Introduction to Neutrosophic Topological Spatial Region, Possible Application to GIS Topological Rules

Published Online December 2014 in MECS

Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. The paper will discuss several possible contributions to the GIS field including remote sensing, object reconstruction from airborne laser scanner, real time tracking, routing applications and modeling cognitive agents. In this paper, a simple neutrosophic region and fundamental concepts for uncertainty and indeterminacy modeling of spatial relationships are analyzed from the viewpoint of neutrosophic (NS) logic. This paper gives fundamental concepts and properties of a neutrosophic spatial region.

• II. R elated W ork

  Algebraic topological models for spatial objects were introduced in (White 1979). Thirteen topological relations between two temporal intervals were identified by J.F.Allen [6]. After the 4-intersection model proposed by M.J Egenhofer [16], M.J. Egenhofer and R.D.Franzosa [17], the 9-intersection approach introduced by M.J Egenhofer and J. R. Herring [18] was proposed as a formalism for topological relations. This approach is based on point-set topological concepts. In the 9-intersection method, a spatial object A is

decomposed into three parts: an interior denoted by Ao, an exterior denoted by AE, and a boundary denoted by ∂A. There are nine intersections between six parts of two objects. The other significant approach known as RCC (Region-Connection Calculus) has been provided by Cohn et al. D.A. Randell, Z. Cui [25]; M.N.Gotts, J.M.Gooday et al. [21]; A.G.Cohn, B.Bennet et al. [12]. During recent years, the topological relations have been extended into fuzzy domains. An example of a fuzzy object was provided by P. Fisher [19]. A number of papers of M. Schneider [27, 28, 29, 30] was presented to model fuzzy set in GIS community and to design a system of fuzzy spatial data types including operations and predicates. M. Molenaar [24] extended the formal model into fuzzy domain and based on this model T. Cheng [8] proposed a process-oriented spatio-temporal data model. The intersection model is extended to vague regions by three main approaches: the work of E. Clementini and Di Felice [9, 11] on regions with “broad boundary”, the work of F. B Zhan [34] who developed a method for approximately analyzing binary topological relations between geographic regions with indeterminate boundaries based on fuzzy sets, and X. Tang and W. Kainz [32] that provided a 3*3, a 4*4, and a 5*5 intersection matrix based on different topological parts of two fuzzy regions. The extension of the RCC schemes to accommodate vague region has been ad-dressed by F.Lehmann and A.G Cohn [23], and by A.G.Cohn and M.N .Gotts [13]. In this direction J.G. Stell and M.F Worboys [31] have used Heyting structures. The notion of intuitionistic ets (IFS) was introduced by K. Atanassov. [1, 2, 3] as generalization of fuzzy sets. The notion of neutrosophic  sets  (NS) was introduced by  F.

Smarandache  [45,  46,  47] as generalization  of intuitionistic fuzzy sets. In other research, A. A Salama [36] introduced the concept of neutrosophic topology. A. A. Salama, et,al [42] proposed a new mathematical model called "Neutrosophic crisp Sets and Neutrosophic crisp Topological Spaces.

• III.    P reliminaries

First we present the fundamental concepts and definitions given by Salama and Smarandache. We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [7, 8], Atanassov in [1, 2, 3] and Salama [9]. Smarandache introduced the neutrosophic components T, I, F which represent the membership, indeterminacy, and non-membership values respectively, where ] 0 - ,1 ⁺[ is nonstandard unit interval.

Definition 3.1[47]

Let T, I, F be real standard or nonstandard subsets of ] 0 - ,1 ⁺[ , with

Sup_T=t_sup, inf_T=t_inf

Sup_I=i_sup, inf_I=i_inf

Sup_F=f_sup, inf_F=f_inf n-sup=t_sup+i_sup+f_sup n-inf=t_inf+i_inf+f_inf,

We shall now consider some possible definitions for basic concepts of the neutrosophic set and its operations due to Salamaet al [ ].

Definition 3.2

Let X be a non-empty fixed set. A neutrosophic set ( NS for short) A is an object having the form A = {( x , m a ( x ) , a A ( x ) , Y a ( x )) : x e X } where m a ( X ) a A ( x ) and Z ( x ) which represent the degree of member ship function (namely m a ( x ) ), the degree of indeterminacy (namely a a ( x ) ), and the degree of non-member ship (namely y_A ( x ) ) respectively of each element x e X to the set A.

