Positive Implicative Artinian and Positive Implicative Noetherian Hyper Bck Algebras

Positive Implicative-Artinian and Positive Implicative

Noetherian Hyper Bck-Algebras

B. Satyanarayana

1,*

_{, L. Krishna}

2

_{, R. Durga Prasad}

3

1_{Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, Andhra Pradesh, India} 2_{Department of Mathematics, Acharya Nagarjuna University Campus Ongole, Ongole-523 002, India}

3_{Department of Basic Engineering, Devineni Venkata Ramana & Dr. Hima Sekhar MIC College of Technology,} Kanchikacharla- 521 180, India

*Corresponding Author: [email protected]

Copyright © 2014 Horizon Research Publishing All rights reserved.

Abstract

The notion of intuitionistic fuzzy positive implicative hyper BCK-ideals of type-1, 2... 8 of hyper BCK-algebras was introduced in 2012 by Durga Prasad, Satyanarayana and Ramesh. In this paper, we investigate some related properties of intuitionistic fuzzy positive implicative hyper BCK-ideals of types-1. We characterize positive implicative Artinian (shortly, PI-Artinian) hyper BCK-algebras of type-1 and positive implicative Noetherian (shortly, PI-Noetherian) hyper BCK-algebra of type-1.

Keywords

PI-Artinian Hyper BCK-Ideals of Type-1 and PI - NoetherianHyper BCK-Algebra of Type-1

1. Introduction

The notion of logical algebras BCK-algebras [7] was initiated by Imai and Iseki in 1966 as a generalization of both classical and non-classical calculus. After the introduction of fuzzy sets by Zadeh [12], there has been a number of generalization of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov [1] is one among them.

The hyperstructuretheory (called also multi-algebras) was introduced in 1934 by F.Marty [10] at the 8th _{congress of} Scandinavian Mathematicians. Around 40’s several authors worked on hyper groups, especially in France and in the United States, but also in Italy, Russia and Japan. Hyperstructers have many applications to several sectors of both pure and applied sciences. In [9], Y.B. Jun, M. M. Zahedi, X. L. Xin and R.A. Borzooei applied the hyperstructers to BCK-algebras, and introduced the notion of hyper BCK-algebras. In [9] Durga Prasad and others applied the concept of intuitionistic fuzzy sets to positive implicative hyper BCK-ideals of type-1,2, ..,8 of hyper BCK-algebras and then introduced the notion of intuitionistic fuzzy positive implicative hyper BCK-ideals of type-1,2,..,8 and related properties are investigated. In this paper, Using collection of positive implicative hyper BCK-ideal of type-1, we state a

characterization of positive implicative Artinian (shortly,

PI -Artinian) hyper BCK-algebra of type-1 and positive implicative Noetherian (shortly, PI -Noetherian) hyper BCK-algebra of type-1, a few results are investigated. We include some elementary aspects of hyper BCK-algebras that are necessary for this paper, and for more details we refer to [6], [8] and [9]. Let us recall definitions and theorems. Let H be a non-empty set endowed with hyper operation “

ο

” that is a function form H H to× _{P H}*( )₌ ( ) \ { }P H ϕ _{. For} two subsets A and B of H, denoted by

A B

ο

the set

, a b a A b B

ο ∪

∈ ∈ . We shall use x yο instead of { }x yο , { }x yο _{or { } { }}xο y _.

Definition 1.1.[5]. By a hyper BCK-algebra, we mean a non-empty set H endowed with a hyper operation " "ο and a constant 0 satisfying the following axioms:

(HK-1) (x z  ) (y z)<<x y , (HK-2) (x y z ) =(x z y ) ,

(HK-3)x H_ <<{ }x _,

(HK-4)x y and y x x y<< << ⇒ = _{for all , ,}x y z H∈ . We can define a relation “<<” on H by letting x y<< if and only if

0

∈

x



y

and for every ,A B⊆H, _A<<_B is defined by ∀ ∈a Athere exists b B∈ such that

a

<<

b

. In such case, we call “

<<

” the hyper order in H. Note that the condition (HK3) is equivalent to the condition: (P1)

{ }

x y_ << x _{, for all ,}x y H∈ _{in any hyper BCK-algebra H the} following hold: (P2) x_0 { } , 0<< x _x<<{ }x _and 0 0 <<{0}, (P3) 0 0 {0}_ = _{, (P4) 0}<<_x, (P5) x x<< _{, (P6)}

A A<< , (P7) A⊆B _{implies A B}<< , (P8)0_x={0}_{, (P9)} 0 { },

x_ = x _(P10)0_A={0}_{, (P11)} A<<{0}_implies A={0}, (P12)A B A << . (P13)x x∈ _0, for all x y z H, , ∈ _{and for all} non empty sets A, B, C of H..

i. a hyper BCK-sub algebra of H, if x y I_ ⊆ ,_{for all} ,

x y I∈ _,

ii. a weak hyper BCK-ideal of H if xy⊆I, y I∈ imply

x I,

∈

,x y H∈ _,

iii. a hyper BCK-ideal of H, if x y <<I and y I∈ imply , ,x I x y H∈ ∈ _{, for ,}x y H∈ _,

iv. a strong hyper BCK-ideal of H, if x y I_ ∩ ≠ϕ _for y I∈

v. reflexive if

x x I



⊆

, for x H∈ , vi. S-reflexive if (x y_ )∩ ≠I ϕ

⇒

x y I_ << ,

, x y H

∀ ∈

vii. closed, if

x

<<

y

and y I∈

⇒

x I

∈

,

, x y H

∀ ∈

It is easy to see that every S-reflexive sub-set of H is reflexive. Let µ and λ be the fuzzy sets of X. For

,

[0,1]

s t

∈

the set (U µ_A; )s = _{x X∈ /µ_A( ) }x s≥ _{is called} upper s- level cut of μ and the set L( ; ) {λ_At = ∈x X/λ_A( ) }x s≤ is called lower t-level Cut level of λ and can used to the characterization of

μ

and

λ

.

