Vague Direct Product In Bck- Algebra

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1

Vague Direct Product in BCK- Algebra

S.JafariP 1

P

, L. MariapresentiP 2

P

and I. ArockiaraniP 3

P

Department of Mathematics

P

1

P

College of Vestsjaelland south, Herrestraede, Denmark

P

2,3

P

Nirmala College for Women, Coimbatore-18

Email: 30TU[email protected]U30T, 30TU[email protected]U30T

Abstract

In this paper, the notion of direct product of vague set in BCK- algebra is introduced and some related properties are investigated. Also we introduce some of the properties of direct product of finite vague ideal and vague subalgebras of BCK- algebras.

Keywords: vague set, direct product of vague sets, direct product of finite vague ideal, direct product of finite vague subalgebras.

1. Introduction: As crisp set theory does not reflect the real life problems exactly, L. A. Zadeh [16] introduced the concept of fuzzy set to generalize the notion of a member belonging to a set X. The concept of fuzzy set has been applied to various algebraic structures. Imai and Iseki [6,7]introduced two classes of abstract algebras, BCK- algebra and BCI- algebras. Al- Shehri [1], Jun et al[8,9], Saeid et al[13,14] and satyanarayana et al[15], applied the concept of fuzzy set to BCK- algebra. Zhan and Tan [17] introduced the concept of fuzzy H- ideal in BCK- algebras. Gau and Buehrer[3] introduced the concept of vague set. The vague set is developed by means of truth membership function and false membership function. Ranjit Biswas[12] initiated the study of vague algebra by studying vague groups. The objective of this paper is to contribute further to the study of direct product of vague set in BCK algebra and vague ideal in BCK- algebras and discuss some of their results.

2.Preliminaries:

Definition 2.1:[5] A BCI algebra is a non-empty set X with a constant 0 and a binary operation “

∗

” satisfying the following axioms for all

x

,

y

,

z

∈

X

.

:

(i)

((

x

∗

y

)

∗

(

x

∗

z

))

∗

(

z

∗

y

)

=

0

(ii)

(

x

∗

(

x

∗

y

))

∗

y

=

0

(iii)

x

∗

x

=

0

(iv)

x

∗

y

=

0

and

y

∗

x

=

0

implies x=y.

We can define a partial ordering “ ≤ ” by x ≤ y if and only if

x

∗

y

=

0

.

If a BCI- algebra X satisfies

0

∗

x

=

0

, for all

X

x

∈

, then we say that X is a BCK- algebra. Any BCK- algebra X satisfies the following axioms :

(i)

(

x

∗

y

)

∗

(

x

∗

z

)

≤

(

z

∗

y

)

(ii)

x

∗

(

x

∗

y

)

≤

y

(iii)

x

≤

x

(iv)

0

≤

x

(v)

x

≤

y

and

y

≤

x

implies x=y. where x ≤

y means

x

∗

y

=

0

.

Definition 2.2:[4] A non empty subset S of X is called a subalgebra of X if

x

∗

y

∈

S

for any

.

,

y

S

x

∈

Definition 2.3:[5] A non empty subset I of X is

called an ideal of X if it satisfies

(IR1R)

0

∈

I

and

(IR2R)

x

∗

y

∈

I

and

y

∈

I

imply

x

∈

I

.

2

(IR3R)

x

∗

(

y

∗

z

)

∈

I

and

y

∈

I

imply

x

∗

z

∈

I

,

for all

x

,

y

,

z

∈

X

.

