T -FUZZY HYPERIDEALS OF HYPERNEAR RINGSxajzkjdx.cn/gallery/144-dec2019.pdf · and T-fuzzy...

T -FUZZY HYPERIDEALS OF HYPERNEAR RINGS

NEHA GAHLOT AND NAGARAJU DASARI Search Schloar, Department of Mathematics & Statistics

Manipal University, Jaipur , VPO-Dehmi Kala, Tehsil-Sanganer, Off Jaipur-Ajmer Expressway, Rajasthan-303007 INDIA

Abstract. In this article we study about T -fuzzy hyperideals in hypernear ring and also talk about “Λ” (meet) and “ν”(join) lattice operations. by using triangular norm we characterize and study T -fuzzy let and T -fuzzy right hyperideals, homomorphism image and preimage, Direct product, T -fuzzy bi- hyperideals and T -fuzzy quasi-hyperideals in hypernear ring.

Key words and phrases. hypernear ring, T -norm, T –fuzzy hyperideals, T - fuzzy bi-hyperideals, T -fuzzy quasi hyperideals, Direct product

1. INTRODUCTION

A idea about fuzzy set was introduced by Zadeh in 1965 [15].

after introducing the fuzzy set many researchers apply the concept of fuzzy set in pure and applied both cases, in a group

theory fuzzy set was introduced by Rosenfeld [16] and in ring

theory introduced by T. K. Mukherjee, M. K. Sen [3]. T. K. Dutta and T. Chanda introduced fuzzy ideal of Γ-Ring [4] and fuzzy ideals

of a near-rings studied by Abou-Zaid[5].

Hyper system is well established algebraic structure in

classical algebraic theory. Hyper system have many applications

in branch of mathematics, computer applications, information science and many more. now days hyper system is studying in

large number of coun- tries of the whole world. in general hyper structure is an extantion of classical algebraic structure.

A idea about hyper structure was firstly introduced by Marty in

1934 [6] firstly he introduced hypergroup after that Marty was started their results with fraction function, applications of

group and many more algebraic structure. Recently the two

author in a monograph of Leoreanu and corsini have collected

last fifteen years Numerous applications of algebraic hyper

structure mainly from the following subjects: Lattices, Fuzzy

sets, binary relation, rough set, geometry, hypergraph, automata,

Journal of Xi'an University of Architecture & Technology

Volume XI, Issue XII, 2019

Issn No : 1006-7930

Page No: 1378

∈

Artificial intelligence, probabilities, codes, algebra, cryptography, relation algebra and median [13] and an

important relationships between algebraic structures and

hyperstructures have been studied by Ameri, Davvaz, Cristea, Leoreanu, Corsini, Zhan and many more researchers [10], [11],

[12]. M. K Sen, Reza Ameri, G. Chawd- hury introduced the

concept of fuzzy hypersemi groups in [14]. V.Leoreanu Foteaa, B.Davvazb studied about Fuzzy hyperrings in [19]. and

Characterization of hyperrings by fuzzy hyperideals with

respect to a t-norm introduced by Kostq Hila, Krisanthi Naka in [20]. fuzzy hyperideals in hypernear-rings characterize by Zhan

and also, took to the forward fuzzy hypernear rings in [17] and Davvaz also obtained some results in connection of fuzzy

hyperideals in hypernear-rings [18]. a algebraic structure

hypernear ring was introduced by Dasic in [21]. the results of hypernear ring is effecting curiously to one who willing to look

for structure where condition of symmetry is not so generous.

Allow to study of hypernear ring is challenging. S. Yamak, O.

Kazanc, B. Davvaz introduced the concept of Normal fuzzy

hyperideals in hypernear ring in [22].

A triangular conorm(t-conorm) and triangular norm(t-norm) is in

order to generalized triangular inequality in matrix space is intro-

duced by Schweizer and Sklar in 1960 [23]. there are many appli-

cation of triangular in field of mathematics, artificial intelligence, taken suggestion and so on. t-norm also play an important role in

fuzzy algebra recently many research studied on t-norm in many

different algebraic structure the reader refer to [28], [27], [26], [25],

[24].

Structure of this paper as follow after introduction, in section 2 we give definition, properties, example of T -fuzzy hyperideals in hypernear ring, in section 3 we study about direct product of t-fuzzy hyperideals in hypernear ring, in section 4 T -fuzzy quasi hyperideals and T -fuzzy bi-hyperideals in hypernear ring and its related properties, and also give idea about hypernear ring. Now recall some basic concepts which is necessary for this article. An algebraic structure (N , +, .) is a near-ring which satisfy: (1) (N , +) is a group (not necessarily commutative), (2) (N , .) is a semigroup, (3) the multiplication is distributive with respect to the + on the left side, i.e., x.(y + z) = x.y + x.z for all x, y, z N .

Precisely speaking, because it satisfies the left distributive that0s



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∈

≤ ∈ ≤ ∈ ≤ ∈

∈ ∈ ∈ ∈

{ } ≤ ∈

<

why it is a left near-ring.

A near-ring is a triple (N , +, .) where (i) (N , +, .) is a near-ring, (ii) (xy)z = x(yz) for all x, y, z N . A subset A of a near-ring N is called a left (resp., right) ideal of N if (i) (N , +, .) is a normal subgroup of N , (ii) xy A (resp y(x + z) - yz A) for all y A, x, z N . A subset A of M is called an ideal of M if it is both a left ideal and right ideal.

