Neutrosophic Hyperideals of Semihyperrings by Mia Amalia

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Neutrosophic Sets and Systems, Vol.12, 2016

University of New Mexico

Neutrosophic Hyperideals of Semihyperrings Debabrata Mandal Department of Mathematics, Raja Peary Mohan College, Uttarpara, Hooghly-712258, India, dmandaljumath@gmail.com

Abstract. In this paper, we have introduced neutrosophic hyperideals of a semihyperring and considered some op-

erations on them to study its basic notions and properties.

Keywords: Cartesian Product, Composition, Ideal, Intersection, Neutrosophic, Semihyperrimg.

1 Introduction Hyperrings extend the classical notion of rings, substituting both or only one of the binary operations of addition and multiplication by hyperoperations. Hyperrings were introduced by several authors in different ways. If only the addition is a hyperoperation and the multiplication is a binary operation, then we say that R is a Krasner hyperring [4]. Davvaz [5] has defined some relations in hyperrings and proved isomorphism theorems. For a more comprehensive introduction about hyperrings, we refer to [9]. As a generalization of a ring, semiring was introduced by Vandiver [17] in 1934. A semiring is a structure ( R;;;0) with two binary operations  and  such that ( R;;0) is a commutative semigroup, ( R;) a semigroup, multiplication is distributive from both sides over addition and 0  x  0  x  0 for all x  R . In [18], Vougiouklis generalizes the notion of hyperring and named it as semihyperring, where both the addition and multiplication are hyperoperation. Semihyperrings are a generalization of Krasner hyperrings. Note that a semiring with zero is a semihyperring. Davvaz in [12] studied the notion of semihyperrings in a general form. Hyperstructures, in particular hypergroups, were introduced in 1934 by Marty [11] at the eighth congress of Scandinavian Mathematicians. The notion of algebraic hyperstructure has been developed in the following decades and nowadays by many authors, especially Corsini [2, 3], Davvaz [5, 6, 7, 8, 9], Mittas [12], Spartalis [15], Stratigopoulos [16] and Vougiouklis [19]. Basic definitions and notions concerning hyperstructure theory can be found in [2]. The concept of a fuzzy set, introduced by Zadeh in his classical paper [20], provides a natural framework for generalizing some of the notions of classical algebraic struc-

tures.As a generalization of fuzzy sets, the intuitionistic fuzzy set was introduced by Atanassov [1] in 1986, where besides the degree of membership of each element there was considered a degree of non-membership with (membership value + non-membership value)≤ 1. There are also several well-known theories, for instances, rough sets, vague sets, interval-valued sets etc. which can be considered as mathematical tools for dealing with uncertainties. In 2005, inspired from the sport games (winning/tie/ defeating), votes, from (yes /NA /no),from decision making(making a decision/ hesitating/not making), from (accepted /pending /rejected) etc. and guided by the fact that the law of excluded middle did not work any longer in the modern logics, F. Smarandache [14] combined the nonstandard analysis [8,18] with a tri-component logic/set/probability theory and with philosophy and introduced Neutrosophic set which represents the main distinction between fuzzy and intuitionistic fuzzy logic/set. Here he included the middle component, i.e., the neutral/ indeterminate/ unknown part (besides the truth/membership and falsehood/non-membership components that both appear in fuzzy logic/set) to distinguish between ’absolute membership and relative membership’ or ’absolute nonmembership and relative non-membership’. Using this concept, in this paper, we have deﬁned neutrosophic ideals of semihyperrings and study some of its basic properties. 2 Preliminaries Let H be a non-empty set and let P(H ) be the set of all non-empty subsets of H . A hyperoperation on H is a map  : H  H  P( H ) and the couple ( H ,) is called a hypergroupoid. If A and B are non-empty subsets of H and x  H , then we denote A  B   a  b ,

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

aA,bB

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Neutrosophic Sets and Systems, Vol. 12, 2016

