Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.018066

ARTICLE

On Some Properties of Neutrosophic Semi Continuous and Almost Continuous Mapping

Bhimraj Basumatary1,*, Nijwm Wary1, Jeevan Krishna Khaklary2 and Usha Rani Basumatary1

1Department of Mathematical Sciences, Bodoland University, Kokrajhar, 783370, India
2Central Institute of Technology, Kokrajhar, 783370, India
*Corresponding Author: Bhimraj Basumatary. Email: brbasumatary14@gmail.com
Received: 26 June 2021; Accepted: 17 August 2021

1  Introduction

After Zadeh [1] created fuzzy set theory (FST), FST was used to define the idea of membership value and explain the concept of uncertainty. Many researchers attempted to apply FST to a variety of other fields of science and technology. Atanassov [2] expanded on the concept of fuzzy set theory and introduced the concept of degree of non-membership, as well as proposing intuitionistic fuzzy set theory (IFST). Chang [3] introduced fuzzy topology (FT), and Coker [4] generalized the concept of FT to intuitionistic fuzzy topology (IFT). Rosenfeld [5] introduced the concept of fuzzy groups and Foster [6] proposed the idea of fuzzy topological groups. Azad [7] went through fuzzy semi-continuity (FSC), fuzzy almost continuity (FAC), and fuzzy weakly continuity (FWC). Smarandache [8,9] suggested neutrosophic set theory (NST) by generalizing FST and IFST and valuing indeterminacy as a separate component. Many researchers have attempted to apply NST to a variety of scientific and technological fields. Kandil et al. [10] studied the fuzzy bitopological spaces. Mwchahary et al. [11] did their work in neutrosophic bitopological space. Neutrosophic topology was proposed by Salama et al. [12,13]. The semi-continuous mapping was investigated by Noiri [14] and the term almost continuous mappings were coined by Singal et al. [15]. The idea of fuzzy neutrosophic groups and a topological group of the neutrosophic set was studied by Sumathi et al. [16,17]. NST was used as a tool in a group discussion framework by Abdel-Basset et al. [18]. Abdel-Basset et al. [19] investigated the use of the base-worst technique to solve chain problems using a novel plithogenic model.

1.1 Motivations

In this current decade, neutrosophic environments are mainly interested by different fields of researchers. In Mathematics also much theoretical research has been observed in the sense of neutrosophic environment. It will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application. Salama et al. [13] studied neutrosophic closed set and neutrosophic continuous functions. The idea of almost continuous functions is done in 1968 [15] in topology. Similarly, the notion of fuzzy almost contra continuous and fuzzy almost contra α-continuous functions was discussed in [20]. Recently, Al-Omeri et al. [21,22] introduced and studied a number of the definitions of neutrosophic closed sets, neutrosophic mapping, and obtained several preservation properties and some characterizations about neutrosophic of connectedness and neutrosophic connectedness continuity. More recently, in [23–26] authors have given how a new trend of Neutrosophic theory is applicable in the field of Medicine and multimedia with a novel and powerful model. From the literature survey, it is noticed that exactly the properties of neutrosophic semi-continuous and almost continuous mapping are not done. To update this research gap, in this research article, we attempt to investigate the neutrosophic semi-continuous and almost continuous mapping and its properties. Also, we study properties of the neutrosophic semi-open set (NSOS), neutrosophic semi-closed set (NSCoS), neutrosophic regularly open set (NROS), neutrosophic regularly closed set (NRCoS), neutrosophic semi-continuous (NSC), and neutrosophic almost continuous mapping (NACM).

2  Methodologies

2.1 Definition [8]

A neutrosophic set (NS) AN on X can be expressed as AN={<x∈X,TAN(x),IAN(x),FAN(x)>}, where T,I,F:X⟶]-0, 1+[. Note that 0≼TAN(x)+IAN(x)+FAN(x)≼3.

2.2 Definition [8]

Complement of AN is expressed as

ANc(x)={<x∈X,TANc(x)=FAN(x),IANc(x)=1−IAN(x),FANc(x)=TAN(x)>}.

2.3 Definition [8]

Let X≠ϕ and AN={<x∈X,TAN(x),IAN(x),FAN(x)>} and BN={<x∈X,TBN(x),IBN(x),FBN(x)>} are NSs. Then

(i) AN⋒BN={<x,min(TAN(x),TBN(x)),min(IAN(x),IBN(x)),max(FAN(x),FBN(x))>}

(ii) AN⋓BN={<x,max(TAN(x),TBN(x)),max(IAN(x),IBN(x)),min(FAN(x),FBN(x))>}

(iii) AN≼BN  if  TAN(x)≼TBN(x),IAN(x)≼IBN(x),FAN(x)≽FBN(x), for  x∈X.

