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Isomorphism of single valued neutrosophic hypergraphs.

1 Introduction

The neutrosophic set (NS) was proposed by Smarandache [8] as a general ization of the fuzzy sets [14], intuitionistic fuzzy sets [12], interval valued fuzzy set [11] and interval-valued intuitionistic fuzzy sets [13] theories, and it is a powerful mathematical tool for dealing with incomplete, indeterminate and inconsistent information in the real world. The neutrosophic sets are characterized by a truth-membership function (t), an indeterminacy-membership function (/) and a falsity membership function (f independently, which are within the real standard or non-standard unit interval ]-0, 1+[. To conveniently use NS in the real-life applications, Wang et al. [9] introduced the single-valued neutrosophic set (SVNS), as a subclass of the neutrosophic sets. The same authors [10] introduced the interval valued neutrosophic set (IVNS), which is even more precise and flexible than the single valued neutrosophic set. The IVNS is a generalization of the single valued neutrosophic set, in which the three membership functions are independent, and their values belong to the unit interval [0, 1]. The hypergraph is a graph in which an edge can connect more than two vertices. Hypergraphs can be applied to analyse architecture structures and to represent system partitions. In this paper, we extend the concept into isomorphism of single valued neutrosophic hypergraphs, and some of their properties are introduced.

2 Preliminaries Definition 2.1

A hypergraph is an ordered pair H = (X, E), where:

(1) X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]} a finite set of vertices;

(2) E = {[E.sub.1], [E.sub.2], ..., [E.sub.m]} a family of subsets of X;

(3) [E.sub.j] are not-empty for j = 1,2,3, ..., m and [U.sub.j] ([E.sub.j]) = X.

The set X is called set of vertices and E is the set of edges (or hyper-edges). Definition 2.2

A fuzzy hypergraph H = (X, E) is a pair, where X is a finite set and E is a finite family of non-trivial fuzzy subsets of X, such that X = [[union].sub.j] Supp([E.sub.j]), j = 1,2,3, ..., m.

Remark 2.3

The collection E = {[E.sub.1], [E.sub.2], [E.sub.3], ..., [E.sub.m]] is the collection of edge sets of H.

Definition 2.4

A fuzzy hypergraph with underlying set X is of the form H = (X, E R), where E = {[E.sub.1], [E.sub.2], [E.sub.3], ..., [E.sub.m]] is the collection of fuzzy subsets of X, that is [E.sub.j] : X [right arrow] [0,1], j = 1, 2, 3, ..., m and R : E [right arrow] [0,1] is a fuzzy relation on fuzzy subsets [E.sub.j], such that:

R([x.sub.1], [x.sub.2], ..., [x.sub.r]) [less than or equal to] min([E.sub.j]([x.sub.1]), ..., [E.sub.j]([x.sub.r])), (1)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X

Definition 2.5

Let X be a space of points (objects) with generic elements in Xdenoted by x. A single valued neutrosophic set A (SVNS A) is characterized by truth membership function [T.sub.A] (x), indeterminacy membership function [I.sub.A] (x), and a falsity membership function [F.sub.A](x). For each point x [member of]X; [T.sub.A](x), [I.sub.A](x), [F.sub.A] (x) [member of] [0, 1].

Definition 2.6

A single valued neutrosophic hypergraph (SVNHG) is an ordered pair H = (X, E), where:

(1) X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]} a finite set of vertices.

(2) E = {[E.sub.1], [E.sub.2], ..., [E.sub.m]} a family of SVNSs of X

(3) [E.sub.j] [not equal to] O = (0, 0, 0) for j = 1, 2, 3, m and [U.sub.j] Supp([E.sub.j]) = X

The set X is called set of vertices and Eis the set of SVN-edges (or SVN-hyperedges).

Proposition 2.7

The SVNHG is the generalization of the fuzzy hypergraphs and of the intuitionistic fuzzy hypergraphs.