 0  AT ( x)  A I ( x)  A F ( x)  3 . From philosophx  A  {x}  A and A  x  A  {x} . A hypergroupoid the value ( H ,) is called a semihypergroup if for all x, y, z  H ical point of view, the neutrosophic set takes   we have ( x  y)  z  x  ( y  z ) which means that from real standard or non-standard subsets of ]0,1[ . But  u  z   xv. in real life application in scientific and engineering probux y vy  z

A semihyperring is an algebraic structure ( R;;) which satisfies the following properties: (i) ( R;) is a commutative semihypergroup

(ii) ( R;) is a semihypergroup (iii) Multiplication is distributive with respect to hyperoperation + that is x  ( y  z)  x  y  x  z ,

(x  y)  z  x  z  y  z (iv) 0  x  0  x  0 for all x  R.

lems it is difficult to use neutrosophic set with value from

real standard or non-standard subset of ]0,1[ . Hence we consider the neutrosophic set which takes the value from the subset of [0, 1]. Throughout this section unless otherwise mentioned R denotes a semihyperring. 

Definition 3.2. Let

 T (x) ,  I (x) or  F (x) not equal to x  R ).Then  is called a neutrosophic

( R;;) is called commutative if and only if a  b  b  a for all a, b  R . of R if Vougiouklis in [18] and Davvaz in [6] studied the notion of semihyperrings in a general form, i.e., both the sum and product are hyperoperations.

( R;;) with identity 1R  R means that 1R  x  x  1R  x for all x  R . An element x  R is called unit if there exists y  R such that 1R  x  y  y  x . A nonempty subset S of a semihyperring ( R;;) is called a sub-semihyperring if a  b  S and a  b  S for all a, b  S . A left hyperideal of a semihyperring R is a non-empty subset I of R satisfying (i) If a, b  I then a  b  I (ii) If a  I and s  R then s  a  I (iii) I  R. A right hyperideal of R is defined in an analogous manner and an hyperideal of R is a nonempty subset which is both a left hyperideal and a right hyperideal of R .

(i) (ii)

A semihyperring

For more results on semihyperrings and neutrosophic sets we refer to [6, 10] and [14] respectively. 3. Main Results

Definition 3.1. [14] A neutrosophic set A on the universe of discourse X is defined as

AT , A I , A F : X ] 0,1[ 

  ( T ,  I ,  F ) be a non empty

neutrosophic subset of a semihyperring R (i.e. anyone of

A semihyperring

A  { x : AT ( x), A I ( x), A F ( x) , x  X }



where and

zero for some left hyperideal

inf  T ( z )  min{  T ( x),  T ( y )},

zx  y

inf  I ( z ) 

 I ( x)   I ( y ) 2

zx  y

(iii) sup

zx  y

 F ( z )  max{ F ( x),  F ( y )},

(iv) inf

 T ( z )   T ( y ),

(v) inf

 I ( z )   I ( y ),

zxy

(vi) sup  ( z )  F

zxy

 F ( y ).

for all x, y  R. Similarly we can define neutrosophic right hyperideal of R.

R  {0, a, b, c} be a set with the hyperoperation  and the multiplication  defined as fol-

Example 3.3. Let lows:



0 0 a b c

a a {a,b} b c

b b b {0,b} c

c c c c {0,c}



0 0 0 0

a 0 a a

b 0 a b

c 0 a c

0 a b c and 0 a b

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

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Neutrosophic Sets and Systems, Vol. 12, 2016

c Then

( R,,) is a semihyperring.

inf  I ( z ) 

 I ( x)   I ( y )

zx  y 2  of R by F sup  ( z )  max{ F ( x),  F ( y )}  (0)  (1,0.6,0.1) ,  (a)  (0.7,0.4,0.3) , zx  y  (b)  (0.8,0.5,0.2)  (c)  (0.6,0.2,0.4) . Then  For the first inequality, choose 1 is a neutrosophic left hyperideal of R . t1  [ inf  T ( z )  min{  T ( x),  T ( y )}] . Then 2 zx  y Theorem 3.4. A neutrosophic set  of R is a neutroinf  T ( z ) t 1  min{  T ( x),  T ( y )} which implies sophic left hyperideal of R if and only if any level subsets zx  y T T , x, y   t but x  y   t - a contradiction. tT : {x  R :  T ( x)  t , t  [0.1]}