Definition 1. [5] A Fuzzy set μ is a fuzzy ideal of a near ring N if it satisfing following axioms: (i) min(μ(u), μ(v) μ(u - v), for u, v N . (ii) μ(v) μ(u + v - u), for u, v, w N . (iii) μ(v) μ(u(v + w) - uw), for u, v, w N . (iv) μ(u) μ(uv), for u, v N. A fuzzy subset with (i), (ii) and (iii) is called a fuzzy right ideal of N whereas a fuzzy subset with (i), (ii) and (iv) is called a fuzzy left ideal of N .

A hyperstructure is a non-empty set R together with a mapping

“ᴼ”: ℘∗( ) where ℘∗( ) is the set of all the non-empty subsets of . A canonical hypergroup (not necessarily commutative) is an algebraic structure( ,+) satisfying the

following conditions: (i) for every x, y, z ∈ , x + (y + z) = (x + y) + z (ii) there exists 0 ∈ such that 0 + x = x = x + 0 for all x ∈ . (iii) for every x ∈ , there exists a unique element x

0 ∈ such

that 0 ∈ (x + x0 ) ∩ (x

0 + x). (we call the element x

0 the inverse of

x); (iv) z ∈ x + y implies y ∈ - x + z and x ∈ z - y.

Definition 2. [17] An algebraic structure (N , +, .) is hypernear-

ring if it satisfying the following axioms: (i) (N , +) is a canonical hypergroup; (ii) with respect to the multiplication (N , +, .) is a semigroup; (iii) satisfying distributive property on the left hand side with respect to the hyperoperation ”+” , i.e.



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{ } ≤ ∈ ≤ ∈

{ } ≤ ∈

x.(y + z) = x.y + x.z for all x, y, z ∈ N .

Precisely speaking, it is a left hypernear-ring because it satisfies the left distributive law. We will use the word “hypernear-ring” to mean “left hypernear-ring”. We note that if x ∈ N and A, B are non-empty subsets of N , then by A + B, A + x and x + B we can deduce that

a∈A∪,,b∈B

a + b, A + {x} and {x} + B, respectively. Also,

for all x, y ∈ N , we have -(-x) = x and 0 = - 0 where 0 is unique and -(x + y) = -y - x. A subhypergroup A ⊆ R is called normal if for all x ∈ N , we have x + A - x ⊆ A. A normal subhypergroup A of the hypergroup N is called (i) a left hyperideal of N if xA ∈ A for all x ∈ N and a ∈ A; (ii) a right hyperideal of N if (x + A)y - xy ∈ A for all x, y ∈ N ; (iii) a hyperideal of N if (x + A)y - xy ∪ zA ∈ A for all x, y, z ∈ N .

Definition 3. [18] Let (N , +, .) be a hypernear-ring. Then we call a fuzzy set µ of N a fuzzy subhypernear-ring of N if it satisfies the following inequalities:

(i) min µ(x), µ(y) infzx-yµ(z) for all x, y N . (ii) µ(x) µ(-x), for all x N , (iii) min µ(x), µ(y) µ(xy) for all x, y N . Furthermore, we call µ a fuzzy hyperideal of N if is a fuzzy subhypernear- ring of N and (3) µ(y) ≤ infzx+y-xµ(z) for all x, y ∈ N , (4) µ(y) ≤ µ(xy) for all x, y ∈ N .

(5) µ(i) ≤ infzx(x+i)-xyµ(z)for all x, y, i ∈ N .

Definition 4. [23] A mapping T : [0, 1] [0, 1] [0, 1] is called

a triangular norm [T -norm] if and only if it satisfies the following conditions: T1) T (x, 1) = T (1, x) = 1, for all x ∈ [0, 1]. T2) If x ≥ xo, y ≥ yo then T (x, y) ≥ T (xo, yo). T3) T (x, y) = T (y, x), for all x, y ∈ [0, 1]. T4) T (x, T (y, z)) = T (T (x, y), z), for all x, y, z ∈ [0, 1].

Note. The T -norm minimum (min T -norm) is defined by T (x, y) = min(x, y). Some other T -norms are Tp(x, y) = xy, Tn(x, y) =

max(x + y - 1, 0) and



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≤ ≤

◦ { } ∧ { } ∩ { }

{ ∈ | }

{ }

→

+ 0 a b c 0 {0} {a} {b} {c} a {a} {0, a} {b} {c} b {b} {b} {0, a, c} {b, c} c {c} {c} {b, c} {0, a, b}

. 0 a b c 0 0 a b c a 0 a b c b 0 a b c c 0 a b c

Every T -norm T∗ follow: Tw(x, y) T∗ min(x, y) and, T (0, x) = 0, T (1, 1) = 1 and T∗(0, 0) = 0. for a t-norm T on [0, 1] there is ET = [ β [0, 1] T(β, β) = β . A t-norm T is generality of mini function. Sherwood and Anthony in [29] replaced the condition min{µ(x), µ(x)} ≤ µ(xy)obtained by the definition fuzzy subgroup by T {µ(x), µ(x)} ≤ µ(xy).

Remark 1. if µ and ν be any two fuzzy ideals in near ring then some fuzzy subset is defined as: 1) (µ ν) = min µ(x), ν(y) 2) (µ ν) = T µ(x), ν(y) 3) (µ ν) = Sup T µ(y), ν(z) (product)

x∈y.z

2. T -fuzzy hyperideals in hypernear-rings

Here we discus only T -fuzzy right hyperideals in hypernear - rings and give some properties related to T -fuzzy hyperideals in hypernear - rings.

