

S

. Then, for any

H \in I^{S}

and

r \in I_{0}

(1): $τ^{T ⭑} (L_{1}^{T I F} \cup I_{2}^{T I F}) = {(τ^{T I F ⭑} (L_{2}^{T I F}))}^{⭑} (L_{1}^{T}) = {(τ^{T I F ⭑} (L_{1}^{T I F}))}^{⭑} (L_{2}^{T})$ ,
(2): $τ^{T ⭑} (L_{1}^{T I F} \cap L_{2}^{T I F}) = τ^{T I F ⭑} (L_{1}^{T I F}) \cap τ^{T ⭑} (L_{2}^{T I F})$ .

Proof.

Straightforward. □

Definition 15.

For an

SVNTS

(S, τ^{T I F})

with a single-valued neutrosophic ideal

I^{T I F},

τ^{T I F}

is said to be single-valued neutrosophic ideal open compatible with

I^{T I F}

, denoted by

τ^{T I F} \sim L^{T I F}

, if for each

H, C \in I^{S}

and

x_{s, t, p} \in H

with

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

, and

L^{F} (C) \leq 1 - r

, there exists

D \in Q_{τ^{T I F}} (x_{t}, r)

such that

T_{D} (κ) + T_{H} (κ) - 1 \leq T_{C} (κ)

I_{D} (κ) + I_{H} (κ) - 1 > I_{C} (κ)

, and

F_{D} (κ) + F_{H} (κ) - 1 > F_{C} (κ)

holds for any

κ \in S

, then

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

and

L^{F} (H) \leq 1 - r

Definition 16.

Let

{R_{j}}_{j \in J}

be an indexed family of a single-valued neutrosophic set of

S

such that

R_{j} q H

for each

j \in J

, where

H \in I^{S}

. Then,

{R_{j}}_{j \in J}

is said to be a single-valued neutrosophic quasi-cover of

H

iff

T_{H} (κ) + T_{⋁_{j \in J} (R_{j})} (κ) \geq 1

I_{H} (κ) + I_{⋁_{j \in J} (R_{j})} (κ) < 1

, and

F_{H} (κ) + F_{⋁_{j \in J} (R_{j})} (κ) < 1

, for every

κ \in S

Further, let

(S, τ^{T I F})

be an

SVNTS

, for each

τ^{T} (R_{j}) \geq r

τ^{I} (R_{j}) \leq 1 - r

, and

τ^{F} (R_{j}) \leq 1 - r

. Then, any single-valued neutrosophic quasi-cover will be called single-valued neutrosophic quasi open-cover of

H .

Theorem 9.

Let

(S, τ^{T I F})

be an

SVNTS

with single-valued neutrosophic ideal

L^{T I F}

S

. Then, the following conditions are equivalent:

(1): $τ \sim L .$
(2): If for every $H \in I^{S}$ has a single-valued neutrosophic quasi open-cover of ${R_{j}}_{j \in J}$ such that for each j, $T_{H} (κ) + T_{R_{j}} (κ) - 1 \leq T_{C} (κ)$ , $I_{H} (κ) + I_{R_{j}} (κ) - 1 > I_{C} (κ)$ , and $F_{H} (κ) + F_{R_{j}} (κ) - 1 > F_{C} (κ)$ for every $κ \in S$ and for some $L^{T} (C) \geq r$ , $L^{I} (C) \leq 1 - r$ , and $L^{F} (C) \leq 1 - r$ , then $L^{T} (H) \geq r$ , $L^{I} (H) \leq 1 - r$ , and $L^{F} (H) \leq 1 - r$ ,
(3): For every $H \in I^{S}$ , $H \land H_{r}^{⭑} = (0, 1, 1)$ implies $L^{T} (H) \geq r$ , $L^{I} (H) \leq 1 - r$ , and $L^{F} (H) \leq 1 - r$ ,
(4): For every $H \in I^{S}$ , $L^{T} (\tilde{H}) \geq r$ , $L^{I} (\tilde{H}) \leq 1 - r$ , and $L^{F} (\tilde{H}) \leq 1 - r$ , where $\tilde{H} = ⋁ x_{s, t, p}$ such that $x_{s, t, p} \in H$ but $x_{s, t, p} \notin H_{r}^{*}$ ,
(5): For every $τ^{T ⭑} (\underset{̲}{1} - H) \geq r$ , $τ^{I ⭑} (\underset{̲}{1} - H) \leq 1 - r$ , and $τ^{F ⭑} (\underset{̲}{1} - H) \leq 1 - r$ we have $L^{T} (\tilde{H}) \geq r$ , $L^{I} (\tilde{H}) \leq 1 - r$ , and $L^{F} (\tilde{H}) \leq 1 - r$ ,
(6): For every $H \in I^{S},$ if $A$ contains no $R \neq (0, 1, 1)$ with $R \leq R_{r}^{⭑}$ , then $L^{T} (H) \geq r$ , $L^{I} (H) \leq 1 - r$ , and $L^{F} (H) \leq 1 - r$ .