Define

neutrosophic

subset

For the second inequality, choose

and  : {x  R :  ( x)  t , t  [0.1]} 1  I ( z )  min{  I ( x),  I ( y )}] . Then  : {x  R :  T ( x)  t , t  [0.1]} are left hyperide- t 2  2 [ zinf x  y als of R .  I ( x)   I ( y ) inf  I ( z )  t 2  which implies 2 Proof. Assume that the neutrosophic set  of R is a neu- zx  y x, y   tI but x  y   tI - a contradiction. trosophic left hyperideal of R . T I F the third inequality, choose Then anyone of  ,  or  is not equal to zero for For 1 T I some x  R i.e., in other words anyone of  t ,  t or t 3  [ sup  F ( z )  max{  F ( x ),  F ( y )}] . Then I t F t

 t F is not empty for some t  [0,1] . So, it is sufficient to consider that all of them are not empty. Suppose x, y   t  (  t ,  t ,  t ) and T

s  R .Then

inf  T ( z )  min{  T ( x),  T ( y )}  min{ t , t}  t

 I ( x)   I ( y )

t t t 2 2 sup  F ( z )  max{ F ( x),  F ( y )}  max{t , t}  t

zx  y



zx  y

which implies x  y 

 t T ,  t I ,  t F i.e., x  y   t .

Also

inf  ( z )   ( x)  t , T

zsx

inf  I ( z )   I ( x)  t , zsx

sup  F ( z )   F ( x)  t , zsx

Hence sx 

t .

 t is a left hyperideal of R . Conversely, suppose  t (  ) is a left hyperideal of R . If possible  is not a neutrosophic left hyperideal. Then for x, y  R anyone of the following inequality is true. inf  T ( z )  min{  T ( x),  T ( y )} Therefore

zx  y

sup  ( z ) t 3  max{ F ( x),  F ( y )} which imF

zx  y

inf  I ( z ) 

plies x, y   t3 but x  y   t3 - a contradiction. F

So, in any case we have a contradiction to the fact that  t is a left hyperideal of R .

Hence the result follows. Definition 3.5. Let  and  be two neutrosophic subsets of R. The intersection of  and is  defined by

(  T  T )( x)  min{  T ( x), T ( x)} (  I  I )( x)  min{  I ( x), I ( x)} (  F  F )( x)  max{ F ( x), F ( x)} for all x  R. Proposition 3.6. Intersection of a nonempty collection of neutrosophic left hyperideals is a neutrosophic left hyperideal of R . Proof. Let { i : i  I } be a non-empty family of neutrosophic left hyperideals of R and

inf (   )( z )

zx  y iI

T i

 inf inf  iT ( z ) zx  y iI

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

x, y  R . Then

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Neutrosophic Sets and Systems, Vol. 12, 2016

 inf {min{  iT ( x),  iT ( y)}}

sup(   iF )( z )

 min{inf  ( x), inf  ( y)}

 sup sup  iF ( z )

iI

T i

iI

T i

 min{   iT ( x),   iT ( y)} iI

iI

zsx iI

 sup  iF ( x)    iF ( x) iI

iI

inf (   iI )( z )

Hence   i is a neutrosophic left hyperideal of R .

zx  y iI

iI

 inf inf  iI ( z ) zx  y iI

 inf

 iI ( x)   iI ( y )

2 inf  ( x)  inf  iI ( y) iI

 

I i

iI

2   ( x)    iI ( y)

iI

I i

iI

sup (   iF )( z )

 sup sup  iF ( z ) zx  y iI

 sup{max{ iF ( x),  iF ( y)}} iI

 max{sup  iF ( x), sup  iF ( y )} iI

iI

 max{  ( x),   iF ( y)} iI

inf (   )( z ) zsx iI

T i

 inf inf  iT ( z ) zsx iI

 inf  iT ( x ) iI

   iT ( x) iI

inf (   iI )( z )

a, b  R (i) f (a  b)  f (a)  f (b) (ii ) f (ab)  f (a) f (b) (iii ) f (0 R )  0 S where 0 R and 0 S are the zeros of R and S respectively. morphism if for all