Remark 3.1

A neutrosophic A = { < x , m ( x ) , a ( x ) , y_a ( x ) > : x e X } can be identified to an ordered triple <  m a , a a , Y a >  in ] " 0,1 ⁺[ on X.

Remark 3.2

For the sake of simplicity, we shall use the symbol A =< x , M a ^{, a} A , Y a >  ^{for the} NS A = { < x , m a ( x ) , a _A ( x ) , y a ( x ) > : x e X }

Example 3.1

Every IFS A a non-empty set X is obviously on NS having the form A = { < x , m ( x ) ,1 - ( Ma ( x ) + y a ( x ) ) , y_a ( x ) > : x e X }

Since our main purpose is to construct the tools for developing neutrosophic set and neutrosophic topology, we must introduce the NSS 0 and 1 in X as follows:

0 may be defined as four types:

( 0 1 ) Type 1. 0 N = { x ,0,0,1): x e X }

( 0 2 ) Type 2. 0 n = {( x ,0,1,1): x e X }

( 0₃ ) Type 3. 0_N = { x ,0,1,0): x e X }

( 0₄ ) Type 4. 0„ = {( x ,0,0,0): x e X }

• 1    may be defined as four types:

• ( 1 1 )    Type 1. 1 n ={( x ,1,0,0): x e X }

• ( 1 2 )    Type 2. 1 n = {( x ,1,0,1): x e X }

• ( 1 3 )    Type 3 1 N = {( x ,1,1,0): x e X }

• ( 1 4 )    Type 4. 1 N = {( x ,1,1,1): x e X }

Definition 3.3

Let A = ( ^M a ^{, a} _A , Z A ) a NS on X , then the complement of the set A ( C ( A ) , for short ) maybe defined as three kinds of complements

( ^C 1 ) ^C ( A ) = {( ^x ,1 ^- M a ( ^x ) ,1 ^- Y a ( ^x )) ^{: x e} X } ,

(C2) C (A ) = {(x,ya ,aA (x ),ma (x )) :x eX }

( ^C з )   ^C ( A ) = {( ^x , Y a ,1 ^{- a} A ( ^x ) . M a ( ^x )} ^{: x e} X }

One can define several relations and operations between NSS follows:

Definition 3.4

Let X be a non-empty set, and NSS A and B in the ^form A = ( ^x , M a ( ^x ) ^{, a} A ( ^x ) , Y a ( ^x )} , B = ( ^x , M b ( ^x ) ^{, а} в ( ^x ) , Y b ( ^x ^ , then we may consider two possible definitions for subsets ( A c B )

( A c B ) may be defined as two types:

• ⁽¹⁾    Type 1: A c B ^ M a ( x ) < m b ( x ) , Y a ( x ) > Y ^and

a_A ( x ) <  a_B ( x ) V x e X

• ⁽²⁾    Type 2: A c B « M a ( x ) < M b ( x ) , Y a ( x ) > Y b ( x ) and a ( x ) >  a ( x )

Proposition 3.1

For any neutrosophic set A the following are holds

• (1)   0 N c A ,  0 N c 0 N

• (2)    A c 1 N , 1 N c 1 N

Definition 3.5

Let X be a non-empty set, and

A =< x , M a ( x ) , Y a ( x ) , a A ( x ) > ,

B =< x , m ( x ) , a ( x ) , Z ( x ) >  are NSS . Then

• (1)    AB may be defined as three types:

• ( 1 1 )    Type¹: A A s =< x , Д а ( x ) ^ ( x ) c ( x С ( x ) ,

Ya (x ) Yb (x ) >

• ( 1 2 )    Type 2: A A B =< x , д а ( x ) л Ц в ( x ) , ° a ( x ) л С в ( x ) ,

Y a ( ^x ) ^v Y b ( ^x ) >

• ( 1 3 )    Type 3: A П в =< x , Д а ( x ) л Д в ( x ) c ( x ) v С в ( x ) ,

Y a ( ^x ) ^v Y b ( ^x ) >

• (2)    AB may be defined as two types :

( u 1 ) Typel: A U B =< x , Д а ( x ) v Д в ( x ) , O a ( x ) v o ^ ( x ) ,

Y a ( ^x ) ^л Y b ( ^x ) >

( U 2 ) Type ²:

A U B =< x , Д а ( x ) v Д в ( x ) , С а ( x ) л С в ( x ) ,

Y a ( ^x ) ^л Y b ( ^x ) >

• ⁽³⁾   [ ] A =< x , Д а ( ^x ) ^{, с} а ( ^x ) ,1 - Д а ( ^x ) >

• <> A =< x ,1 - Y a ( x ) C a ( x ) , Y a ( x ) >

Definition 3.6

Let { Aj : j g J } be a arbitrary family of NSS in X , then

• (1)    Aj may be defined as two types :

• ⁽ⁱ⁾    Type 1; D A/ = x^x , л Д а/ ( x ) , л o ^ ( x ) , ^v Y Aj ( x )^

• ⁽ⁱⁱ⁾    Type 2: Aj j = xX , ^лД а/ ( x ) , V o / x ) , ^vY a/ ( x ))

• (2)    Aj maybe defined as two types:

• (i)    Typel; U Aj = ( x , v , л , л)

• (ii)    Type2; AAj = ( x , v , л , л)

Definition 3.7

Let A and B are neutrosophic sets then

AB may be defined as

A^B = xx , Д а Л Y b ^{, с} а ( x ) ^С в ( x ) , Y a ^v Д в ( x ))

Proposition 3.2

For all A , B two neutrosophic sets then the following are true

(1)  C (А ПВ )= C (A ) UC (B ) (2)  C (A UB )= C (A ) ПС (B ) .

Salama et al. extended the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 7] to the case of neutrosophic sets.

Definition 3.8

A neutrosophic topology ( NT for short) and a non empty set X is a family T of neutrosophic subsets in X satisfying the following axioms

( NT 1 ) O N ,1 N gt ,

( NT^ ) G 1 П G ₂ g t for any G_x G gt ,

( NT ) U G_Z g t V { G. : i g J }_G t

In this case the pair ( X , t ) is called a neutrosophic topological space ( NTS for short) and any neutrosophic set in T is known as neutrosophic open set ( NOS for short) in X . The elements of T are called open neutrosophic sets, A neutrosophic set F is closed if and only if it C (F) is neutrosophic open.

Example 3.2

Any fuzzy topological space ( X, т ) in the sense of Chang is obviously a NTS in the form t = { A : д gt } wherever we identify a fuzzy set in X whose members ship function is ц with its counterpart.

Remark 3.3

Neutrosophic topological spaces are very natural generalizations of fuzzy topological spaces allow more general functions to be members of fuzzy topology.

Example 3.3

Let X = { x } and

A = {( x ,0.5,0.5,0.4): x g X }

B = {( x ,0.4,0.6,0.8): x g X }

D = {( x ,0.5,0.6,0.4): x g X }

C = {( x ,0.4,0.5,0.8): x g X }

Then the family t = { O „ ,1, A , B ,C , D } of SfSs in X is neutrosophic topology on X .

Example 3.4

Let ( X , _{T o} ) be a fuzzy topological space in changes sense such that T o is not indiscrete suppose now that T 0 = { o _N ,1 _N } ^ { v j : j G J } then we can construct two NTSS on X as follows

• ^{a)    T} 0 =^{{ o} N ,1 N ^}^^{<  ^x , V j , ^{c ( x} ),⁰ > ^: j ^{G J }} .

• ^b)    T o = { 0 N ,1 N } o^{< x , V j ,0, c ( x ),1 - V j > : j g J ^} .

Proposition 3.3

Let ( X , t ) be an NTS on X , then we can also construct several NTSS on X in the following way:

• ^a)    T o ,1 = { [ ] G : G g t } ,

• ^b)    T o ,₂ = { <> G : G g t } .

Definition 3.9

Let ( X , т 1 ),( X, т 2 ) be two neutrosophic topological spaces on X . Then T is said be contained in т 2 (in symbols т с т ₂ ) if G ет for each G e т ■ In this case, we also say that т is coarser than т ₂ ■

Proposition 3.4

Let { _T/ : j e J } be a family of NTSS on X ■ Then _от is a neutrosophic topology on X . Furthermore, _{о т} is the coarsest NT on X containing all. _т , s

Definition 3.10

The complement of A (C (A) for short) of NOS . A is called a neutrosophic closed set ( NCS for short) in X .