Proof.

It is proved that most of the equivalent conditions ultimately prove the all the equivalence.

(1)⇒(2): Let

{R_{j}}_{j \in J}

be a single-valued neutrosophic quasi open-cover of

H \in I^{S}

such that for

j \in J

T_{H} (κ) + T_{R_{j}} (κ) - 1 \leq T_{C} (κ)

I_{H} (κ) + I_{R_{j}} (κ) - 1 > I_{C} (κ)

, and

F_{H} (κ) + F_{R_{j}} (κ) - 1 > F_{C} (κ)

for every

κ \in R

and for some

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

, and

L^{F} (C) \leq 1 - r

. Therefore, as

{R_{j}}_{j \in J}

is a single-valued neutrosophic quasi open-cover of

R

, for each

x_{s, t, p} \in H

, there exists at least one

R_{j \circ}

such that

x_{s, t, p} q R_{j \circ}

and for every

κ \in S

T_{H} (κ) + T_{R_{j \circ}} (κ) - 1 \leq T_{C} (κ)

I_{H} (κ) + I_{R_{j \circ}} (κ) - 1 > I_{C} (κ)

, and

F_{H} (κ) + F_{R_{j \circ}} (κ) - 1 > F_{C} (κ)

for every

κ \in S

and for some

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

and

L^{F} (C) \leq 1 - r

. Obviously,

R_{j \circ} \in Q_{τ^{T I F}} (x_{s, t, p}, r) .

By (1), we have

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

, and

L^{F} (H) \leq 1 - r

(2)⇒(1): Clear from the fact that a collection of

{R_{j}}_{j \in J}

, which contains at least one

R_{j \circ} \in Q_{τ^{T} I F} (x_{s, t, p}, r)

of each single-valued neutrosophic point of

H

, constitutes a single-valued neutrosophic quasi-open cover of

H

(1)⇒(3): Let

H \cap H_{r}^{⭑} = (0, 1, 1)

, for every

κ \in S,

x_{t} \in H

implies

x_{s, t, p} \notin H_{r}^{⭑} .

Then, there exists

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

and

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

such that for every

κ \in S

T_{D} (κ) + T_{H} (κ) - 1 \leq T_{C} (κ)

I_{D} (κ) + I_{H} (κ) - 1 > I_{C} (κ)

, and

F_{D} (κ) + F_{H} (κ) - 1 > F_{C} (κ)

. Since

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

, By (1), we have

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

, and

L^{F} (H) \leq 1 - r

(3)⇒(1): For every

x_{s, t, p} \in H

, there exists

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

such that for every

κ \in S

T_{D} (κ) + T_{H} (κ) - 1 \leq 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 some

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

. This implies

x_{s, t, p} \notin H_{r}^{⭑}

. Now, there are two cases: either

H_{r}^{⭑} = (0, 1, 1)