Definition 2.5: [2] A vague set A in the universe of discourse U is characterized by two membership functions given by:

(i) A true membership function

]

1

,

0

[

:

U

→

t

_A and

(ii) A false membership function

]

1

,

0

[

:

U

→

f

_A

where

t

_A

(

x

)

is a lower bound on the grade of membership of x derived from the “evidence for x”,

)

(

x

f

_A is a lower bound on the negation of x derived from the “evidence against x”, and

1

)

(

)

(

x

+

f

x

≤

t

_{A A} . Thus the grade of membership of U in the vague set A is bounded by a subinterval

)]

(

1

),

(

[

t

_A

x

−

f

_A

x

of [0,1]. This indicates that if the actual grade of membership of x is µ(x), then,

)

(

1

)

(

)

(

x

x

f

x

t

_A

≤

µ

≤

−

_A .The vague set A is

written as

A

=

{

x

,

[

t

A

(

x

),

1

−

f

A

(

x

)

]

/

u

∈

U

}

where, the interval

[

t

_A

(

x

),

1

−

f

_A

(

x

)]

is called the vague value of x in A, denoted by

V

_A

(

x

)

.

Definition 2.6:[2] Let A and B be vague sets(VSs) of the form

A

=

{

x

,

[

t

A

(

x

),

1

−

f

A

(

x

)

]

/

x

∈

X

}

and

[

]

{

x

t

x

f

x

x

X

}

B

=

,

_B

(

),

1

−

_B

(

)

/

∈

Then

(i)

A

⊆

B

if and only if

t

_A

(

x

)

≤

t

_B

(

x

)

and

)

(

1

)

(

1

−

f

_A

x

≤

−

f

_B

x

for all x

∈

X

(ii)

A=B if and only if

A

⊆

B

and

B

⊆

A

(iii)

A

{

x

f

A

x

t

A

x

x

X

}

c

=

−

∈

/

)

(

1

),

(

,

(iv)

(

)

(

)

_















∈ −

−

= x X

x B f x A f

x B t x A t x B

A /

) ( 1 ), ( 1 min

, ) ( ), ( min ,



(v)

(

)

(

)

_















∈ −

∨ −

∨

= x X

x B f x A f

x B t x A t x B

A /

) ( 1 ) ( 1 max

, ) ( ) ( max ,



For the sake of simplicity, we shall use the notation

A f A t x

A= , ,1− instead of

[

]

{

x t_A x f_A x x X

}

A= , ( ),1− ( ) / ∈ .

Definition 2.7:[11] A vague set A on X is called a vague subalgebra of x if, for any x∈X , we have

)} ( ), ( min{ )

(xy t_A x t_A y A

t ≥ and

)} ( 1 ), ( 1 min{ ) (

1− f_A xy ≥ − f_A x − f_A y

Definition 2.8:[11] A vague set A of a BCK- algebra X is called a vague ideal of X if the following conditions are true:

(i) (V_A(0)≥V_A(x)), (∀x∈X)

(ii) (V_A(x)≥imin{V_A(x∗y),V_A(y)}

) ,

(∀x y∈X that is,

), ( 1 ) 0 ( 1 ), ( ) 0

( t_A x f_A f_A x A

t ≥ − ≥ − and

)} ( 1 ), ( 1 { min ) ( 1 (

)} ( ), ( { min ) ( (

y A f y x A f x

A f

y A t y x A t x

A t

− ∗ − ≥ −

∗ ≥

for

all x,y∈X .

Definition 2.9:[11] Let A be a vague set of a

universe X with the true- membership function t_A

and the false- membership function fRAR. The (α, β)-

cut of the vague set A is a crisp subset AR(α, β)Rof the set

X given by A_(α,β) ={x∈X/V_A(x)≥[α,β]}.

Clearly AR(0,0)R=X. The (α,β)- cut of the vague set A

are also called vague cuts of A.

Definition 2.10:[11]The α- cut of the vague set A is

a crisp subset ARαR of the set X given by ARαR= AR(α,α)R

.ThusAR0R=X, and if α ≥ β then,

A

β

⊆

A

αand A(α,

β)= ARα.REquivalently, we define the α-cut as }.

) ( , :

{ α

α = x x∈X t_A x ≥ A

3. Direct product of vague ideals in BCK- algebras

Definition 3.1: Let V_A =[t_A,1− f_A] and

] 1 , [t_B f_B B

V = − be two vague sets in BCK- algebras

1

X and X₂respectively. Then the direct product of

vague sets V_Aand V_Bis denoted by

] 1

, [t_{A B} f_{A B} B

A

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3

)}

( ), ( { min ) ,

(x y V_A x V_B y B

A

V _× =

(i.e.,) t_A×B(x,y)=min {t_A(x),t_B(y)} and

)} ( 1 ), ( 1 { min ) , (

1− f_A×B x y = − f_A x − f_B y for all

2 1 ) ,

(x y ∈X ×X .