Proposition 3.8. Let f : R  S be a morphism of semihyperrings. Then (i) If  is a neutrosophic left hyperideal of S , then

zx  y iI

F i

Definition 3.7. Let R , S be semihyperrings and f : R  S be a function. Then f is said to be a homo-

iI

( ) [13] is a neutrosophic left hyperideal of R . (ii) If f is surjective morphism and  is a neutronsophic left hyperideal of R , then f (  ) [13] is a neutrosophic left hyperideal of S . f

1

f : R  S be a morphism of semihyperrings. Let  be a neutrosophic left hyperideal of S and r, s  R . inf f 1 ( T )( z ) Proof. Let

zr  s

 inf  T ( f ( z )) zr  s



inf

f ( z ) f ( r )  f ( s )

 T ( f ( z ))

zsx iI

 min{ T ( f (r )),  T ( f ( s))}

 inf inf  iI ( z )

 min{ f

 inf  iI ( x)

inf f 1 ( I )( z )

zsx iI iI

   iI ( x) iI

1

( T )(r ), f

1

zr  s

 inf  I ( f ( z )) zr  s



Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

inf

f ( z ) f ( r )  f ( s )

 I ( f ( z ))

( T )( s)} .

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Neutrosophic Sets and Systems, Vol. 12, 2016

 ' inf'

 I ( f (r ))   I ( f ( s ))



2 f ( )( r )  f  2 1 F sup f ( )( z ) 1

1

( I )( s )

 .

 sup  ( f ( z )) zr  s

sup

f ( z ) f ( r )  f ( s )

 F ( f ( z ))

 max{ F ( f (r )),  F ( f ( s))}  max{ f

1

( F )(r ), f 1 ( F )( s)} .

Again

inf f ( )( z )  inf  T ( f ( z )) zrs

inf

f ( z ) f ( r ) f ( s )

 T ( f ( z ))

  T ( f ( s))  f 1 ( T )( s) . 1

zrs

( )( z )

{min{  T ( x),  T ( y )}}

(y )

y f 1 ( y ' )

z x  y

 ' inf'

sup  I ( z )

 ' inf'

sup

z x  y ' z f 1 ( z ' )

z x  y ' x f 1 ( x ' ), y f 1 ( y ' )

inf

f ( z ) f ( r ) f ( s )

1

( I )( s) .

 sup  F ( f ( z )) zrs

 F ( f ( z ))

  F ( f ( s))  f 1 ( F )( s). ( ) is a neutrosophic left hyperideal of R . (ii) Suppose  be a neutrosophic left hyperideal of R and

Thus f

1

x , y  S . Then inf' ' ( f (  T ))( z ' ) '

z x  y

 ' inf'

sup  T ( z )

z x  y z f 1 ( z ' )

 sup

zrs

sup

1  [ sup  I ( x)  sup  I ( y )] 2 x f 1 ( x ' ) y f 1 ( y ' ) 1  [( f (  I ))( x ' )  ( f (  I ))( y ' )] . 2 sup ( f (  F ))( z ' )

inf

1 ' z x  y z f ( z )

sup f 1 ( F )( z )

f ( z ) f ( r ) f ( s )

 I ( x)   I ( y )

sup

 I ( f ( z ))

  I ( f ( s))  f

 I ( z)

x f 1 ( x ' ), y f 1 ( y ' )

 sup

zrs

( x ), y f

z ' x '  y '

 inf  I ( f ( z ))



1

 min{( f (  ))( x ), ( f (  T ))( y ' )} . inf' ' ( f (  I ))( z ' ) '

zrs



 T ( z)

 min{ sup  T ( x), sup  T ( y )}

 1

inf f

sup



x f

1

x f 1 ( x ' )

zr  s



sup

z x  y ' x f 1 ( x ' ), y f 1 ( y ' )