H_{r}^{⭑} \neq (0, 1, 1)

but

s > T_{H_{r}^{⭑}} (κ) \neq 0

t \leq I_{H_{r}^{⭑}} (κ) \neq 1

, and

p \leq F_{H_{r}^{⭑}} (κ) \neq 1

. Let, if possible,

x_{s, t, p} \in H

such that

t > T_{H_{r}^{⭑}} (κ) \neq 0

t \leq I_{H_{r}^{⭑}} (κ) \neq 1

, and

t \leq F_{H_{r}^{⭑}} (κ) \neq 1

. Let

s^{'} = T_{H_{r}^{⭑}} (κ) \neq 0

t^{'} = I_{H_{r}^{⭑}} (κ) \neq 1

, and

p^{'} = F_{H_{r}^{⭑}} (κ) \neq 1

. Then,

x_{s^{'}, t^{'}, p^{'}} \in H_{r}^{*} (κ)

. In addition,

x_{s^{'}, t^{'}, p^{'}} \in H .

Thus, for every

V \in Q_{τ^{T I F}} (x_{s, t, p}, r)

, for every

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

, and

L^{F} (C) \leq 1 - r

, there is at least one

κ \in S

such that

T_{V} (κ) + T_{H} (κ) - 1 > T_{C} (κ)

I_{V} (κ) + I_{H} (κ) - 1 \leq I_{C} (κ)

, and

F_{V} (κ) + F_{H} (κ) - 1 \leq F_{C} (κ)

. Since

x_{s, t, p} \in H

, this contradicts the assumption for every single-valued neutrosophic point of

H

. So,

H_{r}^{⭑} = (0, 1, 1)

. That means

x_{s, t, p} \in H

implies

x_{s, t, p} \notin H_{r}^{*}

. Now this is true for every

H \in I^{S}

. So, for any

H \in I^{S}

H \cap H_{r}^{⭑} = (0, 1, 1)

. Hence, by (3), we have

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

L^{F} (H) \leq 1 - r

, which implies

τ^{T I F} \sim L^{T I F}

(3)⇒(4): Let

x_{s, t, p} i n \tilde{H}

. Then,

x_{s, t, p} \in H

but

x_{s, t, p} \notin H_{r}^{⭑}

. So, there exists a

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

such that for every

κ \in S

T_{D} (κ) + T_{H} (κ) - 1 \leq 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 some

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

. Since

\tilde{H} \leq H

, for every

κ \in S

T_{D} (κ) + T_{\tilde{H}} (κ) - 1 \leq T_{C} (κ)

I_{D} (κ) + I_{\tilde{H}} (κ) - 1 > I_{C} (κ)

, and

F_{D} (κ) + F_{\tilde{H}} (κ) - 1 > F_{C} (κ)

, for some

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

and

L^{F} (C) \leq 1 - r

. Therefore,

x_{s, t, p} \notin {\tilde{H}}_{r}^{⭑}

implies that

{\tilde{H}}_{r}^{⭑} = (0, 1, 1)

{\tilde{H}}_{r}^{⭑} \neq (0, 1, 1)

but

s > T_{{\tilde{H}}_{r}^{⭑}}

t \leq I_{{\tilde{H}}_{r}^{⭑}}

, and

p \leq F_{{\tilde{H}}_{r}^{⭑}}

. Let

x_{s^{'}, t^{'}, p^{'}}

S V N P (S)

such that

s^{'} \leq T_{{\tilde{H}}_{r}^{⭑}} (κ) < s

t \leq I_{{\tilde{A}}_{r}^{⭑}} (κ) < t^{'}

, and

p \leq F_{{\tilde{H}}_{r}^{⭑}} (κ) < p^{'}

, i.e.,

x_{s^{'}, t^{'}, p^{'}} \in {\tilde{H}}_{r}^{⭑}

. Then, for each

V \in Q_{τ^{T I F}} (x_{s^{'}, t^{'}, p^{'}}, r)

and for each

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

, there is at least one

κ \in S

such that

T_{V} (κ) + T_{\tilde{H}} (κ) - 1 > T_{C} (κ)

I_{V} (κ) + I_{\tilde{H}} (κ) - 1 \leq I_{C} (κ)