Let H be a hyper BCK-algebra. Then we say that a fuzzy subset

μ

of H is fuzzy closed, if x y≤ _{in H then}

( )x ( )y

µ ≥µ _for x y H, ∈ . _{Let H be a hyper BCK-algebra.} Then we say that a fuzzy subset

λ

of H is anti- fuzzy closed, if x y≤ _{in H then}λ( ) ( )x ≤λ y _.

Definition 1.2.[5] Let H be a hyper BCK-algebra. Then H is said to be a positive implicative hyper BCK- algebra, if for all x y z H x y z x z y z, , ∈ , (  ) =(   ) ( ).

Definition 1.3.[5] Let I be a non-empty sub-set of H and

I

0

∈

. Then I is said to be a positive implicative hyper BCK-ideal of

i. type 1, if (x y z I ) ⊆ and y z I ⊆ imply x z_ ⊆I, ii. type 2, if (x y z I_{ }) << _and

_{y z I}

_

_⊆

_imply

x z I,

_

⊆

iii. type 3, if (x y z I_{ }) << _and y z_ <<I_imply ,

x z_ ⊆I

iv. type 4 , if (x y z I_{ }) ⊆ _and y z_ <<I_imply x z I_ ⊆ , v. type 5, if (x y z I_{ }) ⊆ _and y z_ ⊆I_{imply x z I} << , vi. type 6, if

I, z imply x I z y and I y)

(x z<<  <<  <<

vii. type 7, if (xy)z⊆ Iandyz<< Iimply xz<<I, viii. type 8, if (x y z I_{ }) << _and y z I_ ⊆ _imply

x z I << for all , ,x y z H∈ .

Definition 1.4.[1, 2]. An intuitionist fuzzy set in a non-empty set X is an object having the form

{( , ( ), ( ))/ }

A= xµ_A x λ_A x x X∈ _{, where the function} :X [0,1]

A

µ → and λ_A:X →[0,1] denoted the degree of membership (namelyµ_A( )x _{) and the degree of non} membership (namely λ_A( )x _{of each elementx X}∈ to the set A respectively and 0 ≤ µ_A( ) x +λ_A( ) 1x ≤ ∀ ∈ _{x X} . For the sake of simplicity, we use the symbol form

( , , )

A X A A= µ λ _orA=(µ λ_{A A}, )_.

Definition 1.5.[10] Let A=( ,

µ λ

_{A A}) _{be an intuitionistic}

fuzzy sub-set of H and

µ

_A(0)≥

µ

_A( )x _,

λ

_A(0)≤

λ

_A( )y _for

all x y H, ∈ _{. Then} A=( ,

µ λ

_{A A}) _{is said to be an}

intuitionistic fuzzy positive implicative hyper BCK-ideal of i. type 1, if for all t x z∈ _{ ,}

( ) min{ inf ( ), inf ( )} ( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈ _{ } ∈ _ and

( ) max{ sup ( ), sup ( )} ( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈   ∈  .

ii. type 2, if for all t x z∈  ,

( ) min{ sup ( ), inf ( )}

( )

t a b

A A A

b y z a x y z

µ ≥ µ µ

∈

∈    and

( ) max{ inf ( ), sup ( )}

( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈   ∈ 

iii. type 3, if for all t x z∈ _{ ,}

( ) min{ sup ( ), sup ( )}

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈   ∈  and

( ) max{ inf ( ), inf ( )}

( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈   ∈  .

iv. type 4, if for all t x z∈  ,

( ) min{ inf ( ), sup ( )}

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈   ∈  and

( ) max{ sup ( ), inf ( )} ( )

t c d

A A A

d y z c x y z

λ ≤ λ λ

∈

∈   

, , x y z H∈ _.

Definition 1.6.[10] Let A=( ,

µ λ

_{A A}) _{be an intuitionistic}

fuzzy sub-set of H. Then A=( ,

µ λ

_{A A})_{is said to be an}

intuitionistic fuzzy positive implicative hyper BCK-ideal of i. type 5, if there exists t x z∈ _{ such that}

( ) min{ inf ( ), inf ( )}

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈   ∈  and

( ) max{ sup ( ), sup ( )} ( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈ _{ } ∈ _ .

ii. type 6, if there existst x z∈  such that ( ) min{ sup ( ), sup ( )}

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈   ∈  and

( ) max{ inf ( ), inf ( )}

( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈   ∈  .

iii. type 7, if there existst x z∈  such that ( ) min{ inf ( ), sup ( )}

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈   ∈  and

( ) max{ sup ( ), inf ( )}

( )

t c d

A A A

d y z c x y z

λ ≤ λ λ

∈

∈    .

iv. type 8, if there existst x z∈ _{ such that} ( ) min{ sup ( ), inf ( )}

( )

t a b

A A A

b y z a x y z

µ ≥ µ µ

∈

∈    and

( ) max{ inf ( ), sup ( )}

( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈   ∈  .

Definition 2.1. A hyper BCK- algebra H is said to satisfies the PI-ascending (resp., PI-descending) chain condition (briefly, ACC (resp., DCC)) of type-1, if for every ascending (resp., descending) sequence A₁⊆ A₂⊆...

(

resp., A A₁⊇ ₂⊇....

)

_{of positive implicative hyper} BCK-ideals of type-1of H, there exists a natural number n such that A A_n= _{kfor all n k≥ .}

Definition 2.2.

i. A hyper BCK-algebra H is said to be a PI-Artinian hyper BCK-algebras of type-1, if H satisfies PI-DCC of type-1.

ii. A hyper BCK-algebra H is said to be a PI-Noetherian hyper BCK-algebra of type-1 if H satisfies PI-ACC of type-1.

Theorem 2.3. Let H be a PI-Artinian hyper BCK-algebra of type-1 and A=(

µ λ

_{A A}, ) _{is an intuitionistic fuzzy positive}

implicative hyper BCK-ideal of type-1 of H. If a sequence of elements of Im( )A is strictly intuitionistic increasing, that is, a sequence of elements of Im(

µ

_A) _{is strictly increasing}

and a sequence of elements of Im(

λ

_A) is strictly decree-sing, then A has finite number of intuitionistic values, that is,

µ

_Aand

λ

_A has finite number of values.

Proof: Suppose that Im(

µ

_A) _{is not finite. Let}

{ }

_{sn be a}

strictly increasing sequence of elements of Im(

µ

_A)_{, that is,} 0< < < ≤s s_{1 2} ... 1_{. Define} U(µ_{A r}; )s = {x X∈ / µ_A( )x ≥s_r}_, for r=2,3,4,... _{. By theorem 3.14(i) [10], We have}

( ; )

U

µ

_{A r}s _{is a positive implicative hyper BCK-ideal of}

type-1. Let x U∈ (

µ

_{A r}; )s _then µ_A( )x s s_{≥ > −r r 1 which} implies that x U∈ (

µ

_{A r};s −₁) . Hence U(

µ

A r; )s ⊆

( ; ₁)

U µ_{A r}s ₋ . Since _sr−1∈Im(

µ

_A) _{then there exists} 1

x_r− ∈X _{such that}

µ

_{A r}(x ₋₁)=s_{r−1 . It follows that}

( ; )

1 1

x_r− ∈U

µ

_{A r}s _{− but} x_r−1∉U(

µ

_{A r}; )s _{. Thus} ( ; )

U

µ

_{A r}s _{is a proper sub set of} U(

µ

_{A r};s ₋₁)_{and thus we}

can obtain a strictly descending chain ( ; )₁ ( ; )₂ ( ; ) ..._{3 3}

U µ_A s ⊃U µ_A s ⊃U µ s ⊃ _{of positive} implicative hyper BCK- ideals of type-1of H which is not terminating. This contradicts the assumption that H satisfies PI-DCC of type-1 of H.

Now assume that Im( )λA is not finite.

Let

{ }

_tn be a strictly decreasing sequence of elements of Im( )λ_{A , that is,} 0 ...≤ < < ≤t₂ t₁ 1. _Define

( ; ) { / ( ) }

L µ_{A k}t = x X∈ λ_A x ≤t_k for k=2,3,4,... . By Theorem 3.14(i) [10], We have (L µ_{A k}; )t is a positive implicative hyper BCK- ideal of type-1of H. If

( ; )

y L∈ λ_{A k}t _{, then} λ_A( )y ≤t_{k < −}t_{k 1} and so ( ; ₁)

y L∈ λ_{A k}t ₋ . This shows that (L λ_{A k}; )t ⊆

( ; ₁)

L λ_{A k}t ₋ , since _tk−1∈Im(λ_A) then there exists 1

y_k− ∈X such that λ_{A k}(y ₋₁)=t_k−1. It follows that y_k-1∈L(λ ;t_{A k-1}) but y_k−1∉L(λ_{A k}; )t . Therefore

( ; )

L λ_{A k}t is a proper sub set of (L λ_{A k};t ₋₁) and thus we can obtain a strictly descending chain

( ; )₁ ( ; )₂ ( ; ) ...₃ L λ_A t ⊃L λ_A t ⊃L λ_A t ⊃

of positive implicative hyper BCK-ideals of type-1of H, which is not terminating. This contradicts the assumption that H satisfies the PI-DCC of type-1 of H. Thus

( , )

_{A A}

A

=

µ λ

has finite number of intuitionistic values. Now we consider the converse of the Theorem 2.3.

Theorem 2.4. Let H be a hyper BCK- algebra. If every intuitionistic fuzzy positive implicative hyper BCK- ideal of type-1of H has finite number of intuitionistic fuzzy values, then H is a PI- Artinian hyper BCK-algebra of type-1. Proof: Suppose H does not satisfy PI-DCC of type-1, then there exists a strictly descending chain 0A ⊃ A1⊃

.. 2

A ⊃ of positive implicative hyper BCK- ideals of type-1of H which does not terminates at finite step.