Let be given a SVNHGH = (X, E R), with underlying set X, where E = {[E.sub.1], [E.sub.2], , ..., [E.sub.m]} is the collection of non-empty family of SVN subsets of X and R being SVN's relation on SVN subsets [E.sub.j] such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X

Example 2.8

Consider the SVNHG H = (X, E R) with underlying set X = {a, b, c} where E = {A, B} and R, which is defined in the Tables given below.
H   A               B

a   (0.2,0.3,0.9)   (0.5,0.2,0.7)
b   (0.5,0.5,0.5)   (0.1,0.6,0.4)
c   (0.8,0.8,0.3)   (0.5,0.9,0.8)

R   [R.sub.T]       [R.sub.I]       RF

A   0.2             0.8             0.9
B   0.1             0.9             0.8


By routine calculations, H = (X, E, R) is a SVNHG.

3 Isomorphism of SVNHGs

Definition 3.1

A homomorphism f : H [right arrow] K between two SVNHGs H = (X, E, R) and K = (Y, F, S) is a mapping f: X [right arrow] Y, which satisfies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

for all x [member of] X, and

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [less than or equal to] [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (8)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (9)

[R.sub.F] ([x.sub.i], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (10)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Example 3.2

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined in the Tables given below, and f: X [right arrow] Y defined by f(a) = x, f(b) = y and f(c) = z.
H   A               B

a   (0.2,0.3,0.9)   (0.5,0.2,0.7)
b   (0.5,0.5,0.5)   (0.1,0.6,0.4)
c   (0.8,0.8,0.3)   (0.5,0.9,0.8)

K   C               D

x   (0.3,0.2,0.2)   (0.2,0.1,0.3)
y   (0.2,0.4,0.2)   (0.3,0.2,0.1)
z   (0.5,0.8,0.2)   (0.9, 0.7, 0.1)

R   [R.sub.T]   [R.sub.I]   RF

A   0.2         0.8         0.9
B   0.1         0.9         0.8

S   [S.sub.T]   [S.sub.I]   SF

C   0.2         0.8         0.3
D   0.1         0.7         0.3


By routine calculations, f: H [right arrow] K is a homomorphism between H and K.

Definition 3.3

A weak isomorphism f: H [right arrow] K between two SVNHGs H = (X, E, R) and K = (Y, F, S) is a bijective mapping f: X [right arrow] Y, which satisfies f is homomorphism, such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

for all x [member of] X.

Note

The weak isomorphism between two SVNHGs preserves the weights of vertices.

Example 3.4

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined in the Tables given below, and f: X [right arrow] Y defined by f(a) = x, f(b) = y and f(c) = z.
H   A               B

a   (0.2,0.3,0.9)   (0.5,0.2,0.7)
b   (0.5,0.5,0.5)   (0.1,0.6,0.4)
c   (0.8,0.8,0.3)   (0.5,0.9,0.8)

K   C               D

x   (0.2,0.3,0.2)   (0.2,0.1,0.8)
y   (0.2,0.4,0.2)   (0.1,0.6,0.5)
z   (0.5,0.8,0.9)   (0.9,0.9,0.1)

R   [R.sub.T]   [R.sub.I]   [R.sub.R]

A   0.2         0.8         0.9
B   0.1         0.9         0.9

S   [S.sub.T]   [S.sub.I]   [S.sub.F]

C   0.2         0.8         0.9
D   0.1         0.9         0.8


By routine calculations, f: H [right arrow] K is a weak isomorphism between H and K.

Definition 3.5

A co-weak isomorphism f: H [right arrow] K between two SVNHGs H = (X, E, R) and K = (Y, F, S) is a bijective mapping f: X [right arrow] Y which satisfies f is homomorphism, i.e.:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (14)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (15)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (16)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Note

The co-weak isomorphism between two SVNHGs preserves the weights of edges.