, and

F_{V} (κ) + F_{\tilde{H}} (κ) - 1 \leq F_{C} (κ)

. Since

\tilde{H} \leq H

, then for each

V \in Q_{τ^{T I F}} (x_{s^{'}, t^{'}, p^{'}}, r)

and for each

L^{T} (C) \geq r

L^{I} (C) \leq 1 - r

L^{F} (C) \leq 1 - r

, there is at least one

κ \in S

such that

T_{V} (κ) + T_{H} (κ) - 1 > T_{C} (κ)

I_{V} (κ) + I_{H} (κ) - 1 \leq I_{C} (κ)

, and

F_{V} (κ) + F_{H} (κ) - 1 \leq F_{C} (κ)

. This implies

x_{s^{'}, t^{'}, p^{'}} \in H_{r}^{⭑}

. But as

s^{'} < s

t^{'} < t

, and

p^{'} < p

x_{s, t, p} \in \tilde{H}

implies

x_{s^{'}, t^{'}, p^{'}} \in \tilde{H},

and therefore,

x_{s^{'}, t^{'}, p^{'}} \notin H_{r}^{⭑}

. This is a contradiction. Hence,

H_{r}^{⭑} = (0, 1, 1)

, so that

x_{s, t, p} \in \tilde{H}

implies

x_{s, t, p} \notin {\tilde{H}}_{r}^{⭑}

with

{\tilde{H}}_{r}^{⭑} = (0, 1, 1)

. Thus,

\tilde{H} \cap {\tilde{H}}_{r}^{*} = \underset{̲}{0}

, for every

H \in I^{X}

. Hence, by (3),

L^{T} (\tilde{H}) \geq r

L^{I} (\tilde{H}) \leq 1 - r

, and

L^{F} (\tilde{H}) \leq 1 - r

(4)⇒(5): Straightforward.

(4)⇒(6): Let

H \in I^{S}

and

H \leq R \neq (0, 1, 1)

with

R \leq R_{r}^{⭑} .

Then, for any

H \in I^{S},

H = \tilde{H} \cup (H \cap H_{r}^{⭑}) .

Therefore,

H_{r}^{⭑} = {(\tilde{A} \cup (H \cap H_{r}^{⭑}))}_{r}^{⭑} = {\tilde{H}}_{r}^{⭑} \cup {(H \cap H_{r}^{⭑})}_{r}^{⭑} .

by Theorem 3 (5).

Now, by (4), we have

L^{T} (\tilde{H}) \geq r

L^{I} (\tilde{H}) \leq 1 - r

, and

L^{F} (\tilde{H}) \leq 1 - r

, then

{\tilde{H}}_{r}^{⭑} = (0, 1, 1)

. Hence,

{(H \cap H_{r}^{⭑})}_{r}^{⭑} = H_{r}^{⭑}

but

H \cap H_{r}^{⭑} \leq H_{r}^{⭑}

, then

H \cap A_{r}^{⭑} \leq {(H \cap H_{r}^{⭑})}_{r}^{⭑} .

This contradicts the hypothesis about every single-valued neutrosophic set

H \in I^{S}

, if

(0, 1, 1) \neq R \leq H

with

R \leq R_{r}^{⭑}

. Therefore,

H \cap H_{r}^{⭑} = (0, 1, 1)

, so that

H = \tilde{H}

by (4), we have

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

, and

L^{F} (H) \leq 1 - r

(6)⇒(4): Since, for every

H \in I^{S}

H \cap H_{r}^{⭑} = (0, 1, 1)

. Therefore, by (6), as

H

contains no non-empty single-valued neutrosophic subset

R

with

R \leq R_{r}^{⭑},

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

, and

L^{F} (H) \leq 1 - r

(5)⇒(1): For every

H \in I^{S}

x_{s, t, p} \in H

, there exists an

D \in Q_{τ^{T I F}} (x_{s, t, p}, r)

such that

T_{D} (κ) + T_{H} (κ) - 1 \leq T_{C} (κ)