Define IFS A=( ,

µ λ

_{A A})in H by

, _{\ 1, 1,2,...} 1

( )

1, , 0

r _{if x A Ar r for r} r

x A

if x _Ar r

µ 

 _∈

 + + =

=  _∞

 _{∈ ∩}

 ₌



1 _,

\ 1, 1,2,... 1

( )

0, 0

if x A A_{r r for r} r

x A

if x _Ar r

λ 

 _∈

 + + =

=  _∞

 _{∈ ∩}

 ₌

 _,

Where 0A =H. Now we prove that A=( ,

µ λ

_{A A}) is an intuitionist fuzzy positive implicative hyper BCK- ideal of type-1of H. Clearly

µ

_A(0) 1= ≥

µ

_A( )x and

(0) 0 ( )y

A A

λ

= ≤

λ

for all

x y H

,

∈

.

Let

x y z H

, ,

∈

be such that (x y z_{ }) ⊆A A_r \ _{r+1 and}y z_ ⊆A_k \A_k+1 for

r

=

0,1,2,...;

k

=

0,1,2,...

without loss of generality, we may assume that

r k

≤

. Then obviously

(x y z ) ⊆A y z_r,  ⊆ A_r , because _Ar is a positive implicative hyper BCK- ideal of type-1, so that

x z A



⊆

_r, for all

t x z

∈



,

( ) ₁ min inf ( ), inf ( )

( )

r

t a b

A _r A A

a x y z b y z

µ ≥ = µ µ

+ _{∈ ∈}



















  



and

( )

1₁ max sup ( ), ( )sup

( )

t c d

A _r A A

c x y z d y z

λ ≤ ₊ = λ λ

∈ ∈

















If ( )

0

x y z _Ar

r ∞ ⊆ ∩ =

  and

0

y z _Ar

r ∞ ⊆ ∩ =

 then

0

x z _Ar

r ∞ ⊆ ∩ =

 so that, for all

t x z

∈



( )

1 min inf ( ), inf ( ) ( )

t a b

A A A

a x y z b y z

µ = = µ µ

∈ ∈



















  



and

( )

0 max sup ( ), ( )sup

( )

t c d

A A A

c x y z d y z

λ = = λ λ

∈ ∈



















  



.

If ( )

0

x y z _Ar

r ∞ ⊄ ∩ =

  and

0

y z _Ar

r ∞ ⊆ ∩ =

 then there exists

i N∈ such that (x y z ) ⊆ A A_i \ _i+1 . It follows that x z ⊆_Ai So that, for all

t x z

∈



( )

₁ min inf ( ), inf ( )

( )

i

t a b

A _i A A

a x y z b y z

µ ≥ = µ µ

+ _{∈ ∈}



















  



and

( )

1₁ max sup ( ), ( )sup ( )

t c d

A _i A A

c x y z d y z

λ ≤ = λ λ

+ _{∈ ∈}



















  



Finally

assume that ( )

0

x y z _Ar

r ∞ ⊆ ∩ =

  and

0

y z _Ar

r ∞ ⊄ ∩ =

 . Then

there exists

j N

∈

such that y z ⊆A_j \A_j+1. Hence

x z ⊆_Aj and so

( )

₁ min inf ( ), inf ( )

( )

j

t a b

A _j A A

a x y z b y z

µ ≥ = µ µ

+ _{∈ ∈}



















  



and

( )

1₁ max sup ( ), sup ( ) ( )

t c d

A _j A A

c x y z d y z

λ ≤ = λ λ

+ _{∈ ∈}



















  



Consequently we conclude that A=(

µ λ

_{A A}, ) is an intuitionistic fuzzy positive implicative hyper BCK- ideal of type-1 of H and A=(

µ λ

_{A A}, ) has an infinite number of different values. This is a contradiction and the proof is completed.

Theorem 2.5. Let H be a PI-Noetherian hyper BCK-algebra of type-1 and A=( ,

µ λ

_{A A}) be an intuition-stic fuzzy positive implicative hyper BCK-ideal of type-1of H. If a sequence of elements of Im( )A is strictly intuitionistic decreasing, that is, a sequence of elements of Im(

µ

_A) is strictly decreasing and a sequence of elements of Im(

λ

_A)

is strictly increasing. Then A=( ,

µ λ

_{A A})has finite number of intuitionistic values, that is,

µ

_A and

λ

_A has finite number of values.

Proof: SupposeIm( )

µ

_A is not finite. Let

{ }

s

_n be a strictly

decreasing sequence of elements of Im(µ_A) _{, that is,}

0 ...

≤

<

s

₂

<

s

₁

≤

1

. Define U(

µ

_{A r}; )s = {x X∈ /µ_A( )x ≥s_r} , for

r

=

2,3,4,...

..By Theorem 3.14(i) [10], We have U(

µ

_{A r}; )s is a positive implicative hyper BCK-ideal of type-1 of H. Let x U∈ (

µ

_{A r};s ₋₁) then

( )x s ₁ s

A r r

µ ≥ ₋ > which implies that x U∈ (

µ

_{A r}; )s . Hence U(

µ

_{A r};s ₋₁)⊆U(

µ

_{A r}; )s Since _sr∈Im(

µ

_A)

then there exists

x

_r

∈

X

such that

µ

_{A r}( )x =s_r . It follows that x_r∈U(

µ

_{A r}; )s but x_r∉U(

µ

_{A r};s ₋₁) . ThusU(

µ

_{A r};s ₋₁)is a proper sub-set of U(

µ

_{A r}; )s and thus we can obtain a strictly ascending chain

( ; )₁ ( ; )₂

U µ_A s ⊂U µ_A s ⊂ U( ; ) ...µ_{3 3}s ⊂ _{of positive} implicative hyper BCK- ideals of type-1of H, which is not terminating. This contradicts the assumption that H satisfies the PI-ACC of type-1.