Example 3.6

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S are defined in the Tables given below, and f: X [right arrow] Y defined by f(a) = x, f(b) = y and f(c) = z.
H   A               B

a   (0.2,0.3,0.9)   (0.5,0.2,0.7)
b   (0.5,0.5,0.5)   (0.1,0.6,0.4)
c   (0.8,0.8,0.3)   (0.5,0.9,0.8)

K   C               D

x   (0.3,0.2,0.2)   (0.2,0.1,0.3)
y   (0.2,0.4,0.2)   (0.3,0.2,0.1)
z   (0.5,0.8,0.2)   (0.9, 0.7, 0.1)

R   [R.sub.T]   [R.sub.I]   [R.sub.F]

A   0.2         0.8         0.9
B   0.1         0.9         0.8

S   [S.sub.T]   [S.sub.I]   [S.sub.F]

C   0.2         0.8         0.9
D   0.1         0.9         0.8


By routine calculations, f: H [right arrow] K is a co-weak isomorphism between H and K.

Definition 3.7

An isomorphism f: H [right arrow] K between two SVNHGs H = (X, E, R) and K = (Y, F, S) is a bijective mapping f: X [right arrow] Y, which satisfies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (20)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (21)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (22)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Note

The isomorphism between two SVNHGs preserves both the weights of vertices and the weights of edges.

Example 3.8

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined in the Tables given below, and f: X [right arrow] Y defined by, f(a) = x, f(b) = y and f(c) = z.
H   A               B

a   (0.2,0.3,0.7)   (0.5,0.2,0.7)
b   (0.5,0.5,0.5)   (0.1,0.6,0.4)
c   (0.8,0.8,0.3)   (0.5,0.9,0.8)

K   C               D

x   (0.2,0.3,0.2)   (0.2,0.1,0.8)
y   (0.2,0.4,0.2)   (0.1,0.6,0.5)
z   (0.5,0.8,0.7)   (0.9,0.9,0.1)

R   [R.sub.T]   [R.sub.I]   [R.sub.F]

A   0.2         0.8         0.9
B   0.0         0.9         0.8

S   [S.sub.T]   [S.sub.I]   [S.sub.F]

C   0.2         0.8         0.9
D   0.0         0.9         0.8


By routine calculations, f: H [right arrow] K is an isomorphism between H and K.

Definition 3.9

Let H = (X, E, R) be a SVNHG; then, the order of H is denoted and defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

and the size of H is denoted and defined by:

S(H) = ([SIGMA][R.sub.T]([E.sub.j]), [SIGMA][R.sub.I] ([E.sub.j]), [SIGMA][R.sub.F] ([E.sub.j])). (24)

Theorem 3.10

Let H = (X, E, R) and K = (Y, F, S) be two SVNHGs, such that H is isomorphic to K.

Then:

(1) O(H) = O(K);

(2) S(H) = S(K).

Proof.

Let f: H [right arrow] K be an isomorphism between H and K with underlying sets X and Y respectively.

Then, by definition, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (28)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (29)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (30)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Consider:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

Similarly, [O.sub.I] (H) = [O.sub.I] (K) and [O.sub.F] (H) = [O.sub.F] (K), hence O(H) = O(K).

Next,

[S.sub.T] (H) = [SIGMA][R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [SIGMA][S.sub.T] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])) = [S.sub.T](K) (32)

Similarly, [S.sub.I] (H) = [S.sub.I] (K), [S.sub.F] (H) = [S.sub.F] (K), hence S(H) = S(K).

Remark 3.11

The converse of the above theorem need not to be true in general.

