Vol 6, No 7 (2015)

(1)

Available online throug

ISSN 2229 – 5046

FUZZY TRANSLATIONS AND FUZZY MULTIPLICATIONS

OF INTERVAL-VALUED FUZZY

BG

-ALGEBRAS

S. R. BARBHUIYA*

Department of Mathematics,

Srikishan Sarda College, Hailakandi, Hailakandi-788151, Assam, India.

(Received On: 14-06-15; Revised & Accepted On: 09-07-15)

ABSTRACT

I

n this paper, we introduced the concept of fuzzy translations, fuzzy multiplications and fuzzy extensions of interval-valued fuzzy subalgebras of

BG

-algebras and investigated some of their basic properties.

Keywords:

BG

-algebra, Interval-valued fuzzy set, Interval-valued fuzzy subalgebras, Fuzzy translation, Fuzzy multiplication, Fuzzy extension.

AMS Subject Classification(2010): 06F35, 03E72, 03G25, 08A72.

1. INTRODUCTION

In 1966, Imai and Iseki [4] introduced the two classes of logical algebras viz. BCK-algebras and BCI-algebras. It is known that the notion of BCI-algebra is a generalization of notion of BCK-algebras. In 2002, Neggers and Kim [9] introduced a new notion, called B-algebras which are related to wide classes of algebras such as BCI/BCK-algebras[5]. Kim and Kim [6] introduced the notion of BG-algebra which is a generalisation of B-algebra. The concept of a fuzzy set, was introduced by Zadeh[12]. In [1] Ahn and Lee applied the fuzzy notions to BG-algebras and introduced the notion of fuzzy BG subalgebras. In [13] Zadeh made an extension of the concept of fuzzy set by an interval-valued fuzzy set. Atanassov and Gargov [2] introduced the concept of interval-valued intuitionistic fuzzy set. In [3] different operators over interval-valued intuitionistic fuzzy sets are defined. A.B.Saeid [10] defined interval-valued fuzzy BG-algebras. The concept of fuzzy translation, fuzzy multiplication and fuzzy extension in fuzzy subalgebras and fuzzy ideals in

BCI

BCK

/

-algebras and fuzzy groups has been discussed in [7, 8, 11] . Motivated by this, In this paper we introduced fuzzy translation, fuzzy multiplication and fuzzy extension in interval-valued fuzzy BG-subalgebras.

2.PRELIMINARIES

Definition 2.1: A BG-algebra is a non-empty set X with a constant ‘0 ‘and a binary operation ‘*’ satisfying following axioms:

( ) * = 0,

i x x

( ) * 0 = ,

ii x

x

( ) ( * ) * (0 * ) = ,

iii

x y

y

x

∀

x y

,

∈

X

.

For brevity we also call X a BG-algebra.

Example 2.2: Let X={ 0, 1, 2, 3, 4} with the following caley table

* 0 1 2 3 4 0 0 4 3 2 1 1 1 0 4 3 2 2 2 1 0 4 3

3 3 2 1 0 4

4 4 3 2 1 0 Then ( X, *, 0) is a BG-algebra.

(2)

3. INTERVAL-VALUED FUZZY SETS

The notion of interval-valued fuzzy set was introduced by L.A.Zadeh[13]. To consider the notion of interval-valued fuzzy sets, we need the following definitions. An interval number on

[0,

1]

, denoted by

a

ˆ

, is defined as the closed sub

interval of

[0,

1]

, where

a

ˆ

=

[

a

,

a

]

,satisfying

0 ≤

a

≤

a

≤

1

. Let

D

[0,

1]

denote the set of all such interval

numbers on

[0,

1]

and also denote the interval numbers

[0,

0]

and

[1,

1]

by

0ˆ

and

1ˆ

respectively.