2.4 Definition [12]

Let X≠ϕ, then neutrosophic topology space (NTS) on X is a family TXN of neutrosophic subsets of X satisfying the following axiom:

(i) 0XN,1XN∈TXN

(ii) GN1⋒GN2∈TXN; for GN1,GN2∈TXN

(iii) ⋓GNi∈TXN,∀{GNi:i∈J}≼TXN.

Then the pair (X,TXN) is called a NTS.

2.5 Definition [12]

Let (X,TXN) be NTS. Then for a NS AN={<x,μNi,σNi,δNi>:x∈X}, neutrosophic interior of AN can be defined as N∽Int(AN)={<x,⋓ μNi,⋒ σNi,⋒ δNi>:x∈X}.

2.6 Definition [12]

Let (X,TXN) be NTS. Then for a NS AN={<x,μNi,σNi,δNi>:x∈X}, neutrosophic closure of AN can be defined as N∽Cl(AN)={<x,⋒ μNi,⋓ σNi,⋓ δNi>:x∈X}.

3  Results and Discussion

3.1 Definition

Let A be a NS of NTS (X,TXN), then A is called a N∽ semi-open set (NSOS) of X if ∃ a B∈TXN such that A≼N∽Cl(N∽Int(B)).

3.2 Definition

Let A be a NS of NTS (X,TXN), then A is called a N∽ semi-closed set (NSCoS) of X if ∃ a Bc∈TXN such that N∽Int(N∽Cl(B))≼A.

3.3 Lemma

Let ϕ:X⟶Y be a mapping and {Aα} be a family of NSs of Y, then

     i)  ϕ−1(⋓ Aα)=⋓ ϕ−1(Aα) and (ii) ϕ−1(⋒ Aα)=⋒ ϕ−1(Aα).

Prove is Straightforward.

3.4 Lemma

Let A,B be NSs of X and Y, then 1XN−A×B=(Ac×1XN)⋓(1XN×Bc).

Proof:

Let (p,q) be any element of X×Y, (1XN−A×B)(p,q)=max(1XN−A(p),1XN−B(q))=max{(Ac×1XN)(p,q),(Bc×1XN)(p,q)}={(Ac×1XN)⋓(1XN×Bc)}(p,q), for each (p,q)∈X×Y.

3.5 Lemma

Let ϕi:Xi⟶Yi and Ai be NSs of Yi, i=1,2; we have (ϕ1×ϕ2)−1(A1×A2)=ϕ1−1(A1)×ϕ2−1(A2).

Proof:

For each (p1,p2)∈X1×X2, we have

3.6 Lemma

Let ψ:X⟶X×Y be the graph of a mapping ϕ:X⟶Y. Then, if A,B be NSs of X and Y, ψ−1(A×B)=A ⋒ ϕ−1(B).

Proof:

For each p∈X, we have

3.7 Lemma

For a family {A}α of NSs of NTS (X,TXN), ⋓ N∽Cl(Aα)≼N∽Cl(⋓ (Aα)). In case B is a finite set, ⋓ N∽Cl(Aα)≼N∽Cl(⋓ (Aα)). Also, ⋓ N∽Int(Aα)≼N∽Int(⋓ (Aα)), where a subfamily B of (X,TXN) is said to be subbase for (X,TXN) if the collection of all intersections of members of B forms a base for (X,TXN).

3.8 Lemma

For a NS A of NTS (X,TXN), (a) 1−N∽Int(A)=N∽Cl(1−A), and (b) 1−N∽Cl(A)=N∽Int(1−A).

Prove is Straightforward.

3.9 Theorem

The statements below are equivalent:

     i)  A is a NSCoS,

    ii)  Ac is a NSOS,

   iii)  N∽Int(N∽Cl(A))≼A, and

    iv)  N∽Cl(N∽Int(Ac))≽Ac.

Proof:

(i) and (ii) are equivalent follows from Lemma 3.8, since for a NS A of NTS (X,TXN) such that 1−N∽Int(A)=N∽Cl(1−A) and 1−N∽Cl(A)=N∽Int(1−A).