Example 3.12

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c, d} and Y = {w, x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined in the Tables given below, where f is defined by f(a) = w, f(b) = x, f(c) = y, f(d) = z.
H   A                 B

a   (0.2,0.5,0.33)    (0.16,0.5,0.33)
b   (0.0,0.0,0.0)     (0.2,0.5,0.33)
c   (0.33,0.5,0.33)   (0.2,0.5,0.33)
d   (0.5,0.5,0.33)    (0.0,0.0,0.0)

K   C                 D

w   (0.2,0.5,0.33)    (0.2,0.5,0.33)
x   (0.16,0.5,0.33)   (0.33,0.5,0.33)
y   (0.33,0.5,0.33)   (0.2,0.5,0.33)
z   (0.5,0.5,0.33)    (0.0,0.0,0.0)

R   [R.sub.T]   [R.sub.I]   [R.sub.F]

A   0.2         0.5         0.33
B   0.16        0.5         0.33

S   [S.sub.T]   [S.sub.I]   [S.sub.F]

C   0.16        0.5         0.33
D   0.2         0.5         0.33


Here, O(H) = (1.06, 2.0,1.32) = O(K) and S(H) = (0.36,1.0, 0.66) = S(K), but, by routine calculations, H is not isomorphism to K.

Corollary 3.13

The weak isomorphism between any two SVNHGs preserves the orders. Remark 3.14

The converse of above corollary need not to be true in general.

Example 3.15

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c, d} and Y = {w, x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined in the Tables given below, where f is defined by f(a) = w, f(b) = x, f(c) = y, f(d) = z.
H   A                B

a   (0.2,0.5,0.3)    (0.14,0.5,0.3)
b   (0.0,0.0,0.0)    (0.2,0.5,0.3)
c   (0.33,0.5,0.3)   (0.16,0.5,0.3)
d   (0.5,0.5,0.3)    (0.0,0.0,0.0)

K   C                D

w   (0.14,0.5,0.3)   (0.16,0.5,0.3)
x   (0.0,0.0,0.0)    (0.16,0.5,0.3)
y   (0.25,0.5,0.3)   (0.2,0.5,0.3)
z   (0.5,0.5,0.3)    (0.0,0.0,0.0)


Here, O(H)= (1.0, 2.0,1.2) = O(K), but, by routine calculations, H is not weak isomorphism to K.

Corollary 3.16

The co-weak isomorphism between any two SVNHGs preserves sizes.

Remark 3.17

The converse of above corollary need not to be true in general.

Example 3.18

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c, d} and Y = {w, x, y, z}, where E = {A, B}, F = {C, D}, R and S are defined in the Tables given below, where f is defined by f(a) = w, f(b) = x, f(c) = y, f(d) = z.
H   A               B

a   (0.2,0.5,0.3)   (0.14,0.5,0.3)
b   (0.0,0.0,0.0)   (0.16,0.5,0.3)
c   (0.3,0.5,0.3)   (0.2,0.5,0.3)
d   (0.5,0.5,0.3)   (0.0,0.0,0.0)

K   C                D

w   (0.0,0.0,0.0)    (0.2,0.5,0.3)
x   (0.14,0.5,0.3)   (0.25,0.5,0.3)
y   (0.5,0.5,0.3)    (0.2,0.5,0.3)
z   (0.3,0.5,0.3)    (0.0,0.0,0.0)

R   [R.sub.T]   [R.sub.I]   [R.sub.F]

A   0.2         0.5         0.3
B   0.14        0.5         0.3

S   [S.sub.T]   [S.sub.I]   [S.sub.F]

C   0.14        0.5         0.3
D   0.2         0.5         0.3


Here, S(H) = (0.34,1.0,0.6) = S(K), but, by routine calculations, H is not co-weak isomorphism to K.

Definition 3.19

Let H = (X, E, R) be a SVNHG; then the degree of vertex [x.sub.i] is denoted and defined by:

deg([x.sub.i]) = ([deg.sub.T]([x.sub.i]), [deg.sub.I] ([x.sub.i]), [deg.sub.F] ([x.sub.i])), (33)

where

[deg.sub.T] ([x.sub.i]) = [SIGMA] [R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]), (34)

[deg.sub.I] ([x.sub.i]) = [SIGMA] [R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]), (35)

[deg.sub.F] ([x.sub.i]) = [SIGMA] [R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]), (36)

for [x.sub.i] [not equal to] [x.sub.r].