Let

a

ˆ

₁

=

[

a

₁

,

a

₁

]

and

a

ˆ

₂

=

[

a

₂

,

a

₂

]

∈

D

[0,

1]

. Define on D[0, 1] the relations

≤

,

=,

<,

+

,.

by

1.

a

ˆ

₁

≤

a

ˆ

₂

⇔

a

₁

≤

a

₂ and

a

₁

≤

a

₂

2.

a

ˆ

₁

=

a

ˆ

₂

⇔

a

₁

=

a

₂ and

a

₁

=

a

₂

3.

a

ˆ

₁

<

a

ˆ

₂

⇔

a

₁

<

a

₂ and

a

₁

<

a

₂

4.

a

ˆ

₁

+

a

ˆ

₂

⇔

[

a

₁

+

a

₂

,

a

₁

+

a

₂

]

5.

a

ˆ

₁

.

a

ˆ

₂

⇔

[

min

(

a

₁

a

₂

,

a

₁

a

₂

,

a

₁

a

₂

,

a

₁

a

₂

),

max

(

a

₁

a

₂

,

a

₁

a

₂

,

a

₁

a

₂

,

a

₁

a

₂

)]

=

[

a

₁

a

₂

,

a

₁

a

₂

]

6.

k

a

ˆ

=

[

k

a

,

k

a

]

where

0 ≤

k

≤

1

Now consider two intervals

a

ˆ

₁

=

[

a

₁

,

a

₁

],

a

ˆ

₂

=

[

a

₂

,

a

₂

]

∈

D

[0,

1]

then we define refine minimum rmin as

)]

,

(

),

,

(

[

=

)

ˆ

,

ˆ

(

a

₁

a

₂

min

a

₁

a

₂

min

a

₁

a

₂

rmin

and refine maximum rmax as

)]

,

(

),

,

(

[

=

)

ˆ

,

ˆ

(

a

₁

a

₂

max

a

₁

a

₂

max

a

₁

a

₂

rmax

generally if

a

ˆ

_i

=

[

a

₁

,

a

_i

],

b

ˆ

_i

=

[

b

₁

,

b

_i

]

∈

D

[0,

1]

for i= 1,2,3,...then

we define

rmax

(

a

ˆ

_i

,

b

ˆ

_i

)

=

[

max

(

a

_i

,

b

_i

),

max

(

a

_i

,

b

_i

)]

and

rmin

(

a

ˆ

_i

,

b

ˆ

_i

)

=

[

min

(

a

_i

,

b

_i

),

min

(

a

_i

,

b

_i

)]

and

]

,

[

=

)

ˆ

(

i i i i i

i

a

rinf

∧

and

rsup

i

(

a

ˆ

i

)

=

[

∨

i

a

i

,

∨

i

a

i

]

)

1],

[0,

(

D

≤

is a complete lattice with

∧

=

rmin

,

∨

=

rmax

, 0 = [0, 0 ]

ˆ

and

ˆ1 = [1, 1]

being the least and the greatest element respectively.

Definition 3.1 An interval-valued fuzzy set defined on a non empty set X as an objects having the form

X

x

,

[

(

),

(

)]},

∀

∈

{

=

ˆ

µ

where

µ

and

µ

are two fuzzy sets in X such that

µ

(

x

)

≤

µ

(

x

)

for all

x

∈

X

.

Let

µ

ˆ

(

x

)

=

[

µ

(

x

),

µ

(

x

)],

∀

x

∈

X

, then

µ

ˆ

(

x

)

∈

D

[0

1],

∀

x

∈

X

.

If

µ

ˆ

and

ν

ˆ

be two interval-valued fuzzy sets in X, then we define

•

µ

ˆ

⊂

ν

ˆ

⇔

for all

µ

(

x

)

≤

ν

(

x

)

and

µ

(

x

)

≤

ν

(

x

)

.

•

µ

ˆ

=

ν

ˆ

⇔

for all

µ

(

x

)

=

ν

(

x

)

and

µ

(

x

)

=

ν

(

x

)

.