Now assume thatIm(

λ

_A) is not finite. Let

{ }

_tn be a strictly increasing sequence of elements of Im(

λ

_A) that is,

1 2

0

≤ <

t

t

<

... 1.

≤

Define L(µ_{A k}; )t = {x X∈ / λ_A( )x ≤t_k}_{, for}

k

=

2,3,4,...

_{. By Theorem} 3.14(i) [10], we have (L µ_{A k}; )t is a positive implicative hyper BCK- ideal of type-1of H. Let y L∈ (λ_{A k};t ₋₁). Then λ_A( )y ≤t_k−1<t_k and so y L∈ (λ_{A k}; )t . This shows that (L λ_{A k};t ₋₁)⊆L(λ_{A k}; )t , since _tk∈Im(λ_A) then there exist y_k ∈X such that λ_{A k}( )y =t_k. It follows that y_k ∈L(λ_{A k}; )t _but y_k∉L(λ_{A k};t ₋₁) _{. Therefore}

( ; ₁)

L λ_{A k}t ₋ is a proper sub set of (L λ_{A k}; )t and so we can obtain a strictly ascending chain (L λ_A; )t₁ ⊂L(λ_A; )t₂ ⊂

( ; ) ..₃

L λ_A t ⊂ of positive implicative hyper BCK- ideals of type-1of H, which is not terminating. This contradicts the assumption that H satisfies the PI-ACC of type-1. Thus

( , )

A=

µ λ

_{A A} has finite number of intuitionistic values. Corollary 2.6. Let H be a PI-Artinian hyper BCK-algebra of type-1 and PI-Noetherian hyper BCK-algebra of type-1 and

( , )

_{A A}

A

=

µ λ

is an intuitionistic fuzzy positive implicative hyper BCK-ideal of type-1of H. If a sequence of elements of

Im( )

µ

_A and

Im( )

λ

_A is strictly decreasing. Then A has finite number of intuitionistic values, that is,

A

µ

and

λ

_A have finite number of values. Proof: The proof is straight forward.

implicative hyper BCK- ideal of type-1of H is a well-ordered sub-set of [0, 1].

Proof:

(i)

⇒

(ii)

. Let A=( ,

µ λ

_{A A})be an intuitionistic fuzzy positive implicative hyper BCK-ideal of type-1 of H. Suppose that the set of values of A is not a well- order sub set of [0, 1]. Then there exist a strictly decreasing sequence{ }_sn such that µ_A( )x =s_{n (elements of Im(}

µ

_A) ). Then

0 ...≤ <s₂<s₁≤1. DefineU(µ_{A r}; ) {s = x X∈ / ( )µ_A x ≥s_r}, for r=2,3,4,... _{.. By Theorem 3.14(i) [10], we have}

( ; )

U

µ

_{A r}s is a positive implicative hyper BCK- ideal of type-1 and thus we can obtain a strictly ascending chain

U(μ ;s ) U(μ ;s ) U(μ ;s ) ...._{A 1} ⊂ _{A 2} ⊂ _{3 3} ⊂ of positive implicative hyper BCK- ideals of type-1 of H, which is not terminating. This contradicts the assumption that H satisfies the PI-ACC of type-1. If there exists a strictly increasing sequence

{ }

_tn such that λ_A( )x =t_n (Elements of

Im(

λ

_A) ) that is, 0≤t₁<t₂<... 1.≤ _Define

( ; ) { / ( ) }

L µ_{A k}t = x X∈ λ_A x ≤t_k for k=2,3,4,... By theorem 3.14(i) [10], we have (L µ_{A k}; )t is an positive implicative hyper BCK- ideal of type-1of H and thus we get a strictly ascending chain

( ; )₁ ( ; )₂ ( ; ) ....₃

L λ_A t ⊂L λ_A t ⊂L λ_A t ⊂ _{of positive} implicative hyper BCK- ideals of type-1of H which is not terminating. This contradicts the assumption that H satisfies the PI-ACC of type-1.

Conversely, suppose that there exist a strictly ascending chain G₁⊂G₂⊂G₃⊂...( )∗ of positive implicative hyper BCK- ideals of type1of H, Which does not, terminates at finite step. Define IFS A=( ,

µ λ

_{A A})in H by

1,

min{

/

}

( )

0,

where k

_{r N x Gr}

x

k

A

if x Gr

µ

=

=

∈

∈

∉







1 ,

max{

/

}

( )

1,

where k

_{n N x Gn}

k

x

A

if x Gn

λ



₌

_∈

_∈



= 



_∉



Where

0

H _Gr

r ∞ = ∪

= . We prove that A=( ,

µ λ

A A)is an intuitionistic fuzzy positive implicative hyper BCK- ideal of type-1of H. Since 0∈_Gr, ∀ = 1, 2, 3,...r _{We have}