I_{D} (κ) + I_{H} (κ) - 1 > I_{C} (κ)

, and

F_{D} (κ) + F_{H} (κ) - 1 > F_{C} (κ)

holds for every

κ \in S

and for some

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

, and

L^{F} (H) \leq 1 - r

. This implies

x_{s, t, p} \notin H_{r}^{⭑}

. Let

R = H \cup H_{r}^{*}

. Then,

R_{r}^{*} = {(H \cup H_{r}^{*})}_{r}^{⭑} = H_{r}^{⭑} \cup {(H_{r}^{⭑})}_{r}^{⭑} = H_{r}^{⭑}

by Theorem 3(4). So,

C_{τ^{T I F}}^{⭑} (R, r) = R \cup R_{r}^{⭑} = R

. That means

τ^{T ⭑} (\underset{̲}{1} - R) \geq r

τ^{I ⭑} (\underset{̲}{1} - R) \leq 1 - r

, and

τ^{F ⭑} (\underset{̲}{1} - R) \leq 1 - r

. Therefore, by (5), we have

L^{T} (R) \geq r

L^{I} (R) \leq 1 - r

, and

L^{F} (R) \leq 1 - r

Once again, for any

x_{s, t, p}

S V N P (X)

x_{s, t, p} \notin {\tilde{R}}_{r}^{⭑}

implies

x_{s, t, p} \in R

but

x_{s, t, p} \notin R_{r}^{⭑} = H_{r}^{⭑}

So, as

B = H \lor H_{r}^{⭑},

x_{s, t, p} \in H

. Now, by hypothesis about

H

. Then, for any

x_{s, t, p} \in H_{r}^{⭑}

. So,

\tilde{R} = H

. Hence,

L^{T} (H) \geq r

L^{I} (H) \leq 1 - r

, and

L^{F} (H) \leq 1 - r

, i.e.,

τ^{T I F} \sim L^{T I F}

. □

Theorem 10.

Let

(S, τ^{T I F})

be an

SVNTS

with single-valued neutrosophic ideal

L^{T I F}

S

. Then, the following are equivalent and implied by

τ \sim L

(1): For every $H \in I^{S}$ , $H \land H_{r}^{⭑} = (0, 1, 1)$ implies $H_{r}^{*} = (0, 1, 1)$ ;
(2): For any $H \in I^{S}$ , ${\tilde{H}}_{r}^{⭑} = (0, 1, 1)$ ;
(3): For every $H \in I^{S}$ , $H \land H_{r}^{⭑} = H_{r}^{⭑}$ .

Proof.

Clear from Theorem 9. □

The following corollary is an important consequence of Theorem 10.

Corollary 3.

Let

τ^{T I F} \sim L^{T I F}

. Then,

β (τ^{T I F}, L^{T I F})

is a base for

τ^{T I F ⭑}

and also

β (τ^{T I F}, L^{T I F}) = τ^{T I F ⭑}

Definition 17.

Let

H, R \in SVNS

S

. If

H

is a single-valued neutrosophic relation on a set

S

, then

H

is called a single-valued neutrosophic relation on

B

if, for every

κ, κ_{1} \in S

T_{R} (κ, κ_{1}) \leq min (T_{H} (κ), T_{H} (κ_{1}))

I_{R} (κ, κ_{1}) \geq max (I_{H} (κ), I_{H} (κ_{1}))

, and

F_{R} (κ, κ_{1}) \geq max (F_{H} (κ), F_{H} (κ_{1}))

A single-valued neutrosophic relation

H

S

is called symmetric if, for every

κ, κ_{1} \in S

T_{H} (κ, κ_{1}) = T_{H} (κ_{1}, κ)

I_{H} (κ, κ_{1}) = I_{H} (κ_{1}, κ)

F_{H} (κ, κ_{1}) = F_{H} (κ_{1}, κ)

; and

T_{R} (κ, κ_{1}) = T_{R} (κ_{1}, κ)

I_{R} (κ, κ_{1}) = I_{R} (κ_{1}, κ)

F_{R} (κ, κ_{1}) = F_{R} (κ_{1}, κ)

In the purpose of symmetry, we can replace Definition 3 with Definition 17.