•

(

µ

ˆ

∪

ν

ˆ

)(

x

)

=

µ

ˆ

(

x

)

∨

ν

ˆ

(

x

)

=

[

max

{

µ

(

x

),

ν

(

x

)},

max

{

µ

(

x

),

ν

(

x

)}]

.

•

(

µ

ˆ

∩

ν

ˆ

)(

x

)

=

µ

ˆ

(

x

)

∧

ν

ˆ

(

x

)

=

[

min

{

µ

(

x

),

ν

(

x

)},

min

{

µ

(

x

),

ν

(

x

)}]

.

•

(

µ

ˆ

×

ν

ˆ

)(

x

,

y

)

=

µ

ˆ

(

x

)

∧

ν

ˆ

(

y

)

=

[

min

{

µ

(

x

),

ν

(

y

)},

min

{

µ

(

x

),

ν

(

y

)}]

.

•

µ

ˆ

c

(

x

)

=

[1

−

µ

(

x

),1

−

µ

(

x

)]

.

Definition 3.2 Let

µ

ˆ

be an interval-valued fuzzy set in X. Then for every

[0,

0]

<

t

ˆ

≤

[1,

1]

, the crisp set

}

ˆ

)

(

ˆ

|

{

=

ˆ

t

x

∈

X

µ

x

≥

t

µ

is called the level subset of

µ

ˆ

.

Definition 3.3 ([10]) An interval-valued fuzzy set

µ

ˆ

in BG-algebra X is called an interval-valued fuzzy BG-subalgebra

(3)

Example 3.4 Consider

BG

-algebra

X

=

{0,1,2,3}

with the following cayley table.

* 0 1 2 3

0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0

Define

µ

ˆ

:

X

→

D

[0,1]

by

µ

ˆ

(0)

=

µ

ˆ

(2)

=

[0.4,0.9],

µ

ˆ

(1)

=

µ

ˆ

(3)

=

[0.3,0.5]

then it is easy to verify that

µ

ˆ

is an interval-valued fuzzy BG-subalgebra of X.

Remark 3.5: ([10]) If

µ

ˆ

₁ and

µ

ˆ

₂ be any two interval-valued fuzzy BG-subalgebras of X. Then their intersection

)

ˆ

(

µ

₁

∩

µ

₂ is also an interval-valued fuzzy BG-subalgebra of X.

4. FUZZY TRANSLATION AND FUZZY MULTIPLICATION OF INTERVAL-VALUED FUZZY BG-ALGEBRAS

In what follows X denotes a BG-algebra, and for any interval-valued fuzzy set

µ

ˆ

=

[

µ

,

µ

]

of X, we denote

}

|

)

(

{

1 =

sup

x

X

T

−

µ

∈

where

T

ˆ

=

(

T

,

T

)

such that

T

≤

T

.

Definition 4.1: Let

µ

ˆ

an interval-valued fuzzy subset of X and

0 ≤

α

≤

T

where

α

ˆ

=

[

α

,

α

].

A mapping

1]

[0,

:

ˆ

X

D

T

→

α

µ

is said to be an fuzzy

α

ˆ

translation of

µ

ˆ

if it satisfies

µ

ˆ

_αˆ

(

x

)

=

µ

ˆ

(

x

)

+

α

ˆ

T

for all

x

∈

X

.

Example 4.2: Consider the interval-valued fuzzy set

µ

ˆ

as in Example 3.4,

T

=

1 −

0.9 =

0.1

let

α

ˆ

=

[0.02,

0.09]

∈

[

0ˆ

,

T

ˆ

]

Then the

α

ˆ

translation of interval-valued fuzzy set

µ

ˆ

is given by

].

[0.32,0.59

=

(3)

ˆ

=

(1)

ˆ

],

[0.42,0.99

=

(2)

ˆ

=

(0)

ˆ

T_ˆ T_ˆ T_ˆ T_ˆ

α α

α

µ

ˆ

β

ˆ

∈

D

[0,

1]

.A mapping

µ

ˆ

_βM_ˆ

:

X

→

D

[0,

1]

is

said to be an fuzzy

β

ˆ

multiplication of

µ

ˆ

if it satisfies

ˆ

(

x

)

=

ˆ

.