Definition 3.2: A vague setV_A×B =[t_A×B,1− f_A×B]

of X₁×X₂ is called a vague subalgebra of X₁×X₂

if V_A×B((x₁,y₁)∗(x₂,y₂))≥

(i.e.,)

)} 2 , 2 ( ), 1 , 1 ( { min

)) 2 , 2 ( ) 1 , 1 ((

y x B A t y x B A t y x y x B A t

× ×

≥ ∗

×

)} 2 , 2 ( 1

), 1 , 1 ( 1

{ min

)) 2 , 2 ( ) 1 , 1 (( 1

y x B A f y x B A f

y x y x B A f

× − ×

− ≥ ∗

× −

for all (x₁,y₁),(x₂,y₂)∈X₁×X₂.

Example 3.3: Let X₁={0,1} andX₂ ={0,1,2} are

BCK- algebras by the following tables:

Then X₁×X₂ ={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}is a

BCK algebra. We define vague set V_A =[t_A,1− f_A]

on XR1R as :X →[0,1] A

t and f_A:X →[0,1] by

X 0 1

VRA [0.5,0.6] [0.3,0.4]

and V_B =[t_B,1− f_B]on XR2R as :X →[0,1] B

t and

] 1 , 0 [ :X → B

f by

X 0 1 2

VRB [0.5,0.7] [0.5,0.6] [0.2,0.5]

Then V_A×B =[t_A×B,1− f_A×B]is a vague subalgebra

of BCK- algebra.

Definition 3.4: A vague set

V

_A×B

=

[

t

_A×B

,

1

−

f

_A×B

]

of X₁×X₂ is called a vague H-ideal of X₁×X₂ if,

(i) V_A×B(0,0) ≥V_A×B(x,y) and

(ii) V_A×B((x₁,y₁)∗(x₃,y₃))≥

)} 2 , 2 ( ))), 3 , 3 ( ) 2 , 2 (( ) 1 , 1 ((

min{V_A×B x y ∗ x y ∗ x y V_A×B x y

(i.e.,)

) , ( 1

) 0 , 0 ( 1

) , ( )

0 , 0 (

y x B A f B

A f

and y x B A t B A t

× − ≥ × −

× ≥ ×

)} 2 , 2 ( ))), 3 , 3 ( ) 2 , 2 (( ) 1 , 1 (( { min

)) 3 , 3 ( ) 1 , 1 ((

y x B A t y x y x y x B A t

y x y x B A t

× ∗

∗ ×

≥ ∗

×

)} 2 , 2 ( 1

))), 3 , 3 ( ) 2 , 2 (( ) 1 , 1 (( 1

min{

)) 3 , 3 ( ) 1 , 1 (( 1

y x B A f y x y x y x B A f

y x y x B A f

× − ∗ ∗

× −

≥ ∗

× −

Definition 3.5: A vague set V_A×B =[t_A×B,1− f_A×B]

of X₁×X₂ is called a vague closed H-ideal of

2 1 X

X × if, V_A×B((0,0)∗(x,y))≥V_A×B(x,y)

(i.e.,)

) , ( 1

)) , ( ) 0 , 0 (( 1

) , ( ))

, ( ) 0 , 0 ((

y x B A f y x B

A f

and y x B A t y x B

A t

× − ≥ ∗ ×

−

× ≥ ∗ ×

Theorem 3.6: Let V_A =[t_A,1− f_A] and

] 1 , [t_B f_B B

V = − be two vague subalgebras of BCK-

algebras X₁and X₂respectively. Then

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague sublagebra of

BCK- algebra X₁×X₂.

