Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.017782

ARTICLE

On Single Valued Neutrosophic Regularity Spaces

Yaser Saber1,2, Fahad Alsharari1,6,*, Florentin Smarandache3 and Mohammed Abdel-Sattar4,5

1Department of Mathematics, College of Science and Human Studies, Hotat Sudair, Majmaah University, Majmaah, 11952, Saudi Arabia
2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt
3Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA
4Department of Mathematics, College of Science and Arts King Khaled University, Mhayal Asier, 61913, Saudi Arabia
5Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, 62511, Egypt
6Department of Mathematics, College of Science and Arts, Jouf University, Gurayat, 77455, Saudi Arabia
*Corresponding Author: Fahad Alsharari. Email: f.alsharari@mu.edu.sa
Received: 06 June 2021; Accepted: 03 August 2021

1  Introduction

A neutrosophic set can be practical in addressing problems with indeterminate, imperfect, and inconsistent materials. The concept of neutrosophic set theory was introduced by Smarandache [1] as a new mathematical method that corresponds to the indeterminacy degree (uncertainty, etc.). Bakbak et al. [2] and Mishra et al. [3] applied the soft set theory successfully applied in several areas, such as the smoothness of functions, as well as architecture-based, neuro-linguistic programming. Wang et al. [4] proposed single-valued neutrosophic sets (SVNSs). Meanwhile, Kim et al. [5,6] inspected the single valued neutrosophic relations (SVNRs) and symmetric closure of SVNR, respectively. Recently, Saber et al. [7–9] introduced the concepts of single-valued neutrosophic ideal open local function and single-valued neutrosophic topological space. Many of their applications appear in the studies of Das et al. [10]. Alsharari et al. [11–13]. Riaz et al. [14]. Salama et al. [15–17]. Hur et al. [18,19]. Yang et al. [20]. El-Gayyar [21], AL-Nafee et al. [22]. Muhiuddin et al. [23,24] and Mukherjee et al. [25].

First, we define single-valued neutrosophic θ£s-closed and single-valued neutrosophic δ£s-closed sets as well as some of their core properties. We also present and explore the properties and characterizations of single valued neutrosophic operators namely θ£s-closure (CIτ~ϱ~σ~ς~θ£s) and δ£s-closure (CIτ~ϱ~σ~ς~δ£s) in the single valued neutrosophic ideal topological space (F~,τϱ~σ~ς~,£sϱ~σ~ς~). We then define the concept of single valued neutrosophic regularity spaces. Next, we study single-valued neutrosophic θ£s-separated and single-valued neutrosophic δ£s-separated with giving some definitions and theorems. Furthermore, we also introduce single-valued neutrosophic θ£s-connected and single valued neutrosophic δ£s-connected relying on the single valued neutrosophic θ£s-closure and δ£s-closure operators.

We define a fixed universe F~ to be a finite set of objects and ζ a closed unit interval [0, 1]. Additionally, we denote ζF as the set of all single-valued neutrosophic subsets of F~.

2  Preliminaries

This section provides a complete survey, some previous studies, and concepts associated with this study.

Definition 1. [1] Let F~ be a non-empty set. A neutrosophic set (briefly, NS) in F~ is an object having the form αn={⟨υ,ϱ~αn(υ),σ~σn(ω),ς~αn(υ)⟩:υ∈F~} where

ϱ~:F~→⌋−0,1+⌊,σ~:F~→⌋−0,1+⌊,ς~:F~→⌋−0,1+⌊ and −0≤ϱ~αn(υ)+σ~αn(υ)+ς~αn(υ)≤3+ (1)

Represent the degree of membership (ϱ~αn), the degree of indeterminacy (σ~αn), and the degree of non-membership (ς~αn) respectively of any υ∈F~ to the set αn.

Definition 2. [4] Suppose that F~ is a universal set a space of points (objects), with a generic element in F~ denoted by υ. Then αn is called a single valued neutrosophic set (briefly, SVNS) in F~, if αn has the form αn={⟨υ,ϱ~αn(υ),σ~αn(υ),ς~αn(υ)⟩:υ∈F~}. Now, ϱ~αn,σ~σn,ς~αn indicate the degree of non-membership, the degree of indeterminacy, and the degree of membership, respectively of any υ∈F~ to the set αn.

Definition 3. [4] Let αn={⟨υ,ϱ~αn(υ),σ~σn(υ),ς~αn(υ)⟩:υ∈F~} be an SVNS on F~. The complement of the set αn (briefly, αnc) defined as follows: ϱ~αnc(υ)=ς~αn(υ),σ~αn(υ)=[σ~αn]c(υ),ς~αnc(υ)=ϱ~αn(υ).