(0) 1 ( )x

A A

µ

= ≥

µ

and

λ

_A(0) 0= ≤

λ

_A( )x for all

.

x X

∈

Let

x y z H

, ,

∈

be such that (x y z∗ )∗ n∈G G_r \ _r−1 and y z G G ⊆ _r \ _r−₁ ,

2,3,4,....

r= , then _{x z Gr} ⊆ , since _Gr is a positive implicative hyper BCK- ideal of type 1. So that, for all

t x z∈ 

1

( ) min inf ( ), inf ( )

( )

t a b

A _r A A

a x y z b y z

µ

≥ =

µ

µ

∈ ∈



















  



and

( )

1 max ( ), ( )sup sup ( )

t c d

A _r A A

c x y z d y z

λ

≤ =

λ

λ

∈ ∈



















  



Assume that (_{x y z Gr }) ⊆ _and y z G G_ ⊆ _r \ _{m for all}

m r

<

, since_Gr is a positive implicative hyper BCK- ideal of type-1of H, therefore _{x z Gr} ⊆ , That is, for all

t x z∈ _{ , we get A}

( )

t 1 1₁ inf _A( )b

r m _{b y z}

µ ≥ ≥ ≥ µ

+ _{∈ } and

( )

1 1 sup ( ) 1

t d

A _{r m} A

d y z

λ ≤ ≤ ≤ λ

+ _{∈ } . Hence for all t x z∈  ,

( ) min inf ( ), inf ( )

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈ ∈



















  



and

( )

max sup ( ), ( )sup

( )

t c d

A A A

c x y z d y z

λ ≤ λ λ

∈ ∈



















  



Similarly, for the case (x y z G G_{ }) ⊆ _r \ _{mand y z Gr} ⊆ _,

We have ( ) min inf ( ), inf ( )

( )

t a b

A A A

a x y z b y z

µ ≥ µ µ

∈ ∈



















  



and

( )

max sup ( ), ( )sup ( )

t c d

A A A

c x y z d y z

λ

≤

λ

λ

∈ ∈



















  



_.

Hence A=( ,

µ λ

_{A A}) is an intuitionistic fuzzy positive implicative hyper BCK- ideal of type-1of H. Since the chain

)

(

∗

is not terminating, A has strictly decreasing sequence of values, a contradiction that the values set of any intuitionistic fuzzy positive implicative hyper BCK- ideals of type-1of H is well-ordered.

Notation: Let A=( ,µ λ_{A A}) is an intuitionistic fuzzy positive implicative hyper BCK- ideal of type-1of H, “_{u A}µ ” denotes the family of all upper level positive implicative hyper BCK- ideals of type-1 of H with respect to µ_Aand “v Aλ ” denotes the family of all lower level positive implicative hyper BCK- ideals of type-1of H with respect to

A λ _.

a sequence of elements of Im(

µ

_A) is strictly increasing and a sequence of elements of Im(

λ

_A) is strictly decreasing, then u_µA = Im(µ_A) and v_λA =

Im(λ_A).

Proof: Since H is a PI-Artinian hyper BCK-algebra of type-1. It follows from Theorem 2.3, that

Im( )

A

is finite (i.e.

Im(

µ

_A) and Im( )

λ

_A are finite). Let Im(µ_A)= { , ,... }s s_{1 2 sn} , where s₁<s₂<...<_{sn. It is sufficient to} show that “ u Aµ ” consists of upper level positive implicative hyper BCK- ideals of type-1 of H with respect to

A

µ

for all _si∈Im(µ_A) , That is , u Aµ = { (U µ_{A i}; ) / 1s ≤ ≤i n}. Obviously U(µA i; )s ∈uµ_A for all _si∈Im(µ_A)_{. Let0}≤ ≤s

µ

_A(0) and let (U µ_A; )s _be an upper level positive implicative hyper BCK- ideal of type-1of H with respect to

µ

_A. Assume that s∉Im(

µ

_A). If

1

s s< and x U∈ (

µ

_A; )s₁ then µ_A( )x ≥s₁>s and so

( ; )

x U∈

µ

_A s . Thus (U µ_A; )s₁ ⊆ U(µ_A; )s . Let

( ; )

x U∈

µ

_A s , then µ_A( )x >s _because s∉_Im(

µ

_A) and so

µ

_A( )x ≥s₁⇒ . x U∈ (µ_A; )s_{1 . Therefore}

( ; ) ( ; )₁

U

µ

_A s ⊆U

µ

_A s . Hence U(

µ

_A; )s = (µ_A; )s₁ . And so let s_{i < < +}s s_{i 1} , for some

i

. Then

( ; ₁) ( ; )

U µ_{A i}s₊ ⊆U µ_A s . Let x U∈ (µ_A; )s , then

s

(x)

A

μ

>

becauses∉Im(

µ

_A), and soµ_A( )x _{≥ +}s_{i 1}, that is,x U∈ (µ_{A i};s₊₁). Hence (U µ_A; ) (s = µ_{A i};s₊₁), which shows that “_{u A}µ ” consists of all upper level positive implicative hyper BCK- ideals of type-1 of H with respect to

A

µ

, for all _si∈Im(µ_A). Therefore, u_µA = Im(

µ

_A) . Let Im(λ_A) { , ,... }= t t_{1 2} t_m where t₁>t₂ >...>_tm . We claim that v_λA = Im(

