NEUTROSOPHIC HYPERIDEALS OF Γ-SEMIHYPERRINGS by Mia Amalia

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(2017), 153-164 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p)

NEUTROSOPHIC HYPERIDEALS OF Γ-SEMIHYPERRINGS Debabrata Mandal Abstract. The purpose of this paper is to introduced neutrosophic hyperideals of a Γ-semihyperring and consider some operations on them to investigate some of its basic properties.

1. Introduction Hyperstructures, in particular hypergroups, were introduced in 1934 by Marty [12] 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 [13], Spartalis [16], Stratigopoulos [17] and Vougiouklis [20]. Basic definitions and notions concerning hyperstructure theory can be found in [2]. The classical notion of rings was extended by hyperrings, 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 prove isomorphism theorems. For a more comprehensive introduction about hyperrings, we refer to [9]. As a generalization of a ring, semiring was introduced by Vandiver [18] 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 [19], 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 2010 Mathematics Subject Classification. 16Y99, 20N20, 20N25. Key words and phrases. Hyperideal, Neutrosophic, Semihyperring. 153

154

D. MANDAL

semihyperring. Davvaz in [12] studied the notion of semihyperrings in a general form. The concept of a fuzzy set, introduced by Zadeh in his classical paper [21], provides a natural framework for generalizing some of the notions of classical algebraic structures.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)6 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 [15] combined the non-standard 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 non-membership and relative non-membership’. Using this concept, in this paper, I have defined neutrosophic ideals of Γsemihyperrings 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∈A,b∈B

a ◦ b, x ◦ A = x ◦ A and A ◦ x = A ◦ x

A hypergroupoid (H, ◦) is called a semihypergroup if for all x, y, z ∈ H we have (x ◦ y) ◦ z = x ◦ (y ◦ z) which means that ∪ u◦z =

u∈x◦y

∪ x ◦ v.

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 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.

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155

A semihyperring (R, +, ·) is called commutative if and only if a · b = b · a for all a, b ∈ R. Vougiouklis in [19] and Davvaz in [6] studied the notion of semihyperrings in a general form, i.e., both the sum and product are hyperoperations. A semihyperring (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 ̸= S. 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 ideal and a right ideal of R. We now recall the definition of Γ-semihyperring from [11]. Let R be a commutative semihypergroup and Γ be a commutative group. Then R is called a Γ-semihyperring if there exists a map R × Γ × R → P (R) ( (a, α, b) 7→ aαb) for a, b ∈ R, α ∈ Γ and P (R)−the set of all non-empty subsets of R, satisfying the following conditions: (i) (a + b)αc = aαc + bαc, (ii) aα(b + c) = aαb + aαc, (iii) a(α + β)b = aαb + aβb, (iv) aα(bβc) = (aαb)βc for all a, b, c ∈ S and for all α, β ∈ Γ. We say that R is a Γ-semihyperring with zero, if there exists 0 ∈ R such that a ∈ a + 0 and 0 ∈ 0αa, 0 ∈ aα0 for all a ∈ R and α ∈ Γ. For more results on semirings and neutrosophic sets we refer to [6, 10] and [15] respectively. 3. Main Results Definition 3.1. [15] A neutrosophic set A on the universe of discourse X is defined as A = {< x, AT (x), AI (x), AF (x) >, x ∈ X}, where AT , AI , AF : X →]− 0, 1+ [ and − 0 6 AT (x) + AI (x) + AF (x) 6 3+ . From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ]− 0, 1+ [. But in real life application in scientific and engineering problems 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 Γ-semihyperring.

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D. MANDAL

Definition 3.2. Let µ = (µT , µI , µF ) be a non empty neutrosophic subset of a semihyperring R (i.e. anyone of µT (x), µI (x) or µF (x) not equal to zero for some x ∈ S). Then µ is called a neutrosophic left hyperideal of R if (i) inf µT (z) > min{µT (x), µT (y)}, inf µT (z) > µT (y) z∈x+y

(ii)

z∈xγy

inf µI (z) >

z∈x+y

µI (x)+µI (y) , inf µI (z) 2 z∈xγy

> µI (y)