(i)⇒(iii). By definition ∃ a NCoS B such that N∽Int(B)≼A≼B and hence N∽Int(B)≼A≼N∽Cl(A)≼B. Since N∽Int(B) is the greatest NOS contained in B, we have N∽Int(N∽Cl(B))≼N∽Int(B)≼A.

(iii)⇒(i) follows by taking B=N∽Cl(A).

(ii)⇔(iv) can similarly be proved.

3.10 Theorem

     i)  Arbitrary union of NSOSs is a NSOS, and

    ii)  Arbitrary intersection of NSCoSs is a NSCoS.

Proof:

(i) Let {Aα} be a collection of NSOSs of NTS (X,TXN). Then ∃ a Bα∈TXN such that Bα≼Aα≼N∽Cl(Bα), for each α. Thus, ⋒Bα≼⋓Aα≼⋓N∽Cl(Bα)≼N∽Cl(⋓(Bα)) [Lemma 3.7], and ⋓Bα∈TXN, this shows that ⋓Bα is a NSOS.

(ii) Let {Aα} be a collection of NSCoSs of NTS (X,TXN). Then ∃ a Bα∈TXN such that N∽Int(Bα)≼Aα≼Bα, for each α. Thus, N∽Int(⋒(Bα))≼⋒N∽Int(Bα)≼⋒Aα≼⋒Bα [Lemma 3.7], and ⋓Bα∈TXN, this shows that ⋒Bα is a NSCoS.

3.11 Remark

It is clear that every neutrosophic open set (NOS) (neutrosophic closed set (NCoS)) is a NSOS (NSCoS). The converse is false, it is seen in Example 3.12. It also shows that the intersection (union) of any two NSOSs (NSCoSs) need not be a NSOS (NSCoS). Even the intersection (union) of a NSOS (NSCoS) with a NOS (NCoS) may fail to be a NSOS (NSCoS). It should be noted that the ordinary topological setting the intersection of a NSOS with an NOS is a NSOS.

Further, the closure of NOS is a NSOS and the interior of NCoS is a NSCoS.

3.12 Example

Let X={a,b} and A,B be neutrosophic subsets of X such that

Then, TXN={1XN,0XN,A,B,A⋓B,A⋒B} is a NTS on X.

Let P={<a(0.8,0.2,0.1)>,<b(0.7,0.2,0.3)>} be any neutrosophic set XN, then N∽Int(P)=⋓{G:G is open set,G≼P}=A⋓B=A and N∽Cl(P)=⋒{K≽P:K is closed set inTXN}=1XN. Therefore, P is a NSOS which is not a NOS and also by Theorem 3.9, Pc is a NSCoS which is not an NCoS.

3.13 Theorem

If (X,TXN) and (Y,TYN) are NTSs and X is product related to Y. Then the product A×B of a NSOS A of X and a NSOS B of Y is NSOS of the neutrosophic product space X×Y.

Proof:

Let P≼A≼N∽Cl(P) and Q≼B≼N∽Cl(Q), where P∈TXN and Q∈TYN. Then P×Q≼A×B≼N∽Cl(P)×N∽Cl(Q). For NSs P’s of X and Q’s of Y, we have

a) inf{P,Q}=min{infP,infQ},

b) inf{P×1XN}=(infP)×1XN, and

c) inf{1XN×Q}=1XN×(infQ).

It is sufficient to prove N∽Cl(A×B)≽N∽Cl(A)×N∽Cl(B). Let P∈TNX and Q∈TNY.

Then

Since, inf{(Pc×1XN)⋓(1XN×Qc)|Pc≽A}≽inf{(Pc×1XN)|Pc≽A}=inf{Pc|Pc≽A}×1XN=N∽Cl(A)×1XN

We have, N∽Cl(A×B)≽min{N∽Cl(A)×1XN,1XN×N∽Cl(B)}=N∽Cl(A)×N∽Cl(B). Hence the result.

3.14 Definition

A NS A of NTS X is called a N∽ regularly open set (NROS) of (X,TXN) if N∽Int(N∽Cl(A))=A.

3.15 Definition

A NS A of NTS (X,TXN) is called a N∽ regularly closed set (NRCoS) of X if N∽Cl(N∽Int(A))=A.

3.16 Theorem

A NS A of NTS (X,TXN) is a NRO iff Ac is NRCo.

Proof: It follows from Lemma 3.8.

3.17 Remark

It is obvious that every NROS (NRCoS) is NOS (NCoS). The converse need not be true. For this we cite an example.