Definition 5. A fuzzy hyperideal µ is a hypernear-ring N if it is satisfying following axiom: (i) T {µ(x), µ(y)} ≤ infzx-yµ(z) for all x, y ∈ N ,

(ii) µ(x) ≤ infzy+x-yµ(z) for all x, y ∈ N , (iii) µ(i) ≤ infzx(y+i) - xyµ(z) for all x, y, i ∈ N . (iv) µ(x) ≤ infz xyµ(z) for all x, y ∈ N .

Example 6. Let N = a, b, c, d and hyperoperation (+, .) on N is by the following tables:

clearly (N, +, .) is a near ring, define mapping µ: N [0, 1] by µ(0) = 0.8, µ(a) = 0.6, µ(c) = 0.2, µ(b) = 0.5, µ(c) = 0.2 and a



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≥ ≥∈ −

∈ −

≥ { } ≥ { }

{ }

→

T-norm define by T(x, y) = min(x, y). The routine calculation shows that µ is a T-fuzzy hyperideal of a hypernear-ring.

Proposition 7. Every fuzzy hyperideals in a hypernear ring N is also a T-fuzzy hyperideals.

Proof. suppose that µ be any fuzzy hyperideals in a

hypernear ring N . then (i) infz∈x−yµ(z) min µ(x), µ(y) T µ(x), µ(y) . (ii) infz x+y xµ(z) µ(y). (iii) infz (x+i)y xyµ(z) µ(i). hence N is a T -fuzzy hyperideals.

Note:- Every fuzzy hyperideals in a hypernear ring N is also a T -fuzzy hyperideals. but every T -fuzzy hyperideals is not need to be fuzzy hyperideals in a hypernear ring N . we can show this one contradiction example.

Example 8. Let N = 0, u, v, w, x be a set with two hyper operations (+, .), these two operations define as follow

Since

(N, +, .) is hypernear ring. now if mapping µ: N [0, 1] define as follow µ(0) = 0.9, µ(u) = 0.8, µ(v) = 0.6, µ(w) = 0.5, µ(x) = 0.2. then routine calculation it is clear that it is T-fuzzy hyperideal under Tp(a, b) = ab of N but it is not fuzzy hyperideal of N.

Lemma 9. If A be a fuzzy hyperideals in a hypernear ring N then χA is also a T-fuzzy hyperideals.

Proof. proof is straight forward.

Theorem 10. A fuzzy set µ in a hypernear ring N is a T-fuzzy ideal of N if and only if the level set U (µ; α) = {x ∈ N | µ(x) ≥ α}is an hyperideal of N when it is non empty.


Theorem 11. suppose that µ be a any T-fuzzy hyperideal of a hypernear ring N and α ∈ [0, 1] then,

+ 0 u v w x 0 0 u v w x u u {0, u} v w x v v v {0, u} x w w w w x {0, u} v x x x w v {0, u}

. 0 u v w x 0 0 0 0 0 0 u 0 0 0 0 0 v 0 0 0 0 0 w 0 0 0 0 0 x 0 0 0 0 0



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∧ ∈ { }

} ∧

{ }

{ ∈

(i) If α = 1, then U (µ; α) is either hyperideal of N or an empty set. (ii) If T = min, U (µ; α) is either hyperideal of N or an empty set.

(iii) µ(0) ≥ µ(x) for every x ∈ N.

Theorem 12. Let µ and ν be a two T-fuzzy hyperideals in hypernear ring N then µ ∧ ν is also T-fuzzy hyperideals in N.

Proof. let for any x, y, i ∈ N then, (i) infz∈x−y(µ ∧ ν)(z) ≥ T (infz∈x−yµ(z), infz∈x−yν(z)) ≥ T (T (µ(x), µ(y)), T (ν(x), ν(y)) = T (T (T (µ(x), µ(y)), (ν(x), ν(y)))) ≥ T (T (µ(x), ν(x)), T (µ(y), ν(y))) = T ((µ ∧ ν)(x), (µ ∧ ν)(y)). (ii) infz∈x+y−x(µ ∧ ν)(z) ≥ T (infz∈x+y−xµ(a), infz∈x+y−xν(z)) ≥ T (µ(y), ν(y)) = (µ ∧ ν)(y). (iii) infz∈x(y+i)y−xy(µ ∧ ν)(z) ≥ T (infz∈x(y+i)y−xyµ(z), infz∈x(y+i)y−xyν(z)) ≥ T (µ(i), ν(i))

= (µ ∧ ν)(i). hence µ ∧ ν is T -fuzzy hyperideals of N .

Corollary 13. If Ai for i ∈ I is a T-fuzzy hyperideal of a hypernear ring N then

i∧∈I

Ai is also a T -fuzzy hyperideal of N where i∧∈I

Ai is defined by (

i∧∈I

Ai)(x) = inf{Ai(x) | i ∈ I} for all x ∈ N.