Definition 4. [26] Let F~ be a non-empty set and αn,εn∈ζF~ be in the form: αn={⟨υ,ϱ~αn(υ),σ~αn(υ),ς~αn(υ)⟩:υ∈F~} and εn={⟨υ,ϱ~εn(υ),σ~εn(υ),ς~εn(υ)⟩:υ∈F~} on F~ then,

(a)   αn⊆εn for every υ∈F~; ϱ~αn(υ)≤ϱ~εn(υ),σ~αn(υ)≥σ~εn(υ),ς~αn(υ)≥ς~εn(υ).

(b)   αn=εn iff σn⊆εn and σn⊇εn.

(c)   0~=⟨0,1,1⟩ and 1~=⟨1,0,0⟩.

Definition 5. [20] Let αn,εn∈ζF~. Then,

(a)   αn∩εn is an SVNS, if for every υ∈F~,

αn∩εn=⟨(ϱ~αn∩ϱ~εn)(υ),(σ~αn∪σ~εn)(υ),(ς~αn∪ς~εn)(υ)⟩, (2)

where, (ϱ~αn∩ϱ~εn)(υ)=ϱ~αn(υ)∩ϱ~εn(υ) and (ς~αn∪ς~εn)(υ)=ς~αn(υ)∪ς~εn(υ), for all υ∈F~,

(b)   αn∪εn is an SVNS, if for every υ∈F~,

αn∪εn=⟨(ϱ~αn∪ϱ~εn)(υ),(σ~αn∩σ~εn)(υ),(ς~αn∩ς~εn)(υ)⟩. (3)

Definition 6. [15] For an any arbitrary family {αn}i∈j∈ζF~ of SVNS the union and intersection are given by

(a)   ⋂i∈j[αn]i=⟨∩i∈jϱ~[αn]i(υ),∪i∈jσ~[αn]i(υ),∪i∈jς~[αn]i(υ)⟩,

(b)   ⋃i∈j[αn]i=⟨∪i∈jϱ~[αn]i(υ),∩i∈jσ~[αn]i(υ),∩i∈jς~[αn]i(υ)⟩.

Definition 7. [21] A single-valued neutrosophic topological spaces is an ordered (F~,τ~ϱ~,τ~σ~,τ~ς~) where τ~ϱ~,τ~σ~,τ~ς~:ζF~→ζ is a mapping satisfying the following axioms:

(SVNT1) τ~ϱ~(0~)=τ~ϱ~(1~)=τ~σ~(0~)=τ~σ~(1~)=0 and τ~ς~(0~)=τ~ς~(1~)=1.

(SVNT2)τ~ϱ~(αn∩εn)≥τ~ϱ~(αn)∩τ~ϱ~(εn),τ~σ~(αn∩εn)≤τσ~(αn)∪τ~σ~(εn),τ~ς~(αn∩εn)≤τ~ς~(αn)∪τ~ς~(εn) for every, αn,εn∈ζF~

(SVNT3)τ~ϱ~(∪j∈Γ[αn]j)≥∩j∈Γτ~ϱ~([αn]j), τ~σ~(∪i∈Γ[αn]j)≤∪j∈Γτ~σ~([αn]j),τ~ς~(∪j∈Γ[αn]j)≤∪j∈Γτ~ς~([αn]j), for every [αn]j∈ζF~.

The quadruple (F~,τ~ϱ~,τ~σ~,τ~ς~) is called a single-valued neutrosophic topological spaces (briefly, SVNT, for short). Occasionally write τϱ~σ~ς~ for (τ~ϱ~,τ~σ~,τ~ς~) and it will cause no ambiguity.

Definition 8. [7] Let (F~,τϱ~σ~ς~) be an SVNTS. Then, for every αn∈ζF~ and r∈ζ0. Then the single valued neutrosophic closure and single valued neutrosophic interior of αn are define by:

Cτϱ~σ~ς~(αn,r)=⋂{εn∈ζF~:αn≤εn,τϱ~([εn]c)≥r,τσ~([εn]c)≤1−r,τς~([εn]c)≤1−r} (4)

intτϱ~σ~ς~(αn,r)=⋃{εn∈ζF~:αn≥εn,τϱ~(εn)≥r,τσ~(εn)≤1−r,τς~(εn)≤1−r} (5)