5. Conclusions

In this paper, we defined a single-valued neutrosophic closure space and single-valued neutrosophic ideal to study some characteristics of neutrosophic sets and obtained some of their basic properties. Next, the single-valued neutrosophic ideal open local function, single-valued neutrosophic ideal closure, single-valued neutrosophic ideal interior, single-valued neutrosophic ideal open compatible, and ordinary single-valued neutrosophic base were introduced and studied.

Discussion for further works:

We can apply the following ideas to the notion of single-valued ideal topological spaces.

(a): The collection of bounded single-valued sets [53];
(b): The concept of fuzzy bornology [54];
(c): The notion of boundedness in topological spaces. [54].

Author Contributions

This paper was organized by the idea of Y.S., F.A. analyzed the related papers with this research, and F.S. checked the overall contents and mathematical accuracy. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by Majmaah University.

Acknowledgments

The authors would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No: R-1441-62. The authors would also like to express their sincere thanks to the referees for their useful suggestions and comments.

Conflicts of Interest

The authors declare no conflict of interest.

References

Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
Chang, C.L. Fuzzy topological spaces. J. Math. Anal. Appl. 1968, 24, 182–190. [Google Scholar] [CrossRef] [Green Version]
El-Gayyar, M.K.; Kerre, E.E.; Ramadan, A.A. On smooth topological space II: Separation axioms. Fuzzy Sets Syst. 2001, 119, 495–504. [Google Scholar] [CrossRef]
Ghanim, M.H.; Kerre, E.E.; Mashhour, A.S. Separation axioms, subspaces and sums in fuzzy topology. J. Math. Anal. Appl. 1984, 102, 189–202. [Google Scholar] [CrossRef] [Green Version]
Kandil, A.; El Etriby, A.M. On separation axioms in fuzzy topological space. Tamkang J. Math. 1987, 18, 49–59. [Google Scholar]
Kandil, A.; Elshafee, M.E. Regularity axioms in fuzzy topological space and FRi-proximities. Fuzzy Sets Syst. 1988, 27, 217–231. [Google Scholar] [CrossRef]
Kerre, E.E. Characterizations of normality in fuzzy topological space. Simon Steven 1979, 53, 239–248. [Google Scholar]
Lowen, R. Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 1976, 56, 621–633. [Google Scholar] [CrossRef] [Green Version]
Lowen, R. A comparison of different compactness notions in fuzzy topological spaces. J. Math. Anal. 1978, 64, 446–454. [Google Scholar] [CrossRef] [Green Version]
Lowen, R. Initial and final fuzzy topologies and the fuzzy Tychonoff Theorem. J. Math. Anal. 1977, 58, 11–21. [Google Scholar] [CrossRef] [Green Version]
Pu, P.M.; Liu, Y.M. Fuzzy topology I. Neighborhood structure of a fuzzy point. J. Math. Anal. Appl. 1982, 76, 571–599. [Google Scholar]
Pu, P.M.; Liu, Y.M. Fuzzy topology II. Products and quotient spaces. J. Math. Anal. Appl. 1980, 77, 20–37. [Google Scholar]
Yalvac, T.H. Fuzzy sets and functions on fuzzy spaces. J. Math. Anal. 1987, 126, 409–423. [Google Scholar] [CrossRef] [Green Version]
Chattopadhyay, K.C.; Hazra, R.N.; Samanta, S.K. Gradation of openness: Fuzzy topology. Fuzzy Sets Syst. 1992, 49, 237–242. [Google Scholar] [CrossRef]
Hazra, R.N.; Samanta, S.K.; Chattopadhyay, K.C. Fuzzy topology redefined. Fuzzy Sets Syst. 1992, 45, 79–82. [Google Scholar] [CrossRef]