3.18 Example

From Example 3.12, it is clear that A is NOS. Now N∽Cl(A)=1XN and N∽Int(N∽Cl(A))=1XN. Therefore, N∽Int(N∽Cl(A))≠A, hence A is not NROS.

3.19 Remark

The union (intersection) of any two NROSs (NRCoS) need not be a NROS (NRCoS).

3.20 Example

Let X={a,b,c} and TXN={0XN,1XN,A,B,C} be NTS on X, where

A={<a(0.4,0.5,0.6)>,<b(0.7,0.5,0.3)>,<c(0.5,0.5,0.5)>}

B={<a(0.6,0.5,0.4)>,<b(0.3,0.5,0.7)>,<c(0.5,0.5,0.5)>},

C={<a(0.6,0.5,0.4)>,<b(0.7,0.5,0.3)>,<c(0.5,0.5,0.5)>}.

Then Cl(A)=Bc,Int(Bc)=A

Clearly, Int(Cl(A))=A.

Similarly, Int(Cl(B))=B.

Now, A⋃B=C.

But Cl(A⋃B)=1XN and Int(Cl(A⋃B))=1XN.

Hence, A and B are two NROSs but A⋃B is not NROS.

3.21 Theorem

(i) The intersection of any two NROSs is a NROS, and

(ii) The union of any two NRCoSs is a NRCoS.

Proof:

(i) Let A1 and A2 be any two NROSs of NTS (X,TXN). Since A1⋒A2 is NOS [from Remark 3.17], we have A1⋒A2≼N∽Int(N∽Cl(A1⋒A2)). Now, N∽Int(N∽Cl(A1⋒A2))≼N∽Int(N∽Cl(A1))=A1 and N∽Int(N∽Cl(A1⋒A2))≼N∽Int(N∽Cl(A2))=A2 implies that N∽Int(N∽Cl(A1⋒A2))≼A1⋒A2. Hence the theorem.

(ii) Let A1 and A2 be any two NROSs of NTS (X,TXN). Since A1⋓A2 is NOS [from Remark 3.17], we have A1⋓A2≽N∽Cl(N∽Int(A1⋓A2)). Now, N∽Cl(N∽Int(A1⋓A2))≽N∽Cl(N∽Int(A1))=A1 and N∽Cl(N∽Int(A1⋓A2))≽N∽Cl(N∽Int(A2))=A2 implies that A1⋓A2≼N∽Cl(N∽Int(A1⋓A2)). Hence the theorem.

3.22 Theorem

(i) The closure of a NOS is NRCoS, and

(ii) The interior of a NCoS is NROS.

Proof:

(i) Let A be a NOS of NTS (X,TXN), clearly, N∽Int(N∽Cl(A))≼N∽Cl(A)⇒N∽Cl(N∽Int(N∽Cl(A)))≼N∽Cl(A). Now, A is NOS implies that A≼N∽Int(N∽Cl(A)) and hence N∽Cl(A)≼N∽Cl(N∽Int(N∽Cl(A))). Thus, N∽Cl(A) is NRCoS.

(ii) Let A be a NCoS of a NTS (X,TXN), clearly, N∽Cl(N∽Int(A))≽N∽Int(A)⇒N∽Int(N∽Cl(N∽Int(A)))≽N∽Int(A). Now, A is NCoS implies that A≽N∽Cl(N∽Int(A)) and hence N∽Int(A)≽N∽Int(N∽Cl(N∽Int(A))). Thus, N∽Int(A) is NROS.

3.23 Definition

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is called a N∽ continuous mapping (NCM), if ϕ−1(A)∈TXN for each A∈TYN; or equivalently ϕ−1(B) is a NCoS of X for each NCoS B of Y.

3.24 Definition

Let ϕ:(X,TXN)⟶(Y,TXN) be a mapping from NTS (X,TXN) to another NTS (Y,TYN), then ϕ is said to be a N∽ open mapping (NOM), if ϕ(A)∈TYN for each A∈TXN.

3.25 Definition

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (Y,TYN), then ϕ is said to be a N∽ closed mapping (NCoM) if ϕ(B) is a NCoS of Y for each NCoS B of X.

3.26 Definition

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N∽ semi-continuous mapping (NSCM), if ϕ−1(A) is a neutrosophic semi-open set of X, for each A∈TYN.

3.27 Definition

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N∽ semi-open mapping (NSOM), if ϕ(A) is a NSOS for each A∈TXN.

3.28 Definition

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N∽ semi-closed mapping (NSCoM), if ϕ(B) is a NSCoS for each NCoS B of X.