Definition 9. [7] Let (F~) be a nonempty set and υ∈F~, let s∈(0,1], t∈[0,1) and k∈[0,1), then the single-valued neutrosophic point xs,t,k in F~ given by

We define that, xs,t,p∈αn iff s<ϱ~αn(υ), t≥σ~αn(υ) and k≥ς~~αn(υ). We indicate the set of all single-valued neutrosophic points in F~ as Pxs,t,k(F~). A single-valued neutrosophic set αn is said to be quasi-coincident with another single-valued neutrosophic set εn, denoted by αnqεn, if there exists an element υ∈F~ such that ϱ~αn(υ)+ϱ~εn(υ)>1,σ~αn(υ)+σ~εn(υ)≤1,ς~αn(υ)+ς~εn(υ)≤1.

Definition 10. [7] A mapping £sϱ~,£sσ~,£sς~:ζF~→ζ is called single-valued neutrosophic ideal (SVNI) on F~ if, it satisfies the following conditions:

(£s1) £sϱ~(0~)=1 and  \pounds σ~(0~)=£sς~(0~)=0.

(£s2) If σn≤γn, then £sϱ~(εn)≤£sϱ~(αn), £sσ~(εn)≥£sσ~(αn) and £sς~(εn)≥£sς~(αn), for εn,αn∈ζF~.

(£s3) £sϱ~(αn∪εn)≥£sϱ~(αn)∩£sϱ~(εn), £sσ~(αn∪εn)≤£sσ~(αn)∪£sσ~(εn) and £sς~(αn∪εn)≤£sς~(αn)∪£sς~(εn), for αn,εn∈ζF~.

The tribal (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is called a single valued neutrosophic ideal topological space in Šostak sense (briefly, SVNITS).

Definition 11. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS for each αn∈ζF~. Then, the single valued neutrosophic ideal open local function [αn]r⊙(τϱ~σ~ς~,£sϱ~σ~ς~) of αn is the union of all single-valued neutrosophic points xs,t,k such that if εn∈Qτϱ~σ~ς~(xs,t,k,r) and £sϱ~(ωn)≥r, £sσ~(ωn)≤1−r, £sς~(ωn)≤1−r, then there is at least one υ∈F~ for which

ϱ~αn(υ)+ϱ~εn(ν)−1>ϱ~ωn(υ),σ~αn(υ)+σ~εn(υ)−1≤σ~ωn(υ),ς~αn(υ)+ς~εn(υ)−1≤ς~ωn(υ) (6)

Occasionally, we will write [αn]r⊙ for [αn]r⊙(τϱ~σ~ς~,£sϱ~σ~ς~) herein to avoid ambiguity.

Remark 1. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and αn∈ζF~. Hence, we can write

CIτϱ~σ~ς~⊙(αn,r)=αn∪[αn]r⊙,intτϱ~σ~ς~⊙(αn,r)=αn∩[(αnc)r⊙]c (7)

Clearly, CIϱ~σ~ς~⊙ is a single-valued neutrosophic closure operator and (τϱ~⊙(£s),τσ~⊙(£s),τς~⊙(£s)) is the single-valued neutrosophic topology generated by CIτϱ~σ~ς~⊙, i.e., τ⊙(J)(αn)=⋃{r|CIτ~ϱ~σ~ς~⊙(αnc,r)=αnc}.

Theorem 1. [7] Let {[αn]i}i∈J⊂ζF~ be a family of single-valued neutrosophic sets on F~ and (F~,τ~ϱ~σ~ς~,£sϱ~σ~ς~) be a SVNITS. Then,

(a)   (∪([αn]i)r⊙:i∈J)≤(∪[αn]i:i∈j)r⊙,

(b)   (∩([αn]i):i∈j)r⊙≥(∩([αn]i)r⊙:i∈J).

Theorem 2. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and r∈ζ, αn,εn∈ζF~. Then,

(a)   intτ~ϱ~σ~ς~⊙(αn∨εn,r)≤intτ~ϱ~σ~ς~⊙(αn,r)∨intτ~ϱ~σ~ς~⊙(εn,r),

(b)   intτ~ϱ~σ~ς~(αn,r)≤intτ~ϱ~σ~ς~⊙(αn,r)≤αn≤CIτ~ϱ~σ~ς~⊙(αn,r)≤Cτ~ϱ~σ~ς~(αn,r),

(c)   CIτ~ϱ~σ~ς~⊙([αn]c,r)=[intτ~ϱ~σ~ς~⊙(αn,r)]c,

(d)   [CIτ~ϱ~σ~ς~⊙(αn,r)]c=intτ~ϱ~σ~ς~⊙([αn]c,r),

(e)   intτ~ϱ~σ~ς~⊙(αn∧εn,r)=intτ~ϱ~σ~ς~⊙(αn,r)∧intτ~ϱ~σ~ς~⊙(εn,r).