λ

_A) . It is sufficient to show that “v Aλ ” consists of lower level positive implicative hyper BCK- ideals of type-1 of H with respect to

λ

_Afor all

Im( )

t j ∈ λA , that is, vλ =_A { (L λA j; ) / 1t ≤ ≤j m} . Obviously L(λA j; )t ∈vλ_A for all _{t j}∈Im(λ_A) _{. Let}

(0)

t

1

λ

≤ ≤

and let (L λ_A; )t be a lower level positive implicative hyper BCK- ideal of type-1 of H with respect to

A

λ

. Assumet∉Im( )

λ

_A . Ift₁<t and y L∈ ( ; )

λ

_A t₁ , then

( ) 1y t t A

λ ≤ < and so y L∈ (λ_A; )t . Thus L( ; )

λ

_A t₁ ⊆ ( ; )

L λ_A t . Let y L∈ ( ; )

λ

_A t then λ_A( )y <t because

Im( )

t∉

λ

_A and so λ_A(x)≤t_{1 implies}

x

∈

L(

λ

_A

t;

₁

)

. Therefore L( ; )

λ

_A t ⊆L( ; )

λ

_A t₁ . Hence L( ; )

λ

_A t =

( ; )₁

L λ_A t . Let tj > > +t tj 1 , for some j . Then ( ; ₁)

L λ_{A j}t ₊ ⊆ _L(λ_{A; )t . Let} y L∈ _{( ; )}

λ

_A t then ( )y t

A

λ < _becauset∉_{Im( )}

λ

_A and so λ_A( )y _{≤ +}t_{j 1, that} is, y L∈ (λ_{A j};t ₊₁)_{. Hence (}L λ_A; ) (t = λ_{A j};t ₊₁)_{. which} shows that v Aλ consists of lower level positive implicative hyper BCK- ideals of type-1of H with respect to

λ

_Afor all

Im( )

i A

t

∈

λ

. Therefore v_λA = Im(λ_A) and the proof is completed..

Theorem 2.9. Let H be an PI-Artinian hyper BCK-algebra of type-1 and A=( ,

µ λ

_{A A}) and B=( ,µ λ_{B B}) are intuitionistic fuzzy positive implicative hyper BCK- ideals of type-1of H and a sequence of elements of

Im( )

A

and

Im( )B are strictly intuitionistic increasing, then

i. u

µ

_A=u

µ

_Band Im( ) Im( )

µ

_A =

µ

_B ⇔

µ

_A=

µ

_B, ii. vλ_A =vλ_Band Im( ) Im( )

λ

_A =

λ

_B ⇔

λ

_A =

λ

_B. Proof: (i) Assume that

μ

_A

=

μ

_B

, then clearly uμ_A =uμ_B and Im(μ_A)=Im(μ_B)

Conversely assume that u_µA =u_{µB and} Im(

µ

_A) Im(=

µ

_B). By theorem 3.3 and 3.8, we obtain

µ

_A and

µ

_Bare finite, u_µA = Im(µ_A) and u_µB = Im(µ_B). Let Im(

µ

_A) { , ,... }= s s_{1 2} s_n and

Im( ) { , ,... }

µ

_B

=

s s

_{1 2}

′ ′

s

_n

′

, Where s₁<s₂ <...<_sn and

...

1

2

s s

′

<

′

<

<

_sn

′

. Since Im(

µ

_A) Im(=

µ

_B) hence s_i =s_i′ for all

i

. We now prove that

( ; ) ( ; )

U µ_{A i}s =U µ_{B i}s for all

i

. Consider (U µ_A; )s₁ and

( ; )

₁

U

µ

_B

s

. Suppose U(µ_A; )s₁ ≠ U(µ_B; )s₁ . Then

( ; )₁ ( ; )

U µ_A s =U µ_{B k}s for some k>1 and

( ; ) ( ; )₁

U µ_{A j}s =U µ_B s _{for some}

_j

_>

₁

_{. Since} 1

Im( )

A

s

∈

µ

there exists

x H

∈

such that

µ

_A

( )

x

=

s

₁. Then µ_A( )x <s_{j for all}

_j

>

₁

, since

( ; )₁ ( ; )

U µ_A s =U µ_{B k}s . It follows that x U∈ (µ_{B k}; )s so that µ_B( )x ≥s_k >s_{1 for some}

_k

_>

₁

_{. Thus}

( ; )₁ ( ; )

1

j

>

. This is contradiction. Hence U(

µ

_A; )s U₁ = (

µ

_B; )s_{1 .}

Continuing in this way, we get (U µ_{A i}; )s =U(µ_{B i}; )s for all

i

. Now let

x H

∈

be such thatµ_A( )x =s_ifor some i. Then

x U

∉

(

µ

A j

; )

s

for all

i

+ ≤ ≤

1

j n

imply

( ; )

x U∉ µ_{B J}s for all

i

+ ≤ ≤

1

j n

. Hence µ_B( )x <s_j for all

i

+ ≤ ≤

1

j n

. Suppose µ_B( )x =s_{p , for some} 1≤ p i≤ _{. If} i p p i≠ ( < ) ⇒µ_B( )x =s_p<s_i ⇒ ∉x U(µ_{B i}; )s _.