Ramaden, A.A. Smooth topological spaces. Fuzzy Sets Syst. 1992, 48, 371–375. [Google Scholar] [CrossRef]
Demirci, M. Neighborhood structures of smooth topological spaces. Fuzzy Sets Syst. 1997, 92, 123–128. [Google Scholar] [CrossRef]
Chattopadhyay, K.C.; Samanta, S.K. Fuzzy topology: Fuzzy closure operator, fuzzy compactness and fuzzy connectedness. Fuzzy Sets Syst. 1993, 54, 207–212. [Google Scholar] [CrossRef]
Peeters, W. Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures. Fuzzy Sets Syst. 1999, 104, 423–433. [Google Scholar] [CrossRef]
Peeters, W. The complete lattice (S(X),⪯) of smooth fuzzy topologies. Fuzzy Sets Syst. 2002, 125, 145–152. [Google Scholar] [CrossRef]
Onasanya, B.O.; Hošková-Mayerová, Š. Some topological and algebraic properties of α-level subsets’ topology of a fuzzy subset. An. Univ. Ovidius Constanta 2018, 26, 213–227. [Google Scholar] [CrossRef] [Green Version]
Çoker, D.; Demirci, M. An introduction to intuitionistic fuzzy topological spaces in Šostak’s sense. Busefal 1996, 67, 67–76. [Google Scholar]
Samanta, S.K.; Mondal, T.K. Intuitionistic gradation of openness: Intuitionistic fuzzy topology. Busefal 1997, 73, 8–17. [Google Scholar]
Samanta, S.K.; Mondal, T.K. On intuitionistic gradation of openness. Fuzzy Sets Syst. 2002, 131, 323–336. [Google Scholar]
Šostak, A. On a fuzzy topological structure. In Circolo Matematico di Palermo, Palermo; Rendiconti del Circolo Matematico di Palermo, Proceedings of the 13th Winter School on Abstract Analysis, Section of Topology, Srni, Czech Republic, 5–12 January 1985; Circolo Matematico di Palermo: Palermo, Italy, 1985; pp. 89–103. [Google Scholar]
Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
Lim, P.K.; Kim, S.R.; Hur, K. Intuitionisic smooth topological spaces. J. Korean Inst. Intell. Syst. 2010, 20, 875–883. [Google Scholar] [CrossRef]
Kim, S.R.; Lim, P.K.; Kim, J.; Hur, K. Continuities and neighborhood structures in intuitionistic fuzzy smooth topological spaces. Ann. Fuzzy Math. Inform. 2018, 16, 33–54. [Google Scholar] [CrossRef]
Choi, J.Y.; Kim, S.R.; Hur, K. Interval-valued smooth topological spaces. Honam Math. J. 2010, 32, 711–738. [Google Scholar] [CrossRef]
Gorzalczany, M.B. A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 1987, 21, 1–17. [Google Scholar] [CrossRef]
Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning I. Inform. Sci. 1975, 8, 199–249. [Google Scholar] [CrossRef]
Ying, M.S. A new approach for fuzzy topology(I). Fuzzy Sets Syst. 1991, 39, 303–321. [Google Scholar] [CrossRef]
Lim, P.K.; Ryou, B.G.; Hur, K. Ordinary smooth topological spaces. Int. J. Fuzzy Log. Intell. Syst. 2012, 12, 66–76. [Google Scholar] [CrossRef]
Lee, J.G.; Lim, P.K.; Hur, K. Some topological structures in ordinary smooth topological spaces. J. Korean Inst. Intell. Syst. 2012, 22, 799–805. [Google Scholar] [CrossRef]
Lee, J.G.; Lim, P.K.; Hur, K. Closures and interiors redefined, and some types of compactness in ordinary smooth topological spaces. J. Korean Inst. Intell. Syst. 2013, 23, 80–86. [Google Scholar] [CrossRef]
Lee, J.G.; Hur, K.; Lim, P.K. Closure, interior and compactness in ordinary smooth topological spaces. Int. J. Fuzzy Log. Intell. Syst. 2014, 14, 231–239. [Google Scholar] [CrossRef]
Saber, Y.M.; Abdel-Sattar, M.A. Ideals on Fuzzy Topological Spaces. Appl. Math. Sci. 2014, 8, 1667–1691. [Google Scholar] [CrossRef]