Proof: For any (x₁,y₁),(x₂,y₂)∈X₁×X₂. Then

)} 2 , 2 ( ), 1 , 1 ( { min

)}} 2 ( ) 2 ( min{ )}, 1 ( ), 1 ( min{ { min

)}} 2 ( ) 1 ( min{ )}, 2 ( ), 1 ( min{ { min

)} 2 1 ( ), 2 1 ( { min

) 2 1 , 2 1 (

)) 2 , 2 ( ) 1 , 1 ((

y x B A V y x B A V

y B V x A V y

B V x A V

y B V y B V x

A V x A V

y y B V x x A V

y y x x B A V

y x y x B A V

× ×

≥

∗ =

∗ ≥

∗ ∗

=

∗ ∗ × =

∗ ×

Hence for all (x₁,y₁),(x₂,y₂)∈X₁×X₂, * 0 1 2

0 0 0 0 1 1 0 1

2 2 2 0 * 0 1

0 0 0

4

]

1 ,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague subalgebra of

BCK- algebra X₁×X₂.

Theorem 3.7: Let V_A =[t_A,1− f_A] and

] 1 , [t_B f_B B

V = − be two vague H-ideals of BCK-

algebras X₁and X₂respectively. Then

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague H- ideals of

BCK- algebra X₁×X₂.

Proof: For any (x,y)∈X₁×X₂,

) , ( )}

( ), ( min{

)} 0 ( ), 0 ( min{ ) 0 , 0 (

y x B A V y B V x A V

B V A V B

A V

× = ≥

= ×

Now for any (x₁,y₁),(x₂,y₂),(x₃,y₃)∈X₁×X₂

)}}. 2 , 2 ( min{

)))}, 3 , 3 ( ) 2 , 2 ( ) 1 , 1 (( min{min{

)}} 2 , 2 ( min{

)))}, 3 2 ( 1 ( )), 3 2 ( 1 (( min{min{

)}} 2 ( ), 2 ( min{

))}, 3 2 ( 1 ( )), 3 2 ( 1 ( min{min{

)}} 2 ( )), 3 2 ( 1 ( min{

)}, 2 ( )), 3 2 ( 1 ( min{ { min

)} 3 1 ( ), 3 1 ( { min

) 3 1 , 3 1 (

)) 3 , 3 ( ) 1 , 1 ((

y x B A V

y x y x y x B A V

y x B A V

y y y x x x B A V

y B V x A V

y y y B V x x x A V

y B V y y y B V

x A V x x x A V

y y B V x x A V

y y x x B A V

y x y x B A V

× ∗ ∗

× ≥

×

∗ ∗ ∗

∗ × =

∗ ∗ ∗

∗ =

∗ ∗ ∗ ∗ ≥

∗ ∗

=

∗ ∗ × =

∗ ×

Hence for all (x₁,y₁),(x₂,y₂),(x₃,y₃)∈X₁×X₂,

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague H- ideal of

BCK- algebra X₁×X₂.

Theorem 3.8: Let V_A =[t_A,1− f_A] and

] 1 , [t_B f_B B

V = − be two vague closed H-ideals of

BCK- algebras X₁and X₂,respectively. Then

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague closed H- ideals

of BCK- algebra X₁× X₂.

Proof: Let V_A =[t_A,1− f_A] and V_B =[t_B,1− f_B]

be two vague closed H-ideals of BCK- algebras X₁

and X₂respectively. Using theorem 3.7,

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague H-ideal of

2 1 X

X × .Now for any (x,y)∈X₁×X₂, then

) , ( )}

( ), ( min{

)} 0 ( ), 0 ( min{

)) 0 ( ), 0 (( ))

, ( ) 0 , 0 ((

y x B A V y B V x A V

y B V x A V

y x B A V y x B

A V

× = ≥

∗ ∗

=

∗ ∗ × = ∗ ×

Hence V_A×V_B =[t_A×B,1− f_A×B] is a vague closed

H- ideal of BCK- algebra X₁×X₂.