On other hand

x U

∈

(

µ

_{A i}

; )

s

=

U

( ; )

µ

_{B i}

s

. Because ( )x s

A i

µ = _{. This is contradiction, and thus} i p= _and ( )x s s ( )x

A i p B

µ = = =µ _{. This is true for all}

_{x H}

_∈

_. Consequently,

µ

_A=

µ

_B.

(ii) Assume that

λ

_A =

λ

_B, then clearly vλ_A =vλ_B and )

B Im(λ ) A

Im(λ = _.

Conversely, assume that vμ_A =vμ_B and Im(λ )_A = Im(λ )_{B . Then by Theorem 2.3 and 2.8, we obtain} λ_{A and}

B

λ are finite and vλ_A = Im( )λ_A and vλ_B = Im(λB) .

Let Im(µ_A)= { , , ,... }t t t_{1 2 3 tm} and Im(µ_A)= { , , ,... }t t t_{1 2 3}′ ′ ′ _tm′ _{, where}t₁>t₂ >_{...tm and 1}t′ > ₂t′ >

'₃ ...

t > _tm′ _{. Since Im(}

λ

_{A) Im(}=

λ

_B)then t_i =t_i′

∀

_i

. We now prove that L( ; )

λ

A jt =L( ; )

λ

B jt for all

j

.

Conceder L( ; )

λ

_A t₁ and L( ; )

λ

_B t₁ . Suppose

( ; )₁ ( ; )₁

L

λ

_A t ≠L

λ

_B t . Then (L λ_A; )t₁ =L(λ_{B k}; )t _for some

k

>

1

and (L λ_{A i}; )t =L(λ_B; )t₁ for some

i

>

1

. Since t₁∈Im(

λ

_A) then there exists

y H

∈

such that

( ) 1y t A

λ

= . Then

λ

_A

( )

y

>

t

_i for all

i

>

1

, since ( ; )₁ ( ; )

L λ_A t =L λ_{B k}t _{. It follows that} y L∈ (λ_{B k}; )t _so that λ_B( )y ≤t_k <t_{1for all}

_k

_>

₁

_{. Thus} y L∈ (λ_B; )t₁ =

( ; )

L λ_{A i}t _implies λ_A( )y ≤t_{i for some}

_i

>

₁

. This is contradiction. HenceL( ; )

λ

_A t₁ =L( ; )

λ

_B t₁ . Continuing in this way, we get (L λ_{A j}; )t =L(λ_{B j}; )t _{for all}

_j

_{. Now let}

y H

∈

be such that

λ

A

( )

y

=

t

j for some

j

. Then

( ; )

_{A i}

y L

∉

λ

t

for all

j

+ ≤ ≤

1

i m

, which implies that ( ; )

y L∉ λ_{B i}t _{for all}

j

+ ≤ ≤

₁

i m

. Hence

λ

_B

( )

y

>

t

_i for

1

j

+ ≤ ≤

i m

. Suppose µ_B( )y =t_{q for some}

₁

≤ ≤

_{q j}

. If j q q j≠ ( < ) ⇒λ_B( )y =t_q > t_j ⇒ ∉y L(

λ

_{B j}; )t _.

On the other hand y L∈ (λ_{A j}; )t = L(

λ

_{B j}; )t _because

( )

A y tj

λ = _{which is contradiction and thus}

j q

=

and ( )x t

A j

λ = = t_q=λ_B( )x _{. This is true for all}

_{x H}

_∈

_.

Consequently λ_A=λ_{B. This completes the proof of the} theorem.

Acknowledgements

The authors are thankful to the referee for giving some useful suggestions to improve this paper.

REFERENCES

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, No.1, 87-96, 1983.

[2] K. Atanassov, New operations defined over the intuitionistic fuzzy Sets, Fuzzy Sets and Systems, Vol. 61, No. 2, 137-142, 1994.

[3] K. Atanassov, Intuitionistic fuzzy systems, Springer Physica-Verlag, Berlin, 1999 .

[4] R. A. Borzooei and Y. B. Jun, Intuitionistic fuzzy hyper BCK-ideals of hyper BCK- algebras, Iranian journals of fuzzy systems, Vol.1, No.1, 65-78, 2004.

[5] M. Bakhshi, M. Zahedi and R. A. Borzooei, Fuzzy (Positive, Weak) hyper BCK-ideals, Iranian journals of fuzzy systems, Vol.1, No. 2, 63-79, 2004.

[6] R. A. Borzooei and M. Bakshi, Some results on hyper BCK-algebras, Quasigroups and related systems, Vol. 11, 9-24, 2004.

[7] Y. Imai and K. Iseki, On axiom system of propositional calculus, Proc. Japan Acad., Vol. 42, 19-22, 1966.

[8] Y. B Jun and X L Xin, Implicative hyper BCK-ideals in hyper BCK-algebras, Mathematicae Japanicae, Vol. 52, No. 3, 435-443, 2000.

[9] Y B. Jun, M. M. Zahedi, L. Xin and R. A. Borzooei, On hyper BCK-algebras, Italian Journal of Pure and Appl. Math., Vol. 8, 127-136, 2000.

[10] F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandinavas, Stockholm, 45-49, 1934. [11] R. Durga Prasad, B. Satyanarayana and D. Ramesh, On

intuitionistic fuzzy positive implicative hyper BCK- ideals of BCK-algebras, International J. of Math. Sci. & Engg. Appls, Vol. 6, 175-196, 2012.