Now, we define neutrosophic closure and interior operations in neutrosophic topological spaces:

Definition 3.11

Let ( X , т ) be NTS and A =< x , ^ ( x ) , y₄ ( x )^₄ ( x ) > be a NS in X .

Then the neutrosophic closure and neutrosophic interior of Aare defined by

NCl (A) = о{K: K is an NCS in X and A с K} NInt(A) = u{G: G is an NOS in X and G с A}.

It can be also shown that NCl ( A )  is NCS and

NInt ( A ) is a NOS in X

• a)    A is in x if and only if NCl ( A ) .

• b)    A is NCSin X if and only if NInt ( A ) = A .

Proposition 3.5

For any neutrosophic set A in ( x , т ) we have

• (a)   NCl ( A c ) = ( NIn t ( A )) ^c ,

• (b)   NInt ( A c ) = ( NCL ( A )) ^c ■

Proposition 3.6

Let ( X, т ) be a NTS and A , B be two neutrosophic sets in X . Then the following properties hold:

• (a)   NInt ( A ) с A ,

• (b) A с NCl ( A ),

• (c)   A с B ^ NInt ( A ) с NInt ( B ),

• (d) A с B ^ NCl ( A ) с NCl ( B ),

Zg\   NInt ( NInt ( A )) = NInt ( A ),

NCL ( NCL ( A )) = NCL ( A )

/-  NCl ( A о B ) = NCl ( A ) о NCl ( B ),

NInt ( A П B ) = NInt ( A ) о NInt ( B )

⁽g)   NInt (1 N ) = 1 N ,

^(h) NCl ( On ) = On

• IV. S ome N eutrosophic T opological N otions of

N eutrosophic R egion

In this section, we add some further definitions and propositions for a neutrosophic topological region.

Corollary4.1

Let A =< a a( x ), a _A( x ), v A ( x ) >  and B =< a b ( x ), c b ( x ), v b ( x ) >  are two neutrosophic sets on a neutrosophic topological space ( X , т ) then the following are holds

• i)     N int( A ) о N int( B ) = N int( A о B ),

• ii)    Ncl ( A ) о Ncl ( B ) = N int( A о B ),

• iii)   N int( A ) с A с Ncl ( A ),

• iv)   ( N int( A ) ) ^c = Ncl ( A^c ),   ( Ncl ( A ) ) ^c = N int( A c ).

Definition 4.1

We define a neutrosophic boundary (NB) of a neutrosophic set A =< Na( x ), ^ a( x )Va ( x ) >  by: 5 A = Ncl ( A ) о Ncl ( A c ) .

The following theorem shows the intersection methods no longer guarantees a unique solution.

Corollary 4.2:

5 A о N int( A ) = O N  iff   N int( A )   is crisp (i.e.,

N int( A ) = O_N or N int( A ) = 1 _N ).

Proof obvious

Definition 4.2

Let A =< ца(x),aa(x),Va(x) > be a neutrosophic sets on a neutrosophic topological space(x,t). Suppose that the family of neutrosophic open sets contained in A is indexed by the family < AG, (x),°"Gi (x),vG (x) >: i e I and the family of neutrosophic open subsets containing A are indexed the family < ak (x),^K (x),VK (x) >: j e J .Then two jjj neutrosophic interior, closure and boundaries are defined as following

• a)    N int( A ) [ ] may be defined as two types

  i) Type 1. N int( A ) [ ] =

  
  

< max ( A G ( x ) ), max ( p G ( x ) ), min ( 1

- A G ^{( x} ) ) >

ii) Type 2. N int( A ) [ ] =

< ma

x ^ A Gj^{( x} ) ) , mint' ^ G ⁽ x )], min ^{( 1}

- A G _; ^{( x} ) ) >

• b)    n int( A ) < >  may be defined as two types

i) Type 1. N int( A ) < >  =

< max(1 — vg (x)) max(oG (x)) min Vg (x))> ii. Type 2.

< maXl - v g

N int( A ) <  >

⁽ x ) ) min (<7 G_i⁽ x ) ) min ( v _G . ⁽ x ) ) >

• c)    Ncl ( A )_{[ ]} may be defined as two types

• i)    Type 1. Ncl ( A )_{[ ]} =

< ma^ u K ( x ) ) min (o" K ( x ) ) max ( 1 - ц к . ( x ) ) >

• ii)    Type 2. Ncl ( A ) [ ] =

< max ( te . ( x ), max i m a ( x ) 1 m

। >

• d)      Ncl ( A ) < >  may be defined as two types

• i)    Type 1. Ncl ( A ) < >  =

• < min ( 1 - v K ( x ) ) min i m _G. ( x ) ), max (v _G . ( x ) ) >

• ii)    Type 2 Ncl ( A ) < >  =

• < min ( 1 - v K ( x ) ), max l m G ( x ) ), max ( v _G . ( x ) ) > .