Theorem 3.9: If V_A×B =[t_A×B,1− f_A×B] is a vague

H- ideal of BCK- algebra X₁×X₂. Then we have

) , ( )

, ( )

, ( ) ,

(a b ⊆ x y ⇒V_A×B x y ⊆V_A×B a b for all

2 1 ) , ( ), ,

(ab x y ∈X ×X .

Proof: Let (a,b),(x,y)∈X₁×X₂, such that

) 0 , 0 ( ) , ( ) , ( ) , ( ) ,

(ab ⊆ x y ⇒ ab ∗ x y = . Consider

). , (

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

)} , ( ))), 0 , 0 ( ) , (( ) , (( min{

)) 0 , 0 ( ) , (( )

, (

b a B A V

b a B A V b a y x B A V

b a B A V b

a y x B A V

y x B A V y x B A V

× =

× ∗

× =

× ∗

∗ ×

≥

∗ ×

= ×

Hence the proof.

Definition 3.10: Let V_A×B =[t_A×B,1− f_A×B] be a

vague set of a BCK- algebra X₁×X₂and for any

] 1 , 0 [ ,β∈

α . Then the (α, β)- cut of the vague set

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a crisp subset AR(α, β) Rof

the set X given by

]} , [ ) , ( / 2 1 ) , {( ) ,

(αβ = x y ∈X ×X VA×B x y ≥ α β

A .The

(α,β)- cut of the vague set V_A×B =[t_A×B,1− f_A×B] are also called vague cuts of

] 1

,

[t_{A B} f_{A B} B

A

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5

Theorem 3.11: Let V_A×B =[t_A×B,1− f_A×B] be a

vague subalgebra of BCK- algebra X₁×X₂. Then

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague subalgebra of

BCK- algebra X₁×X₂ if and only if for any

] 1 , 0 [ ,β∈

α is a (α,β)- cut of the vague subalgebra of

BCK- algebra X₁×X₂.

Proof: Let V_A×B =[t_A×B,1− f_A×B] be a vague

subalgebra of BCK- algebra X₁×X₂. Now for any

] 1 , 0 [ ,β∈

α and (x₁,y₁),(x₂,y₂)∈A_(α,β)then

α ≥ ×_B(x₁,y₁) A

t and t_A×B(x₂,y₂)≥α. Since

)) 2 , 2 ( ) 1 , 1

((x y x y B

A

t _× ∗

α α

α =

≥ ×

×

≥min{t_{A B}(x₁,y₁),t_{A B}(x₂,y₂)} min{ , }

. )) 2 , 2 ( ) 1 , 1

(( ∗ ≥α

×_B x y x y A

t

) , ( ) 2 , 2 ( ) 1 , 1

(x y ∗ x y ∈A_αβ

⇒

and for any (x₁,y₁),(x₂,y₂)∈A_(α,β)then

β ≥ ×

− ( ₁, ₁)

1 f_{A B} x y and 1− f_A×B(x₂,y₂)≥β . Since

β β β = ≥

× − ×

− ≥

∗ ×

−

} , min{

)} 2 , 2 ( 1 ) 1 , 1 ( 1

min{

)) 2 , 2 ( ) 1 , 1 (( 1

y x B A f y x B A f

y x y x B A f

. ) , ( ) 2 , 2 ( ) 1 , 1 (

)) 2 , 2 ( ) 1 , 1 (( 1

β α

β

A y x y x

y x y x B A f

∈ ∗

⇒

≥ ∗

× −

Therefore AR(α, β) Ris a vague subalgebra of BCK-

algebra X₁×X₂.

Conversely, suppose that AR(α, β) Ris a vague subalgebra

of BCK- algebra X₁× X₂. Suppose that

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is not a vague subalgebra

of a BCK- algebra X₁×X₂. Then there exist

2 1 ) 2 , 2 ( ), 1 , 1

(x y x y ∈X ×X , such that

)}. 2 , 2 ( ), 1 , 1 ( min{

)) 2 , 2 ( ) 1 , 1 ((

y x B A t y x B A t y x y x B A t

× ×

< ∗ ×

Now let

)}} 2 , 2 (

), 1 , 1 ( min{ )) 2 , 2 ( ) 1 , 1 (( { 2 1 0

y x B A t

y x B A t y

x y x B A t t

×

× +

∗ ×

=

.