(iii) sup µF (x + y) 6 max{µF (x), µF (y)}, sup µF (z) 6 µF (y). z∈x+y

z∈xγy

for all x, y ∈ R and γ ∈ Γ. Similarly we can define neutrosophic right hyperideal of R. Example 3.1. Let R = {a, b, c, d}, Γ = Z2 . Define the hyperoperation ⊕ and the multiplication ⊙ on R as follows: ⊕ a b c d ⊙ a a {a, b} {c, d} {c, d} a b {a, b} b {c, d} {c, d} and b c {c, d} {c, d} c {c, d} c d {c, d} {c, d} {c, d} d d

a {a, b} {a, b} {a, b} {a, b}

b c d {a, b} {a, b} {a, b} {a, b} {a, b} {a, b} {a, b} {c, d} {c, d} {a, b} {c, d} {c, d}

Also, for x, y ∈ R and α ∈ Z2 , define xαy = {a, b}. Then (R, ⊕, ⊙) is a Γ-semihyperring. Define neutrosophic subset µ of R by µ(a) = (0.5, 0.3, 0.4), µ(b) = (0.6, 0.3, 0.4), µ(c) = (0.7, 0.4, 0.2), µ(d) = (0.8, 0.4, 0.1). Then µ is a neutrosophic left hyperideal of R. Theorem 3.1. A neutrosophic set µ of R is a neutrosophic left hyperideal of R if and only if any level subsets µTt := {x ∈ S : µT (x) > t, t ∈ [0, 1]}, F µIt := {x ∈ S : µI (x) > t, t ∈ [0, 1]} and µF t := {x ∈ S : µ (x) 6 t, t ∈ [0, 1]} are left hyperideals of R. Proof. Assume that the neutrosophic set µ of R is a neutrosophic left hyperideal of R. Then anyone of µT , µI or µF is not equal to zero for some x ∈ S i.e., in other words anyone of µTt , µIt or µF t is not equal to zero for all t ∈ [0, 1]. So it is suﬃcient to consider that all of them are not equal to zero. Suppose x, y ∈ µt = (µTt , µIt , µF t ) and s ∈ R. Then inf µT (z) > min{µT (x), µT (y)} > min{t, t} = t

z∈x+y

inf µI (z) >

z∈x+y

µI (x)+µI (y) 2

t+t 2

sup µ (x + y) 6 max{µ (x), µF (y)} 6 max{t, t} = t F

z∈x+y

which implies x + y ⊆ µTt , µIt , µF t i.e., x + y ⊆ µt . Also inf µT (z) > µT (x) > t

z∈sγx

inf µI (z) > µI (x) > t

z∈sγx

sup µF (z) 6 µF (x) 6 t

z∈sγx

NEUTROSOPHIC HYPERIDEALS OF Γ-SEMIHYPERRINGS

157

Hence sγx ⊆ µt . Therefore µt is a left hyperideal of R. Conversely, suppose µt (̸= ϕ) is a left hyperideal of S. If possible µ is not a neutrosophic left hyperideal. Then for x, y ∈ S anyone of the following inequality is true. inf µT (z) < min{µT (x), µT (y)} z∈x+y

inf µI (z) <

z∈x+y

µI (x)+µI (y) 2

sup µF (x + y) > max{µF (x), µF (y)} z∈x+y

For the first inequality, choose t1 = 21 [ inf µT (z) + min{µT (x), µT (y)}]. Then z∈x+y

inf µT (z) < t1 < min{µT (x), µT (y)} which implies x, y ∈ µTt1 but x + y ̸⊆ µTt1 - a

z∈x+y

contradiction. For the second inequality, choose t2 = 12 [ inf µI (z)+min{µI (x), µI (y)}]. Then z∈x+y

inf µ (z) < t2 <

z∈x+y

µI (x)+µI (y) 2

which implies x, y ∈ µIt2 but x + y ̸⊆ µIt2 - a

contradiction. For the third inequality, choose t3 = 12 [ sup µF (x + y) + max{µF (x), µF (y)}]. z∈x+y

Then sup µF (x + y) > t3 > max{µF (x), µF (y)} which implies x, y ∈ µF t3 but z∈x+y

x + y ̸⊆ µF t3 - a contradiction. So, in any case we have a contradiction to the fact that µt is a left hyperideal of R. Hence the result follows. Definition 3.3. Let µ and ν be two neutrosophic subsets of S. The intersection of µ and ν is defined by (µT ∩ ν T )(x) = min{µT (x), ν T (x)}, (µI ∩ ν I )(x) = min{µI (x), ν I (x)}, (µF ∩ ν F )(x) = max{µF (x), ν F (x)} for all x ∈ S. Proposition 3.1. Intersection of a nonempty collection of neutrosophic left hyperideals is a neutrosophic left hyperideal of R. Proof. Let {µi : i ∈ I} be a non-empty family of neutrosophic left hyperideals of S and x, y ∈ S and γ ∈ Γ. Then inf ( ∩ µT )(z) z∈x+y i∈I i