Example 14. Let N = 0, a, b, c is a hypernear ring as in example 7. and We have (µ ν)(x) = T(µ(x), µ(x)) for all x N (by Remark 1). Define a t-norm T by T(p, q) = max(p + q - 1, 0) for all p, q [0, 1] now Define a fuzzy subset in [0, 1] by µ(0) = 0,8 and µ(a) = µ(b) = µ(c) = 0.2, Then µ = (0, 0.8), (a, 0.2), (b, 0.2), (c, 0.2) Again define a fuzzy subset ν in [0, 1] by ν(0) = 0.7 and ν(a) = 0.5 ν(b) = 0.4, ν(c) = 0.2. Then ν = (0, 0.7), (a, 0.5), (b, 0.4), (c, 0.2) . Then (µ ν)(0) = T(µ(0), ν(0)) = T(0.8, 0.7) = max(0.8 + 0.7 - 1, 0) = max(1.5 - 1, 0) = 0.5. simillary (µ ∧ ν)(a) = 0. (µ ∧ ν)(b) = 0. (µ ∧ ν)(c) = 0. So µ ∧ ν = {(0, 0.5), (a, 0), (b, 0), (c, 0)} is a fuzzy subset on N, defined by (µ ∧ ν)(0) = 0.5 (µ ∧ ν)(x) = 0 for all x ∈ N. Then it can be easily verified that µ ∧ ν is a T-fuzzy hyperideal of a hypernear ring N.

Remark 2. if we take t-norm as a min-norm then T fuzzy

hyperideals concide by fuzzy hyperideals.

Corollary 15. Let µ, ν be two fuzzy hyperideals of a hypernear ring (N, +, .) then µ ∩ ν is also a fuzzy hyperideals N.

Proof. for proof by taking t norm as a min norm in theorem

12.



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∈

≥

∈ →

→

Theorem 16. Let µ be a T-fuzzy hyperideal of a hypernear ring N and let µ+ be a fuzzy set in N defined by µ+(x) = µ(x) + 1 - µ(0) for all x N Then µ+ is normal T-fuzzy hyperideal of N containing µ.

Proof. Let µ be a T -fuzzy hyperideal of a hypernear ring R.

For any x, y, z ∈ N , now (i) infx−yµ+(z) = infx−yµ(z) + 1 - A(0) ≥ T (µ(x), µ(y)) + 1 - µ(0) = T (µ(x) + 1 - µ(0), µ(y) + 1 - µ(0)) = T (µ+(x), µ+(y)). (ii) infx+y−yµ+(z) = infx+y−yµ(z) + 1 - A(0) ≥ µ(y)) + 1 - µ(0) = µ(y) + 1 - µ(0) = µ+(y) and (iii) infx(y+i)y−xyµ+(z) = infx(y+i)y−xyµ(z) + 1 - A(0) µ(y)) + 1 - µ(0) = µ(y) + 1 - µ(0) = µ+(y) Hence µ+ is a T -fuzzy hyperideal of a hypernear ring R. Clearly µ+(0) = 1 and µ ⊆ µ+.

Definition 17. If N and M be two hypernear rings, then a mapping θ: N M such that θ(x + y) = θ(x) + θ(y), θ(xy) = θ(x)θ(y) for all x, y N hypernear ring homomorphism. certainly hypernear ring homomorphism θ is an isomorphism hypernear ring if θ is one-one(injective) and (onto)surjective. if N is isomorphism to M then we write N ∼= M .

Theorem 18. Let θ: N M be a hypernear ring homomorphism Then; (i) An onto homomorphic preimage of hypernear ring M is a T- fuzzy hyperideal of hypernear ring N. (ii) An onto homomorphic image of a T-fuzzy hyperideal of hypernear ring N is a T-fuzzy hyperideal of hypernear ring M .


3.direct product of T -fuzzy hyperideals

Definition 19. Let µ and ν be any two T -fuzzy hyperideal in hypernear ring. Then direct product of T-fuzzy hyperideals is defined by (µ × ν)(x, y) = T (µ(x), ν(y)) for every x, y ∈ N .

Theorem 20. Let N and M be hypernear rings. If µ1 and µ2 are T-fuzzy hyperideals of N and M respectively, then µ = µ1 ×

µ2 is a T-fuzzy hyperideal of the direct product N × M.

Proof. Let µ1 and µ2 be T-fuzzy hyperideals of a hypernear rings

N and M respectively. Let (x1, x2), (y1, y2), (i1, i2) ∈ N × M Then



Issn No : 1006-7930

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×

× ×

× ×

µ(x).

(i) infz∈(x1,x2)−(y1,y2)µ(z) = infz ∈(x1−y1),(x2−y2 µ(z) = infz∈(x1−y1),(x2−y2)(µ1

µ2)(z) = T (infz∈(x1−y1)µ1(z), infz∈(x2−y2)µ2(z)) = T (T (µ1(x1), µ1(y1)), T (µ2(x2), µ2(y2))) = T (T (µ1(x1), µ2(x2)), T (µ1(y1), µ2(y2))) = T ((µ1 × µ2)(x1, x2), (µ1 × µ2)(y1, y2)). (ii) infz∈(x1,x2)+(y1,y2)−(x1,x2)µ(z) = infz∈(x1+y1−x1),(x2+y2−x2)µ(z) = infz∈(x1+y1−x1),(x2+y2−x2 (µ1 × µ2)(z) = T (infz∈(x1+y1−x1)µ1(z), infz∈(x2+y2−x2)µ2(z)) = T (µ1(y1), µ2(y2)) = T ((µ1(y1), µ2(y2)) = ((µ1 × µ2)(y1, y2)). (iii)infz∈(x1,x2)((y1,y2)+(i1,i2))−(x1,x2)(y1,y2)

µ(z) = infz∈x1((y1+i1)−x1y1),(x2(y2+i2)−x2y2)µ(z) = infz∈x1((x1+i1)−x1y1),(x2(y2+i2)−x2y2)(µ1

× µ2)(z) = T (infz∈y1(x1+i1)−x1y1)µ1(z), infz∈y2(x2+i2)−x2y2)µ2(z)) = T (µ1(i1), µ2(i2)) = T ((µ1(i1), µ2(i2)) = ((µ1 × µ2)(i1, i2)).

hence µ1 × µ2 is a T -fuzzy hyperideal of N × M .