This implies

)}. 2 , 2 ( ), 1 , 1 ( min{

0 )) 2 , 2 ( ) 1 , 1 ((

y x B A t y x B A t

t y x y x B A t

× ×

<

< ∗

×

So

) , ( ) 2 , 2 ( ), 1 , 1

(x y x y ∉A_{α β} but

) , ( ) 1 , 1

(x y ∈A_αβ

and

) , ( ) 2 , 2

(x y ∈A_αβ , which is contradiction. Hence

] 1

,

[t_{A B} f_{A B} B

A

V _× = _× − _× is a vague subalgebra of

BCK- algebra X₁×X₂.

Definition 3.12: Let [ ,1 ] i A f i A t i A

V = − be n-vague

set of BCK- algebras XRiR, respectively i=1,2,…..n.

Then ∏

= n

i 1VAi is called direct product of finite vague

sets of ∏

= n

i 1Xiif,

))} (

),..., 1 1 (( 1 min{

) 1 (( 1 ,...,

n y n x n A V y x A V

n y n

i VAi x yi xn

∗ ∗

= ∏

= ∗ ∗

for all ∏

= ∈ n

i Xi

n y y y n x x x

1 ) ,..., 2 , 1 ( ), ,..., 2 , 1

( .

Definition 3.13: Let [ ,1 ] i A f i A t i A

V = − be n-vague

set of BCK- algebras XRiR, respectively i=1,2,…..n.

Then ∏

= n

i 1VAi is called direct product of finite vague

subalgebra of ∏

= n

i 1Xi if,

≥ ∏

=₁ (( 1,..., )∗(y1,...,yn) n

i VAi x xn

)} ,..., 1 ( 1 (( 1,..., ), 1

min{ y y_n

n

i

n

i VAi n x x i A V ∏

6

)}

,..., 1 ( 11 (( 1,..., ), 11 min{

) ,..., 1 ( 11 (( 1,..., )

)} ,..., 1 ( 1 (( 1,..., ), 1 min{

) ,..., 1 ( 1 (( 1,..., ) )

(i.e.,

n y y n

i

n

i fAi

n x x i A f

n y y n

i fAi x xn

n y y n

i

n

i tAi n x x i A t

n y y n

i tAi x xn

∏

= − ∏= −

≥ ∏

= − ∗

∏

= ∏=

≥ ∏

= ∗

for all ∏

= ∈ n

i Xi

n y y y n x x x

1 ) ,..., 2 , 1 ( ), ,..., 2 , 1

( .

Definition 3.14: Let [ ,1 ] i A f i A t i A

V = − be n-vague

set of BCK- algebras XRiR, respectively i=1,2,…..n.

Then ∏

= n

i 1VAi is called direct product of finite vague

ideal of ∏

= n

i 1Xiif, (i)

∏ = ≥ ∏

=

n

i VAi x xn

n

i 1VAi(0,...,0) 1 ( 1,..., ) and

(ii) ∏ ≥

= n

i 1VAi(x1,...,xn)

)} ,..., 1 ( 1 (( 1,..., ) ( 1,..., )), 1

min{n y y_n

i

n

i VAi n

y y n x x i A V ∏

= ∗ ∏=

∏ = − ≥ ∏

= −

∏ = ≥ ∏

=

n

i fAi x xn

n

i fAi

and n

i tAi x xn

n

i tAi i

11 ( 1,..., ) 11 (0,...,0)

1 ( 1,..., ) 1 (0,...,0)

) ( ) (i.e.,

)}, ,..., 1 ( 1 (( 1,..., ) ( 1,..., )), 1 min{

1 ( 1,..., ) )

(

n y y n

i

n

i tAi n y y n x x i A t n

i tAi x xn

ii

∏

= ∗ ∏=

≥ ∏

=

)}, ,..., 1 ( 11 (( 1,..., ) ( 1,..., )), 11 min{

11 ( 1,..., )

n y y n

i

n

i fAi

n y y n x x i A f n

i fAi x xn

∏

= − ∗ ∏= −

≥ ∏

= −

for all ∏

= ∈ n

i Xi

n y y n x x

1 ) ,..., 1 ( ), ,..., 1

( .