Salama, A.A.; Albalwi, S.A. Intuitionistic Fuzzy Ideals Topological Spaces. Adv. Fuzzy Math. 2012, 7, 51–60. [Google Scholar]
Sarkar, D. Fuzzy ideal theory fuzzy local function and generated fuzzy topology fuzzy topology. Fuzzy Sets Syst. 1997, 87, 117–123. [Google Scholar] [CrossRef]
Smarandache, F. Neutrosophy, Neutrisophic Property, Sets, and Logic; American Research Press: Rehoboth, DE, USA, 1998. [Google Scholar]
Salama, A.A.; Broumi, S.; Smarandache, F. Some types of neutrosophic crisp sets and neutrosophic crisp relations. I. J. Inf. Eng. Electron. Bus. 2014. Available online: http://fs.unm.edu/Neutro-SomeTypeNeutrosophicCrisp.pdf (accessed on 10 February 2019).
Salama, A.A.; Smarandache, F. Neutrosophic Crisp Set Theory; The Educational Publisher Columbus: Columbus, OH, USA, 2015. [Google Scholar]
Hur, K.; Lim, P.K.; Lee, J.G.; Kim, J. The category of neutrosophic crisp sets. Ann. Fuzzy Math. Inform. 2017, 14, 43–54. [Google Scholar] [CrossRef]
Hur, K.; Lim, P.K.; Lee, J.G.; Kim, J. The category of neutrosophic sets. Neutrosophic Sets Syst. 2016, 14, 12–20. [Google Scholar]
Salama, A.A.; Alblowi, S.A. Neutrosophic set and neutrosophic topological spaces. IOSR J. Math. 2012, 3, 31–35. [Google Scholar] [CrossRef]
Salama, A.A.; Smarandache, F.; Kroumov, V. Neutrosophic crisp sets and neutrosophic crisp topological spaces. Neutrosophic Sets Syst. 2014, 2, 25–30. [Google Scholar]
Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct. 2010, 4, 410–413. [Google Scholar]
Kim, J.; Lim, P.K.; Lee, J.G.; Hur, K. Single valued neutrosophic relations. Ann. Fuzzy Math. Inform. 2018, 16, 201–221. [Google Scholar] [CrossRef]
Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, 6th ed.; InfoLearnQuest: Ann Arbor, MI, USA, 2007. [Google Scholar]
Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 26, 2450–2466. [Google Scholar] [CrossRef]
Yang, H.L.; Guo, Z.L.; Liao, X. On single valued neutrosophic relations. J. Intell. Fuzzy Syst. 2016, 30, 1045–1056. [Google Scholar] [CrossRef] [Green Version]
El-Gayyar, M. Smooth Neutrosophic Topological Spaces. Neutrosophic Sets Syst. 2016, 65, 65–72. [Google Scholar]
Yan, C.H.; Wu, C.X. Fuzzy L-bornological spaces. Inf. Sci. 2005, 173, 1–10. [Google Scholar] [CrossRef]
Lambrinos, P.A. A topological notion of boundedness. Manuscripta Math. 1973, 10, 289–296. [Google Scholar] [CrossRef]

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Saber, Y.; Alsharari, F.; Smarandache, F. On Single-Valued Neutrosophic Ideals in Šostak Sense. Symmetry 2020, 12, 193. https://doi.org/10.3390/sym12020193

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Saber Y, Alsharari F, Smarandache F. On Single-Valued Neutrosophic Ideals in Šostak Sense. Symmetry. 2020; 12(2):193. https://doi.org/10.3390/sym12020193

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Saber, Y., Alsharari, F., & Smarandache, F. (2020). On Single-Valued Neutrosophic Ideals in Šostak Sense. Symmetry, 12(2), 193. https://doi.org/10.3390/sym12020193

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On Single-Valued Neutrosophic Ideals in Šostak Sense

Abstract

1. Introduction

2. Preliminaries

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

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

5. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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