 F ( z)

inf

' ' 1 1 z x  y x f ( x ), y f ( y ) '



x f

1

inf '

1

( x ), y f

{max{ F ( x),  F ( y )}}

(y )

 max{ inf 1

( x' )

x f

 F ( z)

 F ( x), inf

y f 1 ( y ' )

 F ( y)}

 max{( f (  F ))( x ' ), ( f (  F ))( y ' )} . Again

inf ( f (  T ))( z ' )

z ' x ' y '

 inf ' '

sup  T ( z )

z x y z f 1 ( z ' )

 x f

1

sup '

( x ), y f

1

(y )

 T ( z)

 sup  ( y )  ( f (  T ))( y ' ). T

y f 1 ( y ' )

inf ( f (  I ))( z ' )

z ' x ' y '

 inf ' '

sup  I ( z )

z x y ' z f 1 ( z ' )

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

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Neutrosophic Sets and Systems, Vol. 12, 2016

 x f

1

sup '

( x ), y f

1

(y )

 I ( z)



 sup  ( y )  ( f (  I ))( y ' ) . I

1  ( x1 )   ( y1 )  I ( x 2 )   I ( y 2 )  [  ] 2 2 2 1  I ( x1 )   I ( x 2 )  I ( y1 )   I ( y 2 )  [  ] 2 2 2

sup ( f (  F ))( z ' )

z ' x ' y '

z x y z f '



' '

( z' )

 F ( z)

inf

x f 1 ( x ' ), y f 1 ( y ' )

 inf 1 y f

Thus

( y' )

 F ( z)

sup

f (  ) is a neutrosophic left hyperideal of S .

( z1 , z 2 )( x1 , x 2 )  ( y1 , y 2 )

 and  be two neutrosophic subsets of R. Then the Cartesian product of  and  is de-

Definition 3.9. Let fined by

 

 I ( x)   I ( y ) 2

(   )( x, y)  max{ F ( x), F ( y)} for all x, y  R.

(  F  F )( z1 , z 2 )

sup max

{ F ( z1 ), F ( z 2 )}

z1( x1  y1 ), z 2 ( x2  y 2 ) z1( x1  y1 ), z 2 ( x2  y2 )

inf

( z1 , z 2 )( x1 , x2 )( y1 , y 2 )

 and  be two neutrosophic left hyperideals of R . Then   is a neutrosophic left hyperideal of R  R.



inf

z1x1 y1 , z 2 x2 y 2

inf

( z1 , z 2 )( x1 , x2 )  ( y1 , y 2 )

inf

(  T  T )( z1 , z 2 )

inf min

{ T ( z1 ), T ( z 2 )}

z1( x1  y1 ), z 2 ( x2  y 2 )

(  T  T )( z1 , z 2 )

z1x1 y1 , z 2 x2 y 2

 min{  T ( y1 ), T ( y 2 )}  (  T  T )( y1 , y 2 ) . inf

(  T  T )( z1 , z 2 )

( z1 , z 2 )( x1 , x2 )( y1 , y 2 )



z1( x1  y1 ), z 2 ( x2  y 2 )

(  T  T )( z1 , z 2 )

 inf min { T ( z1 ), T ( z 2 )}

Proof. Let ( x1 , x 2 ) , ( y1 , y 2 )  R  R . Then



sup

Theorem 3.10. Let



(  F  F )( z1 , z2 )

 max{max{  F ( x1 ),  F ( y1 )}, max{ F ( x2 ), F ( y 2 )}}  max{max{  F ( x1 ), F ( x 2 )}, max{ F ( y1 ), F ( y 2 )}}  max{(  F  F )( x1 , x2 ), (  F  F )( y1 , y 2 )} .