3.29 Remark

From Remark 3.11, a NCM (NOM, NCoM) is also a NSCM (NSOM, NSCoM). But the converse is not true.

3.30 Example

Let X={a,b}, Y={x,y}, and

Then TXN={0XN,1XN,A} and TYN={0XN,1XN,B,C} are NTSs on X and Y.

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping defined as ϕ(a)=y,ϕ(b)=x.

Then ϕ:(X,TXN)⟶(Y,TYN) is NSCM but not NCM.

3.31 Theorem

Let X1, X2, Y1 and Y2 be NTSs such that X1 is product related to X2. Then, the product ϕ1×ϕ2:X1×X2⟶Y1×Y2 of NSCMs ϕ1:X1⟶Y1 and ϕ2:X2⟶Y2 is NSCM.

Proof:

Let A≡⋓(Aα×Bβ), where Aα’s and Bβ’s are NOSs of Y1 and Y2, respectively, be a NOS of Y1×Y2. By using Lemma 3.3(i) and Lemma 3.5, we have

(ϕ1×ϕ2)−1(A)=⋓[ϕ1−1(Aα)×ϕ2−1(Aβ)].

That (ϕ1×ϕ2)−1(A) is a NSOS follows from Theorem 3.13 and Theorem 3.10(i).

3.32 Theorem

Let X, X1 and X2 be NTSs and pi:X1×X2⟶Xi(i=1,2) be the projection of X1×X2 onto Xi. Then, if ϕ:X⟶X1×X2 is a NSCM, piϕ is also NSCM.

Proof:

For a NOS A of Xi, we have (piϕ)−1(A)=ϕ−1(pi−1(A)). That pi is a NCM and ϕ is a NSCM imply that (piϕ)−1(A) is a NSOS of X.

3.33 Theorem

Let ϕ:X⟶Y be a mapping from NTS X to another NTS Y. Then if the graph ψ:X⟶X×Y of ϕ is NSCM, then ϕ is also NSCM.

Proof:

From Lemma 3.6, ϕ−1(A)=1XN⋒ϕ−1(A)=ψ−1(1XN×A), for each NOS A of Y. Since ψ is a NSCM and 1XN×A is a NOS X×Y, ϕ−1(A) is a NSOS of X and hence ϕ is a NSCM.

3.34 Remark

The converse of Theorem 3.33 is not true.

3.35 Definition

A mapping ϕ:(X,TXN)⟶(Y,TYN) from NTS X to another NTS Y is said to be a N∽ almost continuous mapping (NACM), if ϕ−1(A)∈TXN for each neutrosophic regularly open set A of Y.

3.36 Theorem

Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping. Then the statements below are equivalent:

a) ϕ is a NACM,

b) ϕ−1(F) is a NCoS, for each NRCoS F of Y,

c) ϕ−1(A)≼N∽Int(ϕ−1(N∽Int(N∽Cl(A))), for each NOS A of Y,

d) N∽Cl(ϕ−1(N∽Cl(N∽Int(F))))≼ϕ−1(F), for each NCoS F of Y.

Proof:

Consider that ϕ−1(Ac)=(ϕ−1(A))c, for any NS A of Y, (a) ⇔ (b) follows from Theorem 3.16.

(a) ⇒ (c). Since A is a NOS of Y, A≼N∽Int(Cl(A)) and hence ϕ−1(A)≼ϕ−1(N∽Int(N∽Cl(A))). From Theorem 3.22(ii), N∽Int(N∽Cl(A)) is a NROS of Y, hence ϕ−1(N∽Int(N∽Cl(A))) is a NOS of X. Thus, ϕ−1(A)≼ϕ−1(N∽Int(N∽Cl(A)))=N∽Int(ϕ−1(N∽Int(N∽Cl(A)))).

(c) ⇒ (a). Let A be a NROS of Y, then we have ϕ−1(A)≼N∽Int(ϕ−1(N∽Int(N∽Cl(A))))=N∽Int(ϕ−1(A)). Thus, have ϕ−1(A)=N∽Int(ϕ−1(A)). This shows that ϕ−1(A) is a NOS of X.

(b) ⇔ (d) similarly can be proved.

3.37 Remark

Clearly, a NCM is NACM. But the converse needs not be true.

3.38 Example

Let X={a,b}, Y={x,y}, and

A={<a(0.6,0.5,0.3)>,<b(0.4,0.5,0.5)>}

B={<a(0.2,0.5,0.7)>,<b(0.4,0.5,0.5)>},