= inf inf µTi (z) z∈x+y i∈I

> inf {min{µTi (x), µTi (y)}} i∈I

= min{inf µTi (x), inf µTi (y)} i∈I

i∈I

= min{( ∩ µTi )(x), ( ∩ µTi )(y)} i∈I

i∈I

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D. MANDAL

inf ( ∩ µI )(z) z∈x+y i∈I i

= inf inf µIi (z) z∈x+y i∈I µIi (x)+µIi (y) 2 i∈I

> inf = =

inf µIi (x)+inf µIi (y) i∈I

i∈I

2 ∩ µIi (x)+ ∩ µIi (y)

i∈I

F sup ( ∩ µF i )(z) = sup sup µi (z)

z∈x+y i∈I

F 6 sup {max{µF i (x), µi (y)}} i∈I

F = max{sup µF i (x), sup µi (y)} i∈I

i∈I

F = max{( ∩ µF i )(x), ( ∩ µi )(y)} i∈I

i∈I

inf ( ∩ µTi )(z) = inf inf µTi (z) > inf µTi (x) = ( ∩ µTi )(x).

z∈sγx i∈I

inf ( ∩

z∈sγx i∈I

µIi )(z)

= inf inf

z∈sγx i∈I

i∈I

µIi (z)

> inf

i∈I

µIi (x)

= ( ∩ µIi )(x). i∈I

F F F sup ( ∩ µF i )(z) = sup sup µi (z) 6 sup µi (x) = ( ∩ µi )(x).

z∈sγx i∈I

i∈I

Hence ∩ µi is a neutrosophic left hyperideal of R. i∈I

Definition 3.4. Let R, S be Γ-semihyperrings and f : R → S be a function. Then f is said to be a homomorphism if for all a, b ∈ R and γ ∈ Γ (i) f (a + b) ⊆ f (a) + f (b) (ii) f (aγb) ⊆ f (a)γf (b) (iii) f (0R ) = 0S where 0R and 0S are the zeroes of R and S respectively. Proposition 3.2. Let f : R → S be a morphism of Γ-semihyperrings. Then (i) If ϕ is a neutrosophic left hyperideal of S, then f −1 (ϕ) [14] is a neutrosophic left hyperideal of R. (ii) If f is surjective morphism and µ is a neutrosophic left hyperideal of R, then f (µ) [14] is a neutrosophic left hyperideal of S. Proof. Let f : R → S be a morphism of Γ-semihyperrings. (i) Let ϕ be a neutrosophic left hyperideal of S and r, s ∈ R and γ ∈ Γ. inf f −1 (ϕT )(z) = inf ϕT (f (z)) >

z∈r+s

inf

ϕT (f (z))

f (z)⊆f (r)+f (s)

> min{ϕT (f (r)), ϕT (f (s))} = min{(f −1 (ϕT ))(r), (f −1 (ϕT ))(s)}. inf f −1 (ϕI )(z)

z∈r+s

= inf ϕI (f (z)) > >

inf ϕI (f (z)) z∈r+s f (z)⊆f (r)+f (s) −1 I −1 ϕI (f (r))+ϕI (f (s)) (ϕI ))(s) = (f (ϕ ))(r)+(f . 2 2

NEUTROSOPHIC HYPERIDEALS OF Γ-SEMIHYPERRINGS

sup f −1 (ϕF )(z)

= sup ϕF (f (z)) 6

z∈r+s

159

ϕF (f (z))

sup f (z)⊆f (r)+f (s)

6 max{ϕF (f (r)), ϕF (f (s))} = max{(f −1 (ϕF ))(r), (f −1 (ϕF ))(s)}. Again inf (f −1 (ϕT ))(z) = inf ϕT (f (z)) >

z∈rγs

z∈rγs T

> ϕ (f (s)) = (f inf (f −1 (ϕI ))(z)

(ϕ ))(s)