Corollary 21. Let N1, N2...Nn be hypernear rings and If µ1, µ2...µn are T-fuzzy hyperideals of N1, N2...Nn respectively, then µ

= µ1 × µ2...× µ2 is a T-fuzzy hyperideal of the direct product N1

× N2...× Nn.

Lemma 22. let µ be a T- fuzzy ideal of hyper near ring N then

µ(0) ≥ µ(x) for all x ∈ N.

Proof. for every x ∈ N , µ(0) ≥ infz∈x−xµ(z) ≥ T {µ(x), µ(-x)} =

Theorem 23. Let µ1 and µ2 be any two T-fuzzy set of hypernear

rings N1 and N2. if µ1 µ2 be a direct product of N1 N2, then atleast one of the following condition hold: (i) µ1(0) ≥ µ2(y) for all y ∈ N2.

(ii) µ2(0) ≥ µ1(x) for all x ∈ N1.

Proof. let assume that µ1 µ2 be a direct product of N1 N2. by contradiction assume that none of following condition is true then there exist x0 ∈ N1 and y0 ∈ N2 such that µ1(0) < µ2(y0) and µ2(0) < µ1(x0) now by definition (µ1 ᵪ µ2)(x0, y0) = T (µ1(x0), µ2(y0)) > T (µ1(0), µ2(0)) = (µ1 ᵪ µ2)(0, 0) which is contradiction

by lemma(22) therefore our assumption is not true.

Theorem 24. Let (N, +, .) be hypernear ring, then the family of T-fuzzy hyperideals in N is completely distributive property lattice with respect to meet ”∧” and join ∨.

Proof. As with respect to the usual ordering in [0, 1], [0, 1] is completely distributive property lattice so, it is sufficient to

show



Issn No : 1006-7930

Page No: 1386

{µα | α ∈ Λ} for all x, y ∈ N then, (i) infz∈x−y( α∈Λ

µα are T -fuzzy hyperideals of N .

µα(yV)) | α ∈ VΛ} ≥ T (inf{µα(x) | α ∈VΛ}, inf{µα(y) | α ∈ Λ}) =

2 0 0 0 0

W

µα)(z) =

Hence

that V

µα and W

µα are T -fuzzy hyperideals of N for family of

α∈Λ α∈Λ W sup {infz∈x−yµα(z) | α ∈ Λ} ≥ sup{T (µα(x), µα(y)) | αW∈ Λ}

( W

)(y)). (ii) infz∈x+y−x( µα)(z) = sup {infz∈x+y−xµα(z) | α ∈ Λ} α∈Λ

α∈Λ α∈ΛW W ≥ sup{µα(y) | α ∈ Λ} = (

α∈Λ )(y). (iii) infz∈x(y+i)−xy(

α∈Λ

µα)(z) = W

sup {infz∈x(y+i)−xyµα(z) | α ∈ Λ} ≥ sup{µα(i) | α ∈ Λ} = ( W

α∈Λ )(i).

similarly we can prove V

µα are T -fuzzy hyperideals of N . (i) infz∈x−y(

α

V

∈Λ

α∈Λ

µα)(z) = inf{infz∈x−yµα(z) | α ∈ Λ} ≥ inf{T (µα(x),

T (( α∈Λ

)(x), ( α∈Λ

)(y)). (ii) infz∈x+y−x( V α∈Λ

µα)(z) = inf{infz∈x+y−xµα(z) V

| α ∈ Λ} ≥ inf{µα(y) | α ∈ Λ} = ( α∈Λ

)(y). (iii) infz∈x(y+i)−xy( αV∈Λ

µα)(z)

= inf{infz∈x(y+i)−xyµα(z) | α ∈ Λ} ≥ inf{µα(i) | α ∈ Λ} = ( V α∈Λ

)(i).

Hence µα are T -fuzzy hyperideals of N . α∈Λ

Definition 25. Let (N , +, .) be a hypernear ring and t be a t-norm. A fuzzy set µ in N is said to satisfy imaginable condition if Im ⊆ ET = {α ∈ [0, 1] | T (α, α) = α}

Theorem 26. suppose that (N, +, .) be any hypernear ring, T be a t-norm and µ be any imaginable fuzzy set in N. if every non empty upper level set U(µ: α) of µ is a hyperideal of N then µ is also imaginable T-fuzzy hyperideal of N.

Proof. Let every not empty subset upper level subset U (µ: α) of µ is a hyper ideal of N . then, first infz∈x−yµ(z) ≥ min{µ(x), µ(y)} for every x, y ∈ N . suppose that it is not true then there exist x0, y0 ∈ N then infz∈x0−y0 µ(z) < min{µ(x0), µ(y0)}, Now take S1 = 1 (infz∈x −y µ(z) + min{µ(x0), µ(y0)}) then we get infz∈x −y µ(z) < S1 < min{µ(x0), µ(y0)} since x0, y0 ∈ U (µ: α) but x0 - y0 ∈/ U (µ: α) which is contradiction therefore infz∈x−yµ(z) ≥ min{µ(x),