Definition 3.15: Let [ ,1 ] i A f i A t i A

V = − be n-vague

set of BCK- algebras XRiR, respectively i=1,2,…..n.

Then ∏

= n

i 1VAi is called direct product of finite vague

H- ideal of ∏

= n

i 1Xi if,

∏ = ≥ ∏

=

n

i VAi x xn

n

i VAi i

1 ( 1,..., ) 1 (0,...,0)

)

( and

)} ,..., 1 ( 1

1 (( 1,..., ) (( 1,..., ) ( 1,..., )), min

1 (( 1,.., ) ( 1,..., ) )

(

n y y n

i VAi n

i VAi x xn y yn z zn

n

i VAi x xn z zn

ii

∏ = ∏

= ∗ ∗

≥ ∏

= ∗

∏ = − ≥ ∏

= −

∏ = ≥ ∏

=

n

i fAi x xn

n

i fAi

and n

i tAi x xn

n

i tAi i

11 ( 1,..., ) 11 (0,...,0)

1 ( 1,..., ) 1 (0,...,0)

) ( ) (i.e.,

)} ,..., 1 ( 1

1 (( 1,..., ) (( 1,..., ) ( 1,..., )), min

1 (( 1,.., ) ( 1,..., ) )

(

n y y n

i tAi n

i tAi x xn y yn z zn

n

i tAi x xn z zn

ii

∏ = ∏

= ∗ ∗

≥ ∏

= ∗

)} ,..., 1 ( 11

11 (( 1,..., ) (( 1,..., ) ( 1,..., )), min{

11 (( 1,.., ) ( 1,..., ) )

(

n y y n

i fAi

n

i fAi x xn y yn z zn

n

i fAi x xn z zn

ii

∏ = − ∏

= − ∗ ∗

∏

= − ∗ ≥

for all ∏

= ∈ n

i Xi

n y y n x x

1 ) ,..., 1 ( ), ,..., 1

( .

Theorem 3.16: Let [ ,1 ] i A f i A t i A

V = − be n-vague

subalgebras of BCK- algebras XRiR, respectively

i=1,2,…..n. Then ∏

= n

i 1VAi is vague subalgebras of

∏ = n

i 1Xi.

Proof: Let [ ,1 ] i A f i A t i A

V = − be n-vague

subalgebras of BCK- algebras XRiR, respectively. Let

∏ = ∈ n

i Xi

n y y y n x x x

1 ) ,..., 2 , 1 ( ), ,..., 2 , 1

www.ijiset.com

7

))} ( ),..., 1 ( 1 min{ )},..., ( ),..., 1 ( 1 min{min{ ))} ( ), ( min{ )},..., 1 ( 1 ), 1 ( 1 min{min{ ))} ( ),..., 1 1 (( 1 min{ ) 1 (( 1 ,...,

)) ,..., 1 ( 1 (( 1,..., )

n y n A V y A V n x n A V x A V n y n A V n x n A V y A V x A V n y n x n A V y x A V n y n

i VAi x yi xn

n y y n

i VAi x xn

= ≥ ∗ ∗ = ∏ = ∗ ∗ = ∏ = ∗ )} ,..., 1 ( 1 , 1 ) ,..., 1 ( min{ )) ,..., 1 ( 1 ) ,..., 1 (( n y y n i Ai

V n i n x x i A V n y y n i n x x i A V ∏ = ∏ = ≥ ∏ = ∗

This completes the proof.