= inf ϕI (f (z)) >

z∈rγs

z∈rγs I

> ϕ (f (s)) = (f z∈rγs F

6 ϕ (f (s)) = (f Thus f

−1

ϕI (f (z))

inf

f (z)⊆f (r)γf (s) −1 I

(ϕ ))(s)

sup (f −1 (ϕF ))(z) = sup ϕF (f (z)) 6

z∈rγs

ϕT (f (z))

inf

f (z)⊆f (r)γf (s) −1 T

ϕF (f (z))

sup

f (z)⊆f (r)γf (s) −1 F

(ϕ ))(s)

(ϕ) is a neutrosophic left hyperideal of R. ′

′

(ii)Suppose µ be a neutrosophic left hyperideal of R and x , y ∈ S. Then ′

inf

z ′ ∈x′ +y ′

(f (µT ))(z )

= >

sup µT (z)

inf

z ′ ∈x′ +y ′ z∈f −1 (z ′ )

sup µT (z)

inf

z ′ ∈x′ +y ′ x∈f −1 (x′ ),y∈f −1 (y ′ )

> sup{min{µT (x), µT (y)}} ′

′

= min{sup µT (x), sup µT (y)} = min{(f (µT ))(x ), (f (µT ))(y )} x∈f −1 (x′ )

′

inf ′

z ∈x +y

′

(f (µI ))(z ) = > =

′

z ∈x +y

sup µI (z)

inf

z ′ ∈x′ +y ′ z∈f −1 (z ′ ) ′

inf ′

sup µI (z)

′

> sup µ

(x)+µI (y) 2

z ∈x +y x∈f −1 (x′ ),y∈f −1 (y ′ ) ′ 1 1 I I I 2 [sup µ (x) + sup µ (y)] = 2 [(f (µ ))(x ) ′ ′ x∈f −1 (x ) y∈f −1 (y )

′

sup (f (µF ))(x + y ) ′

y∈f −1 (y ′ )

′

sup ′

′

+ (f (µI ))(y )]

inf µF (z)

z ∈x +y z∈f −1 (x′ +y ′ ) F

sup

inf µ (z)

z ′ ∈x′ +y ′ x∈f −1 (x′ ),y∈f −1 (y ′ ) F F

6 inf{max{µF (x), µF (y)}} ′

′

= max{inf µ (x), inf µ (y)} = max{(f (µF ))(x ), (f (µF ))(y )} x∈f −1 (x′ )

y∈f −1 (y ′ )

Again inf

z ′ ∈x′ γ ′ y ′

′

f (µT )(z ) =

inf

sup µT (z) >

z ′ ∈x′ γ ′ y ′ z∈f −1 (z ′ ) T T

sup µT (z) x∈f −1 (x′ ),y∈f −1 (y ′ )

′

> sup µ (y) = f (µ )(y ) y∈f −1 (y ′ )

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D. MANDAL

′

inf f (µI )(z ) = ′ ′ ′

z ∈x γ y

inf

sup µI (z)

z ′ ∈x′ γ ′ y ′ z∈f −1 (z ′ ) I

sup µ (z)

x∈f −1 (x′ ),y∈f −1 (y ′ ) I I

′

> sup µ (y) = f (µ )(y ) y∈f −1 (y ′ )

sup

z ′ ∈x′ γ ′ y ′

′

f (µF )(z ) =

sup

inf µF (z) 6

z ′ ∈x′ γ ′ y ′ z∈f −1 (z ′ ) F F

inf µF (z) x∈f −1 (x′ ),y∈f −1 (y ′ )

′

6 inf µ (y) = f (µ )(y ) y∈f −1 (y ′ )

Thus f (µ)is a neutrosophic left hyperideal of S.

Definition 3.5. Let µ and ν be two neutrosophic subsets of R. The cartesian product of µ and ν is defined by (µT × ν T )(x, y) = min{µT (x), ν T (y)} µI (x) + ν I (y) 2 (µF × ν F )(x, y) = max{µF (x), ν F (y)} (µI × ν I )(x, y) =

for all x, y ∈ R. Theorem 3.2. Let µ and ν be two neutrosophic left hyperideals of R. Then µ × ν is a neutrosophic left hyperideal of R × R. Proof. Let (x1 , x2 ), (y1 , y2 ) ∈ R × R. Then inf(µT × ν T )(z1 , z2 )

= inf(µT × ν T )(z1 , z2 ) z1 ∈x1 +y1 ,z2 ∈x2 +y2

(z1 ,z2 )∈(x1 ,x2 )+(y1 ,y2 )