(  T  T )( x, y)  min{  T ( x), T ( y)}

1  [(  I  I )( x1 , x 2 )  (  I  I )( y1 , y 2 )] . 2

 F ( y)  ( f (  F ))( y ' )

(  I  I )( x, y ) 

z1( x1  y1 ), z 2 ( x2  y 2 ) I

y f 1 ( y ' )

 sup inf 1

 I ( z1 )   I ( z 2 )

inf

z1x1 y1 , z 2 x2 y 2

inf

(  I  I )( z1 , z 2 )

 I ( z1 )   I ( z 2 ) 2

z1x1 y1 , z 2 x2 y 2

 min{min{  ( x1 ),  ( y1 )}, min{ ( x2 ), ( y 2 )}}  ( y1 )   ( y 2 )   (  I  I )( y1 , y 2 ) . T T T T  min{min{  ( x1 ), ( x 2 )}, min{  ( y1 ), ( y 2 )}} 2 T T T T sup (  F  F )( z1 , z 2 )  min{(   )( x1 , x2 ), (   )( y1 , y 2 )} . T

( z1 , z 2 )( x1 , x2 )( y1 , y 2 )

inf

( z1 , z 2 )( x1 , x2 )  ( y1 , y 2 )



inf

(  I  I )( z1 , z 2 )

z1( x1  y1 ), z 2 ( x2  y 2 )

(   )( z1 , z 2 ) I



sup

z1x1 y1 , z 2 x2 y2

(  F  F )( z1 , z 2 )

 sup max { F ( z1 ), F ( z 2 )}

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

z1x1 y1 , z 2 x2 y 2

111

Neutrosophic Sets and Systems, Vol. 12, 2016

 max{ F ( y1 ), F ( y 2 )}  (  F  F )( y1 , y 2 ) . Hence

 

is a neutrosophic left hyperideal of

 min{ sup{min { T (ci ), T (d i )}} ,

 aibi

x



R  R.

Definition 3.11. Let  and  be two neutrosophic sets of a semiring R . Define composition of  and  by (  T  T )( x)  sup{min {  T (ai ), T (bi )}}

 0 if x cannot be expressed as above n  I (ai )   I (bi ) (  I  I )( x)  sup  n 2n i 1 x ai bi

i 1



ei f i

i 1

 min{(   T )( x), (  T  T )( y)} T

sup  zx  y n i 1 x  y ai bi

 inf

 I (ai )   I (bi ) 2n

i 1



x



sup



(   )( z )  inf{ max{  (ai ), (bi )}} n

y

zx  y

 0 if x cannot be expressed as above F

sup{min { T (ei ), T ( f i )}}}

i 1

ci d i

inf (  I  I )( z )

i 1

x

i 1



ci d i , y

 I ( c i )   I ( d i )   I ( ei )   I ( f i ) 4n

i 1

ei f i

i 1

 aibi

x

i 1

n  I (c i )   I ( d i ) , 1  [ sup  2 x n c d , i 1 2n i ii

 0 if x cannot be expressed as above where x, ai , bi  R for

i  1,..., n.

i 1

 and  be two neutrosophic left hyperideals of , R then  is a neutrosophic left hyperideal of R .

sup  n i 1 y ei f i , i

Theorem 3.12. If

 ,

Proof. Suppose

be two neutrosophic hyperideals of

R and x, y  R . If x  y



i 1

i i

for

ai , bi  R, then there is nothing to proof. So, assume that n

x  y   ai bi for ai , bi  R, . Then



zx  y

 inf sup{min {  T (ai ), T (bi )}} zx  y



x  y

ai bi

i 1

sup{min { T (ci ), T (d i ),  T (ei ), T ( f i )}}



 ci di , y ei f i

x

i 1

]

(  I  I )( x)  (  I  I )( y ) 2

sup (  F  F )( z )

zx  y

 sup inf{ max{  F (ai ), F (bi )}} zx  y

 aibi

x  y

i 1

inf (  T  T )( z )

i 1

a b

 I (e i )   I ( f i )