• e)    Neutrosophic boundaries defined as i)    8 A [ ] = Ncl ( A[ ]) о Ncl ( A c [ ])

• ii)   5 A < >  = Ncl ( A < > ) о Ncl ( A c < > )

Proposition 4.1

• a)   N int( A ) [ ] £ N int( A ) £ N int( A ) < >  ,

• b) Ncl ( A ) [ ] £ Ncl ( A ) Ncl ( A )< >

• c)   N int( A {[ ], < > } ) = { [ ], < > } N int( A ) and

Nc^{l ( A} {[ ], < > } ) = ^{ [ ]. < >^} Nc^{l (} A )

Proof

We shall only prove (c), and the others are obvious.

[ ] N int( A ) = <  max// G ( x ) ) max ( o G_; ( x ) ), ( 1 - max^, ( x ) ) >  or = <  ma\ ( u _G; ( x ) ) min (<7 G. ( x ) ) ( 1 - max ^ q ( x ) ) >

Based on knowing that ( 1 - max ^ G ( x ) ) = min(1 - u _G ) then [ ] N int( A ) = <  ma^ o G. ( x ) ), mai ( r G. ( x ) ), min ( 1 - uq. ( x ) ) > or       <  ma^ u G. ( x ) ) min i m _G; ( x ) ), min ( 1 - uq, ( x ) ) >      =

[ ] N int( A )

In a similar way the others can prove.

Proposition 4.2

• ^a)     N int( ^A {[ ], < > } ) = ( N int( A )){ [ ], <   .

• ^{b)     Ncl} ( ^A {[ ], < > } ) { [ ], < > } =^{( Ncl} ( ^A ) ) { [ ], < > }

Proof

Obvious

Definition 4.3

Let A =< д а( x ), о д ( x ), v a ( x ) >  be a neutrosophic sets on a neutrosophic topological space ( X , _T ) . We define neutrosophic exterior of A as follows: A NNE = 1 N о A C

Definition 4.4

Let A =< ц а( x ), ct a( x ), v a ( x ) >  be a neutrosophic open sets and B =< ц в ( x ), ^ b( x ), v g ( x ) >  be a neutrosophic set on a neutrosophic topological space ( X , _T ) then

• a)    A is called neutrosophic regular open iff

A = N int( Ncl (A)).

• b)    If B e NCS ( X ) then B is called neutrosophic regular closed iff A = Ncl ( N int( A )).

Now, we shall obtain a formal model for simple spatial neutrosophic region based on neutrosophic connectedness.

Definition 4.5

Let A =< ц а( x ),c a( x ), v a ( x ) >  be a neutrosophic sets on a neutrosophic topological space ( X , _T ) . Then A is called a simple neutrosophic region in connected NTS, such that

• i)     n cl ( A ), n cl ( A ) [ ] , and Ncl ( A )   are neutrosophic

regular closed.

• ii)    n int( a ), n int( a )[ ] , and N int( A )   are neutrosophic

regular open

• iii)    d ( A ), d ( A ) [ ] , and d ( A ) < >  are neutrosophic connected.

Having Ncl ( A ), Ncl ( A ) [ ] ,    Ncl ( A ) < >  , N int( A ), N int( A ) [ ] ,

Nint(A)< > are   d(A),  d(A)[ ], and   5(A)< > for two neutrosophic regions, we enable to find relationships between two neutrosophic regions

• V. C onclusion

Neutrosophic logic is well equipped to deal with missing data. By employing NSs in spatial data models, we can express a hesitation concerning the object of interest. This article has gone a step forward in developing methods that can be used to define neutrosophic spatial regions and their relationships. The main contributions of the paper can be described as the following: Possible applications have been listed after the definition of NS. Links to other models have been shown. We are defining some new operators to describe objects, describing a simple neutrosophic region. This paper has demonstrated that spatial object may profitably be addressed in terms of neutrosophic logic. Implementation of the named applications is necessary as a proof of concept.