ˆ

(

x

)

M

β

µ

_β for all

x

∈

X

.

Example 4.4: Consider the interval-valued fuzzy set

µ

ˆ

as in Example 3.4, let

β

ˆ

=

[0.3,0.6]

Then the

β

ˆ

multiplication of interval-valued fuzzy set

µ

ˆ

is given by

].

[0.09,0.30

=

(3)

ˆ

=

(1)

ˆ

],

[0.12,0.54

=

(2)

ˆ

=

(0)

ˆ

M_ˆ M_ˆ M_ˆ M_ˆ

β β

β

µ

ˆ

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

and

β

ˆ

∈

D

[0,

1].

A mapping

1]

[0,

:

ˆ

X

D

MT

→

α β

µ

is said to be a fuzzy magnified

(

β

ˆ

α

ˆ

)

translation of

µ

ˆ

if it satisfies

µ

ˆ

_β_α

(

)

=

β

ˆ

.

µ

ˆ

(

)

α

ˆ

x

+

MT

for all

x

∈

X

.

Theorem 4.6: Let

µ

ˆ

be an interval-valued fuzzy subset of a BG-algebra X. Let

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

and

β

ˆ

∈

D

[0,

1],

then

µ

ˆ

is interval-valued fuzzy BG-subalgebra X iff

µ

_βMT_α

ˆ ˆ

ˆ

is an interval-valued fuzzy BG-subalgebra X.

Proof: Let

µ

ˆ

be an interval-valued fuzzy BG-subalgebra X, then for

x y

,

∈

X

,

we have

α

µ

β

α

µ

β

µ

ˆ

_βMT_ˆ_α_ˆ

(

x

∗

y

)

=

ˆ

.

ˆ

(

x

∗

y

)

+

ˆ

≥

ˆ

.

rmin

{

ˆ

(

x

),

ˆ

(

y

)}

+

ˆ

=

rmin

{ . ( )

β µ

ˆ

x

+

α β µ

ˆ

, . ( )

ˆ

y

+

α

ˆ

}

=

rmin

{

ˆ

ˆˆ

(

x

),

ˆ

ˆˆ

(

y

)}.

MT MT

α β α

β

µ

Hence

µ

_βMT_α

ˆ ˆ

(4)

Conversely, assume

µ

_βMT_α

ˆ ˆ

ˆ

is an interval-valued fuzzy BG-subalgebra X, then for

x y

,

∈

X

,

we have

)}

(

ˆ

),

(

ˆ

{

)

(

ˆ

=

ˆ

)

(

ˆ

.

ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ

x

y

rmin

x

y

x

MT MT MT

α β α

β α

β

µ

α

µ

β

∗

+

∗

≥

=

rmin

{ . ( )

β µ

ˆ

x

+

α β µ

ˆ

, . ( )

ˆ

y

+

α

ˆ

}

=

β

ˆ

.

rmin

{

µ

ˆ

(

x

),

µ

ˆ

(

y

)}

+

α

ˆ

.

ˆ

)}

(

ˆ

),

(

ˆ

{

.

ˆ

)

(

ˆ

.

ˆ

_µ

_α

_β

_µ

_α

β

∗

+

≥

+

⇒

x

y

rmin

x

y

which implies

µ

ˆ

(

x

∗

y

)

≥

rmin

{

µ

ˆ

(

x

),

µ

ˆ

(

y

)}

for all

x

,

y

∈

X

.

Hence

µ

_βMT_α

ˆ ˆ

ˆ

is an interval-valued fuzzy BG-subalgebra X.