)(x), ≥ T (sup{µα(x) | α ∈ Λ}, sup{µα(y) | α ∈ Λ}) = T ((

α∈Λ



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∈ ≥

⊇ ◦ ∧ ◦ ∧ ∈

≥ ◦ ∧ ◦ ∧

0 0 0 2 0 0 0

2 0 0 0 0 0 0 0 0 0 0

sup

µ(y)} for every x, y ∈ N . similarly if condition infz∈x+y−xµ(z) ≥ µ(y) is not true for every x, y ∈ N . then there exist x0, y0 ∈ N

then infz∈x +y −x µ(z) < µ(y0), Now take S2 = 1 (infz∈x +y −x µ(z) + µ(y0)) then we get infz∈x0+y0−x0 µ(z) < S2 < µ(y0) since y0 ∈ U (µ: α) but x0 + y0 - x0 ∈/ U (µ: α) which is contradiction therefore infz∈x+y−xµ(z) ≥ µ(y) for every x, y ∈ N . Now if condition infz∈x(y+i)−xyµ(z) ≥ µ(i) is not true for every x, y, i ∈ N . then there exist x0, y0, i0 ∈ N then infz∈x0(y0+i0)y0−x0y0 µ(z) < µ(i0), Now take

S3 = 1 (infz∈x (y +i )−x y µ(z) + µ(i0)) then we get infz∈x (x +i )−x y µ(z) < S3 < µ(i0) since i0 ∈ U (µ: α) but x0(y0 + i0) - x0y0 ∈/ U (µ: α) which is contradiction therefore infz∈x(y+i)−xyµ(z) µ(i) for every x, y, i N . Hence µ is a imaginable T -fuzzy hyperideal of N .

1. T -fuzzy quasi hyperideals and T -fuzzy

bi-hyperideals in near rings

Definition 27. Let a T -fuzzy set µ of a hypernear ring N is a quasi hyperideals if it satisfy following condition

(i) infz∈x−yµ(z) ≥ T {µ(x), µ(y)} for all x, y ∈ N .

(ii) µ (µ N N µ N ∗µ) for all x N . here N ∗µ is define as

N ∗µ(x) = x=a(b+i)−ab

µ(i) if x = a(b+i)-ab, if a, b, i ∈ N

0 otherwise

Remark 28. If Q is a quasi-hyperideal of N, then χQ is a T-fuzzy quasi-ideal of N.

Theorem 29. A fuzzy subset µ is a T-fuzzy quasi hyperideal of a hypernear ring if and only if

(i) infz∈x−yµ(z) ≥ T {µ(x), µ(y)} for all x, y ∈ N. (iii) µ(x) ≥ T [ supµ(y), supµ(z), sup µ(i)] for all x, y, z, i ∈

x∈y.z

N. x∈y.z x∈y(z+i)−yz

Proof. Let µ be a T - fuzzy quasi hyperideal of N then condition (i) is obvious, Now for condition (ii) µ(x) T ( (µ N )(x) (N µ)(x) (N ∗µ)(x)) = T ( sup T (µ(y),

x∈y.z



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◦ ∧

χN (z)), sup T (χN (y), µ(z)), N ∗µ(y(z + i) - yz)) ≥ T ( sup T (µ(y), x∈y.z

sup (µ(z)), sup µ(i)). x∈y.z

x∈y.z x∈y(z+i)−yz

conversely µ(x) ≥ T ( sup T (µ(y), sup (µ(z)), sup µ(i)) = T ( sup T (µ(y), x∈y.z x∈y.z x∈y(z+i)−yz x∈y.z

χN (z), sup T (χN (y), µ(z)), N ∗µ(y(z + i) - yz)) = T ( (µ N )(x) x∈y.z

(N ◦µ)(x) ∧ (N ∗µ)(x)) = T ( (µ◦N ) ∧ (N ◦µ) ∧ (N ∗µ))(x).

Corollary 30. A fuzzy subset µ is a fuzzy quasi hyperideal of a hypernear ring if and only if

(i) infz∈x−yµ(z) ≥ min{µ(x), µ(y)} for all x, y ∈ N.

(ii) µ(x) ≥ min[ supµ(y), supµ(z), sup µ(i)] for all x, y, z,

i ∈ N. x∈y.z x∈y.z x∈y(z+i)−yz

Proof. for proof by taking t norm as a min norm in theorem

28.

Theorem 31. Every fuzzy quasi hyperideal of hypernear ring N is a T- fuzzy quasi hyperideal of N.

Proof. Let µ be a fuzzy quasi hyperideal of hypernear ring N then for all x, y, z, i ∈ N , Now (i) infz∈x−yµ(z) ≥ min{µ(x), µ(y)} ≥ T {µ(x), µ(y)}. (ii) µ(x) ≥ min( supµ(y), supµ(z), sup N ∗µ(i))

x∈y.z x∈y.z x∈y(z+i)−yz

≥ T ( sup (µ(y), supµ(z)), sup N ∗µ(i)). x∈y.z x∈y.z x∈y(z+i)−yz

Corollary 32. Let Q be a quasi hyperideal of hypernear ring N then its characteristic function χN is T-fuzzy hyperideal of N. Proof.

proof is similar as theorem (30).

Theorem 33. Every T-fuzzy quasi hyperideal in a zero-symmetric hypernear ring is an T fuzzy subhypernear ring.