Theorem 3.20

If H and K are two isomorphic SVNHGs, then the degree of their vertices is preserved.

Proof.

Let f: H [right arrow] K be an isomorphism between H and K with underlying sets X and Y respectively; then, by definition, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

for all x [member of]X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (40)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (41)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (42)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Consider:

[deg.sub.T]([x.sub.i]) = [SIGMA] [R.sub.T]([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [SIGMA][S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])) = [deg.sub.T] (f([x.sub.i])). (43)

Similarly:

[deg.sub.I]([x.sub.i]) = [deg.sub.I](f([x.sub.i])), [deg.sub.F]([x.sub.i]) = [deg.sub.F](f([x.sub.i])) (44)

Hence:

deg([x.sub.i]) = deg(f([x.sub.i])). (45)

Remark 3.21

The converse of the above theorem may not be true in general.

Example 3.22

Consider the two SVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b} and Y = {x, y}, where E = {A, B}, F = {C, D}, R and S are defined in the Tables given below, where f is defined by, f(a)=x, f(b)=y, here deg(a) = (0.8, 1.0, 0.6) = deg(x) and deg(b) = (0.45,1.0, 0.6) = deg(y).
H   A                B

a   (0.5,0.5,0.3)    (0.3,0.5,0.3)
b   (0.25,0.5,0.3)   (0.2,0.5,0.3)

K   C               D

x   (0.3,0.5,0.3)   (0.5,0.5,0.3)
y   (0.2,0.5,0.3)   (0.25,0.5,0.3)

S   [S.sub.T]   [S.sub.I]   [S.sub.F]

C   0.2         0.5         0.3
D   0.25        0.5         0.3

R   [R.sub.T]   [R.sub.I]   [R.sub.F]

A   0.25        0.5         0.3
B   0.2         0.5         0.3


But H is not isomorphic to K, i.e. H is neither weak isomorphic nor co-weak isomorphic to K.

Theorem 3.23

The isomorphism between SVNHGs is an equivalence relation.

Proof.

Let H = (X, E, R), K = (Y, F, S) and M = (Z, G, W) be SVNHGs with underlying sets X, Y and Z, respectively:

--Refle[x.sub.1]ve.

Consider the map (identity map) f: X [right arrow] X defined as follows: f(x) = x for all x [member of] X, since identity map is always bijective and satisfies the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [R.sub.T] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (49)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [R.sub.I] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (50)

[R.sub.F] ([x.sub.1], [X.sub.2], ..., [x.sub.r]) = [R.sub.F] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (51)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Hence f is an isomorphism of SVNHG H to itself.

--Symmetric.

Let f: X [right arrow] Y be an isomorphism of H and K, then f is bijective mapping, defined as f(x) = y for all x [member of] X.

Then, by definition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (52)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (53)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (54)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (55)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (56)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (57)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Since f is bijective, then we have [f.sup.-1](y) = x for all y [member of] Y.

Thus, we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (59)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (60)

for all x [member of] X, and:

[R.sub.T] ([f.sup.-1]([y.sub.1]), [f.sup.-1]([y.sub.2]), ..., [f.sup.-1]([y.sub.r])] = [S.sub.T]([y.sub.1], [y.sub.2], ..., [y.sub.r]), (61)

[R.sub.I] ([f.sup.-1]([y.sub.1]), [f.sup.-1]([y.sub.2]), ..., [f.sup.-1]([y.sub.r])) = [S.sub.I]([y.sub.1], [y.sub.2], ..., [y.sub.r]), (62)

[R.sub.F] ([f.sup.-1]([y.sub.1]), [f.sup.-1]([y.sub.2]), ..., [f.sup.-1]([y.sub.r])) = [S.sub.F] ([x.sub.1], [y.sub.2], ..., [y.sub.r]), (63)

for all {[y.sub.1], [y.sub.2], ..., [y.sub.r]} subsets of Y.