= inf min{µT (z1 ), ν T (z2 )} z1 ∈x1 +y1 ,z2 ∈x2 +y2

> min{min{µT (x1 ), µT (y1 )}, min{ν T (x2 ), ν T (y2 )}} = min{min{µT (x1 ), ν T (x2 )}, min{µT (y1 ), ν T (y2 )}} = min{(µT × ν T )(x1 , x2 ), (µT × ν T )(y1 , y2 )}. inf(µI × ν I )(z1 , z2 ) (z1 ,z2 )∈(x1 ,x2 )+(y1 ,y2 )

= inf(µI × ν I )(z1 , z2 ) = > = =

sup(µF × ν F )(z1 , z2 ) (z1 ,z2 )∈(x1 ,x2 )+(y1 ,y2 )

z1 ∈x1 +y1 ,z2 ∈x2 +y2 I I (z2 ) inf µ (z1 )+ν 2 z1 ∈x1 +y1 ,z2 ∈x2 +y2 I I I I (y2 ) 1 µ (x1 )+µ (y1 ) + ν (x2 )+ν } 2{ 2 2 I I I I µ (y1 )+ν (y2 ) 1 µ (x1 )+ν (x2 ) { + } 2 2 2 1 I I I I {(µ × ν )(x , x ) + (µ × ν )(y1 , y2 )}. 1 2 2

= sup(µF × ν F )(z1 , z2 ) z1 ∈x1 +y1 ,z2 ∈x2 +y2

= sup max{µF (z1 ), ν F (z2 )} z1 ∈x1 +y1 ,z2 ∈x2 +y2

6 max{max{µF (x1 ), µF (y1 )}, max{ν F (x2 ), ν F (y2 )}} = max{max{µF (x1 ), ν F (x2 )}, max{µF (y1 ), ν F (y2 )}} = max{(µF × ν F )(x1 , x2 ), (µF × ν F )(y1 , y2 )}.

NEUTROSOPHIC HYPERIDEALS OF Γ-SEMIHYPERRINGS

161

inf(µT × ν T )((z1 , z2 )) = inf(µT × ν T )(z1 , z2 ) = inf min{µT (z1 ), ν T (z2 )} (z1 ,z2 )∈(x1 ,x2 )γ(y1 ,y2 )

(z1 ,z2 )∈(x1 γy1 ,x2 γy2 )

z1 ∈x1 γy1 ,z2 ∈x2 γy2

> min{µT (y1 ), ν T (y2 )} = (µT × ν T )(y1 , y2 ). µI (z1 )+ν I (z2 ) 2 z1 ∈x1 γy1 ,z2 ∈x2 γy2 ν I )(y1 , y2 ).

inf(µI × ν I )((z1 , z2 )) = inf(µI × ν I )(z1 , z2 ) = inf (z1 ,z2 )∈(x1 ,x2 )γ(y1 ,y2 )

(z1 ,z2 )∈(x1 γy1 ,x2 γy2 ) µI (y1 )+ν I (y2 ) = (µI 2

sup(µF × ν F )((z1 , z2 )) = sup(µF × ν F )(z1 , z2 ) = sup max{µF (z1 ), ν F (z2 )} (z1 ,z2 )∈(x1 ,x2 )γ(y1 ,y2 )

(z1 ,z2 )∈(x1 γy1 ,x2 γy2 )

z1 ∈x1 γy1 ,z2 ∈x2 γy2

6 max{µF (y1 ), ν F (y2 )} = (µF × ν F )(y1 , y2 ).

Hence µ × ν is a neutrosophic left hyperideal of R × R.

Theorem 3.3. Let µ be a neutrosophic subset of R. Then µ is a neutrosophic left hyperideal of R if and only if µ × µ is a neutrosophic left hyperideal of R × R. Proof. The proof follows by routine verification.

Definition 3.6. Let µ and ν be two neutrosophic sets of a semiring R. Define composition of µ and ν by

µT ◦ ν T (x) =

sup

n ∑ x∈

{min{µT (ai ), ν T (bi )}} i

a i γi b i

i=1

= 0, if x cannot be expressed as above µI ◦ ν I (x)

sup n ∑ x∈ ai γi bi

∑n i=1

µI (ai )+ν I (bi ) 2n

i=1

= 0, if x cannot be expressed as above µF ◦ ν F (x)

inf {max{µF (ai ), ν F (bi )}} n i ∑ x∈ ai γi bi i=1

= 0, if x cannot be expressed as above where x, ai , bi ∈ R for i = 1, ..., n. Theorem 3.4. If µ and ν be two neutrosophic left hyperideals of R then µoν is also a neutrosophic left hyperideal of R.