i 1

inf{ max { F (ci ), F (d i ),  F (ei ), F ( f i )}}





x

 ei fi

ci d i , y

i 1

 max{inf{ max{ F (ci ), F (d i )}} , i



x

ci d i

i 1

inf{ max{ F (ei ), F ( f i )}}} n



y

i 1

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

ei f i

112

Neutrosophic Sets and Systems, Vol. 12, 2016

 max{(  F  F )( x), (  F  F )( y)} inf (  T  T )( z ) zxy

 inf sup{min {  (ai ), (bi )}} T

zxy

 ai bi

xy

i 1

 sup{min {  T ( xei ), T ( f i )}} i

 xei f i

zxy

i 1

 sup{min {  (ei ), ( f i )}} T

 ei f i

y

inf (   )( z ) I

zxy

 inf sup  zxy n i 1 xy ai bi

 I (ai )   I (bi ) 2n

i 1



sup  n i 1 zxy xei f i

 I ( xei )   I ( f i ) 2n

i 1

 sup  n i 1 y ei f i , i

 I (ei )   I ( f i ) 2n

]

i 1

 (  I  I )( y)

sup(  F  F )( z ) zxy

 sup inf{ max{  F (ai ), F (bi )}} zxy



xy

ai bi

i 1

 inf{ max{ F ( xei ), F ( f i )}} n



xy

xei f i

i 1

 inf{ max{ F (ei ), F ( f i )}}} n



y

ei f i

i 1

 (   F )( y) . F



is a neutrosophic left hyperideal of R .

Conclusion This is the introductory paper on neutrosophic hyperideals of semihyperrings in the sense of Smarandache[14]. Our next aim to use these results to study some other properties such prime neutrosophic hyperideal, semiprime neutrosophic hyperideal,neutrosophic bi-hyperideal, neutrosophic quasi-hyperideal, radicals etc. Acknowledgement: The author is highly thankful to the learned Referees and the Editors for their valuable comments. References

i 1

 (  T  T )( y) I

Hence

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87 - 96. [2] P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani Editore, Italy, 1993. [3] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory" Adv. Math., Kluwer Academic Publishers, Dordrecht, 2003. [4] M. Krasner, A class of hyperrings and hyperfields, Internat. J. Math. Math. Sci. 6(2)(1983), 307-312. [5] B. Davvaz, Isomorphism theorems of hyperrings, Indian J. Pure Appl. Math. 35(3)(2004), 321-331. [6] B. Davvaz, Rings derived from semihyperrings, Algebras Groups Geom. 20 (2003), 245- 252. [7] B. Davvaz, Some results on congruences in semihypergroups, Bull. Malays. Math. Sci. Soc. 23(2) (2000), 53-58. [8] B. Davvaz, Polygroup Theory and Related Systems", World scientific publishing Co. Pte. Ltd., Hackensack, NJ, 2013. [9] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications", International Academic Press, Palm Harbor, USA, 2007. [10] B. Davvaz and S. Omidi, Basic notions and ptoperties of ordered semihyperrings, Categories and General Algebraic Structure with Applications, 4(1) (2016), In press. [11] F. Marty, Sur une generalisation de la notion de groupe, 8iem Congress Math. Scandinaves, Stockholm (1934), 45-49. [12] J. Mittas, Hypergroupes canoniques, Math. Balkanica 2 (1972), 165-179. [13] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. [14] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Int.J. Pure Appl. Math. 24 (2005) 287 - 297. [15] S. Spartalis, A class of hyperrings, Rivista Mat. Pura Appl. 4 (1989), 56-64.

Debabrata Mandal, Neutrosophic Hyperideals of Semihyperrings

Neutrosophic Sets and Systems, Vol. 12, 2016

[16] D. Stratigopoulos, Hyperanneaux, hypercorps, hypermodules, hyperspaces vectoriels etleurs proprietes elementaires, C. R. Acad. Sci., Paris A (269) (1969), 489-492. [17] H.S. Vandiver, Note on a simple type of algebra in which cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40 (1934), 914-920. [18] T. Vougiouklis, On some representations of hypergroups, Ann. Sci. Univ. Clermont Ferrand II Math. 26 (1990), 21-29. [19] T. Vougiouklis, Hyperstructures and Their Representations", Hadronic Press Inc., Florida, 1994. [20] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338 - 353. Received: April 04, 2016. Accepted: May 13, 2016

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113