Hence, we have a bijective map [f.sup.-1] : Y [right arrow] X, which is an isomorphism from K to H.

--Transitive.

Let f : X [right arrow] Y and g : Y [right arrow] Z be two isomorphism of SVNHGs of H onto K and K onto M, respectively. Then gof is a bijective mapping from X to Z, where gof is defined as (gof)(x) = g(f(x)) for all x [member of] X.

Since f is an isomorphism, then, by definition, f(x) = y for all x EX, which satisfies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (64)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (65)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (66)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.T](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (67)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (68)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) = [S.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (69)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Since g - Y [right arrow] Z is an isomorphism, then, by definition, g(y) = z for all y [member of] Y, satisfying the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (70)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (71)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (72)

for all x [member of] X, and:

[S.sub.T] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) = [W.sub.T](g([y.sub.1]), g([y.sub.2]), ..., g([y.sub.r])), (73)

[S.sub.I] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) = [W.sub.I](g([y.sub.1]), g([y.sub.2]), ..., g([y.sub.r])), (74)

[S.sub.F] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) = [W.sub.F](g([y.sub.1]), g([y.sub.2]), ..., g([y.sub.r])), (75)

for all {[y.sub.1], [y.sub.2], ..., [y.sub.r]} subsets of Y.

Thus, from above equations, we conclude that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (76)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (77)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (78)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], ..., [x.sub.r]) = [W.sub.T](g(f([x.sub.1])), ..., g(f([x.sub.r]))), (79)

[R.sub.I] ([x.sub.1], ..., [x.sub.r]) = [W.sub.I](g(f([x.sub.1])), ..., g(f([x.sub.r]))), (80)

[R.sub.F] ([x.sub.1], ..., [x.sub.r]) = [W.sub.F](g(f([x.sub.1])), ..., g(f([x.sub.r]))), (81)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Therefore, gof is an isomorphism between H and M. Hence, the isomorphism between SVNHGs is an equivalence relation.

Theorem 3.24

The weak isomorphism between SVNHGs satisfies the partial order relation.

Proof.

Let H = (X, E, R), K = (Y, F, S) and M = (Z, G, W) be SVNHGs with underlying sets X, Y and Z, respectively.

--Reflexive.

Consider the map (identity map) f : X [right arrow] X, defined as follows f(x) = x for all x [member of] X, since the identity map is always bijective and satisfies the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (82)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (83)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (84)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [less than or equal to] [R.sub.T] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (85)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [R.sub.I](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (86)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [R.sub.F](f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (87)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Hence f is a weak isomorphism of SVNHG H to itself.

--Anti-symmetric.

Let f be a weak isomorphism between H onto K, and g be a weak isomorphic between K and H, that is f : X [right arrow] Y is a bijective map defined by f(x) = y for all x [member of] X, satisfying the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (88)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (89)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (90)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [less than or equal to] [S.sub.T] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (91)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [S.sub.I] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (92)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [S.sub.F] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (93)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Since g is also a bijective map g(y) = x for all y [member of] Y satisfying the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (95)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (96)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (97)

for all y [member of] Y, and:

[R.sub.T] (y, [y.sub.2], ..., [y.sub.r]) [less than or equal to] [S.sub.T](g([y.sub.1]), g([y.sub.2]), ..., g([y.sub.r])), (98)

[R.sub.I] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) [greater than or equal to] [S.sub.I](f([y.sub.1]), f([y.sub.2]), ..., f([y.sub.r])), (99)

[R.sub.F] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) [greater than or equal to] [S.sub.F](f([y.sub.1]), f([y.sub.2]), ..., f([y.sub.r])), (100)

for all {[y.sub.1], [y.sub.2], ..., [y.sub.r]} subsets of Y.