Corollary 4.7: Let

µ

ˆ

be an interval-valued fuzzy subset of a BG-algebra X and

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

then

µ

ˆ

is an interval - valued fuzzy BG-subalgebra X iff

µ

ˆ

_αT_ˆ is an interval-valued fuzzy BG-subalgebra X.

Proof: Put

β

ˆ

=

1ˆ

in Theorem 4.6.

Corollary 4.8: Let

µ

ˆ

be an interval-valued fuzzy subset of a BG-algebra X and

β

ˆ

∈

D

[0,

1]

then

µ

ˆ

is an interval - valued fuzzy BG-subalgebra X iff

µ

ˆ

_βM_ˆ is an interval-valued fuzzy BG-subalgebra X.

Proof: Put

α

ˆ

=

0ˆ

in Theorem 4.6.

µ

ˆ

be an interval-valued fuzzy BG-subalgebra X, then _ˆ _ˆ

ˆ ˆ

ˆ

_βαMT

(0)

ˆ

_βαMT

( )

x

µ

≥

µ

for all

x

∈

X

.

Proof: Here

µ

ˆ

be an interval-valued fuzzy BG-subalgebra X therefore

µ

_βMT_α

ˆ ˆ

ˆ

is an interval-valued fuzzy

BG-subalgebra X.

α

µ

β

µ

ˆ

_βMT_ˆ_α_ˆ

(0)

=

ˆ

_βMT_ˆ_α_ˆ

(

x

∗

x

)

=

ˆ

.

ˆ

(

x

∗

x

)

+

ˆ

≥

β

ˆ

.

rmin

{

µ

ˆ

(

x

),

µ

ˆ

(

x

)}

+

α

ˆ

=

rmin

{ . ( )

β µ

ˆ

x

+

α β µ

ˆ

, . ( )

ˆ

x

+

α

ˆ

}

.

ˆ

(

)

ˆ

]}

ˆ

,

ˆ

)

(

ˆ

.

ˆ

[

],

ˆ

)

(

ˆ

.

ˆ

,

ˆ

)

(

ˆ

.

ˆ

{[

=

rmin

β

µ

x

+

α

β

µ

x

+

α

β

µ

x

+

α

β

µ

x

+

α

ˆ

_ˆ

ˆ

_ˆ

ˆ

_ˆ

ˆ

_ˆ

= [

min

{ . ( )

β µ

x

+

α β µ

, . ( )

x

+

α

},

min

{ . ( )

β µ

x

+

α β µ

, . ( )

x

+

α

}]

=

ˆ

ˆˆ

(

x

)

MT

α β

µ

).

(

ˆ

(0)

ˆ

ˆ ˆ ˆ

ˆ

x

MT MT

α β α

β

µ

≥

⇒

µ

_βMT_α ˆ ˆ

ˆ

be an interval-valued fuzzy BG-subalgebra X where

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

and

β

ˆ

,

t

ˆ

∈

D

[0,

1]

with

α

ˆ

_≥

t

, then the level subset

U

_{β α}MT_{ˆ ,}

( , ) = {

µ

ˆ

t

ˆ

x

∈

X

| . ( )

β µ

ˆ

x

≥ −

t

ˆ

α

ˆ

},

∀ ∈

t

ˆ

Im

( )

µ

ˆ

is a subalgebra of X.

Proof: Let

x

,

y

∈

U

_βˆ_,_α

(

µ

ˆ

,

t

ˆ

)

⇒

β

ˆ

.

µ

ˆ

(

x

)

≥

t

ˆ

−

α

ˆ

MT

and

β

ˆ

.

µ

ˆ

(

y

)

≥

t

ˆ

−

α

ˆ

t

x

)

ˆ

(

ˆ

.

ˆ

₊

_≥

⇒

β

µ

α

and

β

ˆ

.