Proof. Let µ be an T -fuzzy quasi hyperideal in a zero-

symmetric hypernear ring N . Choose i, x, y, a, b ∈ N such that a = bc = x(y + i) - xy. Then µ(a) ≥ T {(µ◦N )(a), (N ◦µ)(a), (N ∗◦µ)(a)} = T {sup[min{µ(b), N (c)}], sup[min{N (b), µ(c)}], sup µ(i)} ≥

a=bc a=bc a=x(y+i)−xy

T {sup[min{µ(b), N (c)}], sup[min{N (b), µ(c)}], sup µ(c)} = a=bc a=bc a=b(0+c)−b0



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{ } ≥ { }

⊇ ◦ ∩ ◦ ∩ ◦ ≥ ◦ ∩ ◦ ∩ ◦

◦ ∩ ◦ ∩ ◦

} ◦ } { ◦ }

≥ ◦ ∩ ◦ ∩ ◦ ⊇ ≤ ≤ { }

T supµ(b), supµ(c), supµ(c) T supµ(b), supµ(c) . therefore N is an T fuzzy subhypernear ring.

Theorem 34. Every T-fuzzy left hyperideal of a hypernear ring N then is an T-fuzzy quasi hyperideal of N.

Proof. Suppose that µ be an T -fuzzy left hyperideal of N .

then Choose for any a, b, c, x, y, i ∈ N thus let a = bc = x(y + i) - xy. Then (µ◦N )(a) ∩ (N ◦µ)(a) ∩ (N ∗◦µ)(a) = T {(µ◦N )(a), (N ◦µ)(a), (N ∗◦µ)(a)} = T {sup[min{µ(b), N (c)}], sup[min{N (b),

a=bc a=bc

µ(c)}], infz∈x(y+i)−xyN ∗◦µ(z)} = T {supµ(b), supµ(c), supµ(i)} (as we know by (iii) property of T fuzzy hyper ideals that µ(i) ≤ µ(x(y + i) - xy) then) ≤ T {supN (b), supN (c), infz∈x(y+i)−xyµ(z)} = infz∈x(y+i)−xyµ(z) = µ(a) since µ(a) (( µ N ) (N µ) (N ∗ µ))(a) therefore µ (µ N ) (N µ) (N ∗ µ). Hence µ be a T -fuzzy quasi hyperideal of N .

Theorem 35. Every T-fuzzy right hyperideal of a hypernear ring N then is an T-fuzzy quasi hyperideal of N.

Proof. Suppose that µ be an T -fuzzy right hyperideal of N . then

Choose for any a, b, c, x, y, i ∈ N thus let a = bc = x(y + i) - xy. Then (µ◦N )(a) ∩ (N ◦µ)(a) ∩ (N ∗◦µ)(a) = T {(µ◦N )(a), (N ◦µ)(a), (N ∗◦µ)(a)} = T {sup[min{µ(b), N (c)}], sup[min{N (b),

a=bc a=bc

µ(c) ], infz∈x(y+i)−xyN ∗ µ(z) = T supµ(b), supµ(c), infz∈x(y+i)−xyN ∗ µ(z) (as we know by property of T fuzzy right hyperideals that µ(bc)

µ(b) then) T supµ(bc), supN (c), infz∈x(y+i)−xyN (z) = µ(bc) = µ(a) since µ(a) (( µ N ) (N µ) (N ∗ µ))(a) therefore µ (µ N ) (N µ) (N ∗ µ). Hence µ be a T -fuzzy quasi hyperideal of N .

Theorem 36. Every T-fuzzy hyperideal of N is an T-fuzzy quasi hyperideal of N.

Proof. Proof is straight forward of theorem (35) and (36).

Definition 37. A fuzzy set µ of a hypernear ring N is a T -fuzzy bi-hyperideal if it satifying following condition: (i) inf(a∈x−y)µ(a) ≥ T {µ(x), µ(y)} for all x, y ∈ N .

(ii) inf(a∈x.y.z)µ(a) ≥ T {µ(x), µ(z)} for all x, y, z ∈ N .



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{ }

→

∈

(

∈

∈ − { } ≥ { }

Remark - for T = min then T -fuzzy bi-hyperideal become

fuzzy bi-hyperideal of hyper near ring. Remark - Every hyperideals is T -norm hyperideals but converse is not true we can show this by example.

Example 38. Let N = 0, u, v, w, x be a set with two hyper operations (+, .), these two operations define as follow

+ 0 u v w x . 0 u v w x 0 0 u v w x 0 0 0 0 0 0 u u {0, u} v w x u 0 {0, u} {0, u} {0, u} {0, u} v v v {0, u} x w v 0 {0, u} {0, u} {0, u} {0, u} w w w x {0, u} v w 0 {0, u} {0, u} {0, u} {0, u} x x x w v {0, u} x 0 {0, u} {0, u} {0, u} {0, u}

Since (N, +, .) is hypernear ring. now if mapping µ: N [0, 1] define as follow µ(0) = 0.9, µ(u) = 0.8, µ(v) = 0.6, µ(w) = 0.5, µ(x) = 0.2. then by routine calculation it is clear that it T-fuzzy hyperideal under t-norm Tp = ab is bi-hyperideal of N but it is not bi-hyperideal of N.

Theorem 39. Let B be any bi-hyperideal of a hypernear ring N, then for any α (0, 1) there exist a T-fuzzy hyper bi-ideal of N such that µα = B.