The above inequalities hold for finite sets X and Y only when H and K SVNHGs have same number of edges and the corresponding edge have same weight, hence H is identical to K.

--Transitive.

Let f : X [right arrow] Y and g : Y [right arrow] Z be two weak isomorphism of SVNHGs of H onto K and K onto M, respectively. Then gof is a bijective mapping from X to Z, where gof is defined as (gof)(x) = g(f(x)) for all x [member of] X.

Since f is a weak isomorphism, then, by definition, f(x) = y for all x [member of] X, which satisfies the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (101)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (102)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (103)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [less than or equal to] [S.sub.T] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (104)

[R.sub.I] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [S.sub.I] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (105)

[R.sub.F] ([x.sub.1], [x.sub.2], ..., [x.sub.r]) [greater than or equal to] [S.sub.F] (f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.r])), (106)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Since g : Y [right arrow] Z is a weak isomorphism, then, by definition, g(y) = z for all y [member of] Y satisfying the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (107)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (108)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (109)

for all x [member of] X, and:

[S.sub.T] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) [less than or equal to] [W.sub.T] (g([y.sub.1]), g([y.sub.2]), ..., g([y.sub.r])), (110)

[S.sub.I] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) [greater than or equal to] [W.sub.I] (g([y.sub.1]), ..., g([y.sub.r])), (111)

[S.sub.F] ([y.sub.1], [y.sub.2], ..., [y.sub.r]) [greater than or equal to] [W.sub.F] (g([y.sub.1]), g([y.sub.2]), ..., g([y.sub.r])), (112)

for all {[y.sub.1], [y.sub.2], ..., [y.sub.r]} subsets of Y.

Thus, from above equations, we conclude that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (113)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (114)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (115)

for all x [member of] X, and:

[R.sub.T] ([x.sub.1], ..., [x.sub.r) [less than or equal to] [W.sub.T] (g(f([x.sub.2])), ..., g(f([x.sub.r]))), (116)

[R.sub.I] ([x.sub.1], ..., [x.sub.r]) [greater than or equal to] [W.sub.I] (g(f([x.sub.2])), ..., g(f([x.sub.r]))), (117)

[R.sub.F] ([x.sub.1], ..., [x.sub.r]) [greater than or equal to] [W.sub.F] (g(f([x.sub.2])), ..., g(f([x.sub.r]))) (118)

for all {[x.sub.1], [x.sub.2], ..., [x.sub.r]} subsets of X.

Therefore gof is a weak isomorphism between H and M.

Hence, a weak isomorphism between SVNHGs is a partial order relation.

4 Conclusion

Theoretical concepts of graphs and hypergraphs are highly used by computer

science applications. Single valued neutrosophic hypergraphs are more flexible than fuzzy hypergraphs and intuitionistic fuzzy hypergraphs. The concepts of single valued neutrosophic hypergraphs can be applied in various areas of engineering and computer science.

In this paper, the isomorphism between SVNHGs is proved to be an equivalence relation and the weak isomorphism to be a partial order relation. Similarly, it can be proved that a co-weak isomorphism in SVNHGs is a partial order relation.

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Muhammad Aslam Malik (1), Ali Hassan (2), Said Broumi (3), Assia Bakali (4), Mohamed Talea (5), Florentin Smarandache (6)

(1) Department of Mathematics, University of Punjab, Lahore, Pakistan aslam@math.pu.edu.pk

(2) Department of Mathematics, University of Punjab, Lahore, Pakistan alihassan.iiui.math@gmail.com

(3,5) University Hassan II, Sidi Othman, Casablanca, Morocco broumisaid78@gmail.com

(4) Ecole Royale Navale, Casablanca, Morocco assiabakali@yahoo.fr

(6) University of New Mexico, Gallup, NM, USA smarand@unm.edu
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Date:Jan 1, 2016
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