162

D. MANDAL

Proof. Suppose ∑ µ, ν be two neutrosophic hyperideals of R and x, y ∈ R. If n x + y is not equal to i=1 ai γi bi , for ai , bi ∈ R∑ and γi ∈ Γ, then there is nothing n to proof. So, assume that x + y is not equal to i=1 ai γi bi . Then inf (µT oν T )(z)

z∈x+y

{µT (ai ), ν T (bi )}}

sup{min i

n ∑ z∈x+y∈

ai γi bi

i=1

n ∑ x∈

sup

n ∑

ci γi di , y ∈

i=1

{min{µT (ci ), ν T (di ), µT (ei ), ν T (fi )}} i

ei δi fi

i=1

= min{

{min{µT (ci ), ν T (di )}},

sup

n ∑

x∈

ci γi di

{min{µT (ei ), ν T (fi )}}}

sup

n ∑

y∈

i=1

ei δi fi

i=1

= min{(µT oν T )(x), (µT oν T )(y)} inf (µI oν I )(z)

z∈x+y

sup n ∑ z∈x+y∈

∑n i=1

µI (ai )+ν I (bi ) 2n

ai γi bi

i=1

n ∑ x∈

sup ci γi di , y ∈

i=1

> 21 [

sup n ∑ x∈ ci γi di

∑n i=1

n ∑

µI (ci )+ν I (di )+µI (ei )+ν I (fi ) 4n

ei δ i f i i=1 ∑n µI (ci )+ν I (di ) , i=1 2n

i=1 (µI oν I )(x)+(µI oν I )(y) 2

sup n ∑ y∈ ei δi fi

∑n i=1

µI (ei )+ν I (fi ) ] 2n

i=1

sup (µF oν F )(z) z∈x+y

inf n ∑ z∈x+y∈

{max{µF (ai ), ν F (bi )}} i

a i γi b i

i=1

n ∑ x∈

inf ci γi di , y ∈

i=1

= max{

n ∑

{max{µF (ci ), ν F (di ), µF (ei ), ν F (fi )}} i

ei δi fi

i=1

inf {max{µF (ci ), ν F (di )}}, inf {min{µF (ei ), ν F (fi )}}} n n i i ∑ ∑ x∈ ci γi di y∈ ei δ i f i i=1

= max{(µF oν F )(x), (µF oν F )(y)}

i=1

NEUTROSOPHIC HYPERIDEALS OF Γ-SEMIHYPERRINGS

inf (µT oν T )(z)

z∈xγy

sup n ∑ z∈xγy∈

163

{min{µT (ai ), ν T (bi )}} i

ai γi bi

i=1

sup {min{µT (xδi ei ), ν T (fi )}} i n ∑ z∈xγy∈ xδi ei γi fi

sup {min{µT (ei ), ν T (fi )}} = (µT oν T )(y) i n ∑ y∈ ei γ i f i

i=1

inf (µI oν I )(z) =

sup n ∑

z∈xγy

z∈xγy∈

∑n i=1

ai γi bi

i=1

sup n ∑ z∈xγy∈

∑n i=1

µI (xδi ei )+ν I (fi ) 2n

xδi ei fi

i=1

µI (ai )+ν I (bi ) 2n

sup n ∑ y∈ ei γ i f i

∑n i=1

µI (ei )+ν I (fi ) 2n

= (µI oν I )(y)

i=1

sup (µF oν F )(z) =

inf n ∑

z∈xγy z∈xγy∈

ai γi bi

i=1

inf n ∑ z∈xγy∈

{max{µF (ai ), ν F (bi )}}

{max{µF (xei ), ν F (fi )}} i

xδi ei fi

i=1

inf {max{µF (ei ), ν F (fi )}} = (µF oν F )(y) n i ∑ y∈ ei γ i f i i=1

Hence µoν is a neutrosophic left hyperideal of R.

Conclusion: This is the introductory paper on neutrosophic hyperideals of semihyperrings in the sense of Smarandache[15]. 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..

164

D. MANDAL

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Department of Mathematics, Raja Peary Mohan College, Uttarpara, Hooghly712258, India E-mail address: dmandaljumath@gmail.com