Definition 12. [8] Let (F~,τϱ~σ~ς~) be an SVNITS. For every αn,εn,ωn∈ζF~, αn and εn are called r-single-valued neutrosophic separated if for r∈ζ0,

CIτϱ~σ~ς~(αn,r)∩εn=CIτϱ~σ~ς~(εn,r)∩αn=0~ (8)

An SVNS, ωn is called r-single-valued neutrosophic connected if r-SVNSEP  αn,εn∈ζF~−{0~} such that ωn=αn∪εn does not exist. A SVNS  αn is said to be r-single-valued neutrosophic connected if it is r-single-valued neutrosophic connected for any r∈ζ0. A (F~,τϱ~σ~ς~) is said to be r-single-valued neutrosophic connected if 1~ is r-single-valued neutrosophic connected.

3  Single Valued Neutrosophic δ£s-Cluster Point and Single Valued Neutrosophic θ£s-Cluster Point

In this section, we introduce the r-single-valued neutrosophic δ£s-cluster point (abbreviated SVNδ£s-cluster point) and r-single-valued neutrosophic £s-closed set (abbreviated SVN£sC). Furthermore, we analyze the single-valued neutrosophic δ£s-closure operator (δ£s-closure operator for brevity) and single-valued neutrosophic θ£s-closure operator (θ£s-closure operator for brevity).

Definition 13. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and αn∈ζF~, r∈ζ0. Then,

(a)   αn is said to be r-single valued neutrosophic £s-open (briefly, r-SVN£sO), if and only if αn≤intτ~τ~ϱ~σ~ς~([αn]r⊙,r),

(b)   αn is said to be r-single valued neutrosophic semi-£s-open (briefly, r-SVNS£sO) if and only if αn≤CIτ~ϱ~σ~ς~⊙(intτ~ϱ~σ~ς~([αn]r⊙,r),r),

(c)   αn is called r-single valued neutrosophic pre-£s-open (briefly, r-SVNP£sO) if and only if αn≤intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~([αn]r⊙,r),r),

(d)   αn is called r-single valued neutrosophic regular-£s-open (briefly, r-SVNR£sO) if and only if αn=intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~⊙([αn]r⊙,r),r),

(e)   αn is said to be r-single valued neutrosophic α£s-open (briefly, r-SVNα£sO) if and only if αn≤intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~⊙(intτ~ϱ~([αn]r⊙,r),r),

(f)   αn is said to be r-single valued neutrosophic ⋆-open set (briefly, r-SVN ⋆ O) if and only if αn=CIτ~ϱ~σ~ς~⊙(αn,r).

The complement of an r−SVN\pounds O (resp, r-SVNS£sO, r-SVNP£sO, r-SVNR£sO, r-SVNα£sO, r-SVN⋆O) is said to be an r−SVN\pounds C (resp, r-SVNS£sC, r-SVNP£sC, r-SVNR£sC, r-SVNα£sC, r-SVN⋆C) respectively.

Remark 2. r-single valued neutrosophic open set (r−SVNO) and r-SVN£sO are independent notions as shown by the following example.

Example 1. Let F~={a,b,c} be a set. Define εn,πn,ωn∈ζF~ as follows:

We define an SVNITS (τϱ~σ~ς~,£sϱ~σ~ς~) on F~ as follows: for each αn∈ζF~,

Based on εn=⟨(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3)⟩, it’s clear that, 23−SVNO is set because τϱ~(⟨(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3)⟩)≥23,τσ~(⟨(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3)⟩)≤13,τς~(⟨(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3)⟩)≤13.

However εn is not an r-SVN£sO set, and for that, we must prove that εn≰intτ~ϱ~σ~ς~([εn]23⊙,23). So, we must first obtain [εn]23⊙. Based on Eq. (11), 1~,εn,πn∈Qτϱ~σ~ς~(xs,t,k,23) and £sϱ~(⟨(0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1)⟩)≥23, £sσ~(⟨(0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1)⟩)≤13, £sς~(⟨(0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1)⟩)≤13,

such that by using Eqs. (2), (3) and (6) we obtain,

(0.3,0.3,0.3)(υ)+(0,0,0)(υ)−1≤(0.1,0.1,0.1)(υ),