Proof. suppose that B be any bi-hyper ideal of a hypernear ring

N . Now define a mapping µ: N → [0, 1] by

µ(x) = α ifx ∈ B

0 ifx ∈/ B

here α is fixed in between (0, 1). it is clear that µα = B. Now (i) let x, y ∈ B then inf(a∈x−y)µ(a) = α = min{µ(x), µ(y)} ≥ T {µ(x), µ(y)}. if atleast one of x, y ∈/ B then inf(a∈x−y)µ(a) = 0 = min{µ(x),

µ(y)} ≥ T {µ(x), µ(y)}. (ii) let x, z ∈ B then inf(a∈x.y.z)µ(a) = α = min{µ(x), µ(z)} ≥ T {µ(x), µ(y)}. if atleast one of x, z ∈/ B then inf(a x y)µ(a) = 0 = min µ(x), µ(y) T µ(x), µ(y) . Hence µ be a T -fuzzy bi-hyperideal of hyper near ring.

Corollary 40. Let B be any bi-hyperideal of a hypernear ring N, then for any α (0, 1) there exist a fuzzy hyper bi-ideal of N such that µα = B.



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◦ ∈ ∈ ≥

{ } ≥ { }

∈ ∈ ∈ ≥ ◦

≥ ≥ ◦ ◦ ◦

Proof. for proof raking T norm as a min norm.

Theorem 41. Let a non-empty subset B is a bi-hyperideal of a hypernear ring if and only if χB is a T-fuzzy bi-hyperideal of N.

Proof. suppose that B be a bi-ideal of a hyper near ring N , Now for any x, y, z ∈ N case(1) case(i) let for any x, y ∈ B. then χB(x) = 1 and χB(y) = 1 since inf(a∈x−y)µ(a) = 1 = min{µ(x), µ(y)} ≥ T {µ(x), µ(y)}. case(ii) let for x ∈ B and y ∈/ B. then χB(x) = 1 and χB(y) = 0 since inf(a∈x−y)µ(a) = 0 = min{µ(x), µ(y)} ≥ T {µ(x), µ(y)}. case(iii) let for y ∈ B and x ∈/ B. then χB(x) = 0 and χB(y) = 1 since inf(a∈x−y)µ(a) = 0 = min{µ(x), µ(y)} ≥ T {µ(x), µ(y)}. case(iv) let for x, y ∈/ B. then χB(x) = 0 and χB(y) = 0 since inf(a∈x−y)µ(a) = 0 = min{µ(x), µ(y)} ≥ T {µ(x), µ(y)}. case(1) case(i) let for any x, z ∈ B. then χB(x) = 1 and χB(z) = 1 since inf(a∈x.y.z)µ(a) = 1 = min{µ(x), µ(z)} ≥ T {µ(x), µ(z)}. case(ii) let for x ∈ B and z ∈/ B. then χB(x) = 1 and χB(z) = 0 since inf(a∈x.y.z)µ(a) = 0 = min{µ(x), µ(z)} ≥ T {µ(x), µ(z)}. case(iii) let for z ∈ B and x ∈/ B. then χB(x) = 0 and χB(z) = 1 since inf(a∈x.y.z)µ(a) = 0 = min{µ(x), µ(z)} ≥ T {µ(x), µ(z)}. case(iv) let for x, z ∈/ B. then χB(x) = 0 and χB(z) = 0 since inf(a∈x−y)µ(a) = 0 = min µ(x), µ(z) T µ(x), µ(z) . Hence χB is T -fuzzy hyper ideal of N . for conversely χB is two valued charactertick function then proof is similler to theorem(38).

Corollary 42. Let a non-empty subset B is a bi-hyperideal of a hypernear ring if and only if χB is a fuzzy bi-hyperideal of N.

Proof. for proof raking T norm as a min norm.

Theorem 43. Let µ be any T-fuzzy bi-hyperideal of a hypernear ring N if and only if µ ◦ χN ◦ µ ⊆ µ.

Proof. suppose that µ be any T -fuzzy bi-hyperideal of a hypernear ring N . let x N if x / a.b.c for a, b, c N then, µ(x) (µ χN µ)(x) = 0 but if x a.b.c for a, b, c N then, µ(x) T (µ(a), µ(c)) = T (µ(a), T (1, µ(c))) = T (µ(a), T (χN (b), µ(c))) Hence µ(x)

sup T (µ(a), T (χN (b), µ(c))) µ χN µ(x). therefore µ χN x∈a.y

◦ µ ⊆ µ.

for conversely, it is obvious infa∈x.y.zµ(a) ≥ T (µ(x), µ(z)).



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∈ ∈

∈ ∧ ≥

∧ ∧ ∧

Theorem 44. Every T-fuzzy quasi-hyperideal of hypernear ring N is a T-fuzzy bi-hyperideal of N.

Proof. suppose that µ be any T -fuzzy quasi hyperideal of hyper-

near ring. let x, y, z ∈ N then infa∈x.y.zµ(a) ≥ T (µ(x), infq∈y.zµ(q)) ≥ T (z) hence µ be a T -fuzzy bi-hyperideal of N .

Theorem 45. let µ and ν be any two T-fuzzy bi-hyperideal of hypernear ring N, then µ ∧ ν is also T-fuzzy bi-hyperideal of N.

Proof. Let us suppose that µ and ν be any two T -fuzzy bi- hyperideal of hypernear ring N . Then inf(a x.y.z)(µ ν)(a) T (T (µ(x), µ(z)), T (µ(x), µ(z))) = T (T (µ(x), ν(x)), T (µ(z), ν(z))) = T (µ ν(x), (µ ν)(z)). hence µ ν is T -fuzzy bi-hyperideal of N .

Corollary 46. let µ and ν be any two fuzzy bi-hyperideal of hypernear ring N, then µ ∧ ν is also fuzzy bi-hyperideal of N.



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manipal university jaipur Email address: [email protected]

manipal university jaipur Email address: [email protected]



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