Theorem 3.17: Let [ ,1 ] i A f i A t i A

V = − be n-vague

ideals of BCK- algebras ∏

= n

i 1Xi, respectively. If

) ( )

( x_i

i A V i y i x i A

V ∗ ≥ where i=1,2,…..n, then

∏ = n

i 1VAi is vague H- ideal of ∏= n

i 1Xi

Proof: Let [ ,1 ] i A f i A t i A

V = − be n-vague ideals of

BCK- algebras∏

= n

i 1Xi, respectively and

) ( )

( x_i

i A V i y i x i A

V ∗ ≥ for any

∏ = ∈ n

i Xi

n y y y n x x x 1 ) ,..., 2 , 1 ( ), ,..., 2 , 1

( . Then we

have, ) ,...., 1 (( 1 )} ( ),...., 1 ( 1 min{ )} ( ),...., 1 1 ( 1 min{ ) ,...., 1 1 ( 1 )) ,...., 1 ( ) ,...., 1 (( 1 n x x n

i VAi

n x n A V x A V n y n x n A V y x A V n y n x y x n

i VAi

n y y n x x n

i VAi

∏ = = ≥ ∗ ∗ = ∗ ∗ ∏ = = ∗ ∏ =

Hence ∏

= n

i 1VAi is vague H- ideal of ∏= n

i 1Xi

Theorem 3.18: Let [ ,1 ] i A f i A t i A

V = − be n-vague

ideals of BCK- algebras ∏

= n

i 1Xi, respectively. If

i z i y i

x ∗ ≤ where i=1,2,…..n ,holds in ∏

= n

i 1Xi . Then

)} ,...., 1 ( 1 ), ,...., 1 ( 1 min{ ) ,...., 1 ( 1 n z z n

i VAi n y y n

i VAi n x x n

i VAi

∏ = ∏ = ≥ ∏ = .

Proof: Let

∏ = ∈ n

i Xi

n z z z n y y y n x x x 1 ) ,..., 2 , 1 ( ), ,..., 2 , 1 ( ), ,..., 2 , 1 (

be such that, x_i∗y_i ≤ z_i. Then (x_i∗y_i)∗z_i =0, and thus )} ,..., 1 ( 1 )), ,..., 1 ( ) ,..., 1 (( 1 min{ ) ,..., 1 ( 1 n y y n

i VAi n y y n x x n

i VAi n x x n

i VAi

∏ = ∗ ∏ = ≥ ∏ = )} ,..., 1 ( 1 ), ,..., 1 ( 1 min{ )} ,..., 1 ( 1 )}, ,..., 1 ( 1 ), 0 ,..., 0 ( 1 min{min{ ,..., 1 ( 1 , 1 ( 1,..., )} )), ,..., 1 ( )) ,..., 1 ( ) ,..., 1 ((( 1 min{min{ n z z n

i VAi n y y n

i VAi

n y y n

i VAi n z z n

i VAi n

i VAi n y y n

i VAi

n

i VAi z zn

n z z n y y n x x n

i VAi

∏ = ∏ = = ∏ = ∏ = ∏ = = ∏ = ∏ = ∗ ∗ ∏ = ≥

Theorem 3.19: Let

[

,

1

]

i A i A i

A

t

f

V

=

−

be n-vague

ideal of BCK- algebras XRiR, respectively, If

i i

y

x

≤

,

whenever

x

_i

∗

y

_i

=

0

where i=1,2,…..n. Then

). ,... 1 ( 1 ) ,...., 1 (

1 y yn

n

i VAi n

x x n

i∏=VAi ≥ ∏=

Proof: Let ∏

= ∈ n

i Xi

n y y y n x x x 1 ) ,..., 2 , 1 ( ), ,..., 2 , 1 (

be such that x_i ≤ y_i, whenever x_i∗y_i =0 and then

8

)

,..., 2 , 1 ( 1

)} ,..., 2 , 1 ( 1 ), 0 ,..., 0 , 0 (( 1 min{

) ,..., 2 , 1 ( 1

)), ,..., 2 , 1 ( ) ,..., 2 , 1 (( 1 min{

) ,..., 2 , 1 ( 1

n y y y n

i VAi

n y y y n

i VAi n

i VAi

n y y y n

i VAi

n y y y n x x x n

i VAi n x x x n

i VAi

∏ = =

∏ = ∏

= =

∏ =

∗ ∏

= ≥

∏ =

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