µ

ˆ

(

y

)

+

α

ˆ

≥

t

ˆ

t

x

MT

_ˆ

)

(

ˆ

≥

(5)

Now

ˆ_ˆ ˆ_ˆ ˆ_ˆ

ˆ

_βαMT

(

x y

)

rmin

{

ˆ

_βαMT

( ),

x

ˆ

_βαMT

( )}

y

µ

∗

≥

µ

≥

rmin t t

{ , }

ˆ ˆ

=

rmin

{[

t

ˆ

,

t

ˆ

],

[

t

ˆ

,

t

ˆ

]}

=

[

min

{

t

ˆ

,

t

ˆ

},

min

{

t

ˆ

,

t

ˆ

}]

=

[

t

ˆ

,

t

ˆ

]

=

t

ˆ

t

y

x

)

ˆ

(

ˆ

.

ˆ

_∗

₊

_≥

⇒

β

µ

α

).

ˆ

,

ˆ

(

)

(

, ˆ

t

U

y

x

∗

∈

_βMT_α

µ

⇒

Theorem 4.11: Intersection and union of any two fuzzy translation of an interval-valued fuzzy BG-subalgebra of X is also an interval-valued fuzzy BG-subalgebra of X.

Proof: Let

µ

ˆ

_αT_ˆ and

µ

ˆ

_δT_ˆ be two fuzzy translatons of an interval-valued fuzzy BG-subalgebra

µ

ˆ

, where

]

ˆ

,

0ˆ

[

ˆ

,

ˆ

δ

∈

T

α

. Assume that

α

ˆ

≤

δ

ˆ

By Corollary 4.7

µ

ˆ

_αT_ˆ and

µ

ˆ

_δT_ˆ are interval-valued fuzzy BG-subalgebras of X.

Now

ˆ ˆ ˆ ˆ

ˆ

(

µ

_αT

∪

µ

_δT

)( ) =

x

rmax

{

µ

_αT

( ),

x

µ

_δT

( )}

x

=

rmax

{ ( )

µ

ˆ

x

+

α µ

ˆ

, ( )

ˆ

x

+

δ

ˆ

}

]}

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

[

],

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

{[

=

rmax

µ

x

+

α

µ

x

+

α

µ

x

+

δ

µ

x

+

δ

}]

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

{

},

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

{

[

=

max

µ

x

+

α

µ

x

+

δ

max

µ

x

+

α

µ

x

+

δ

]

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

[

=

µ

x

+

δ

µ

x

+

δ

=

ˆ

(

).

ˆ

)

(

ˆ

=

µ

x

+

δ

µ

_δT_ˆ

x

Also

ˆ ˆ ˆ ˆ

ˆ

(

µ

_αT

∩

µ

_δT

)( ) =

x

rmin

{

µ

_αT

( ),

x

µ

_δT

( )}

x

=

rmin

{ ( )

µ

ˆ

x

+

α µ

ˆ

, ( )

ˆ

x

+

δ

ˆ

}

]}

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

[

],

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

{[

=

rmin

µ

x

+

α

µ

x

+

α

µ

x

+

δ

µ

x

+

δ

}]

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

{

},

ˆ

)

(

ˆ

,

ˆ

)

(

ˆ

{

[

=

min

µ

x

+

α

µ

x

+

δ

min

µ

x

+

α

µ

x

+

δ

=

[

µ

ˆ

(

x

)

+

α

ˆ

,

µ

ˆ

(

x

)

+

α

ˆ

]

=

ˆ

(

x

)

ˆ

=

ˆ

(

x

).

T α

µ

α

µ

+

5. FUZZY TRANSLATIONS AND FUZZY MULTIPLICATIONS OF INTERVA-VALUED FUZZY

BG

-ALGEBRAS UNDER HOMOMORPHISM

X

and

Y

be two BG-algebras. Then a mapping

f

:

X

→

Y

is said to be homomorphism if

X

y

x

y

f

x

f

y

x

f

(

∗

)

=

(

)

∗

(

),

∀

,

∈

.

f

:

X

→

Y

be a homomorphism of BG-algebras. If

µ

ˆ

be an interval-valued fuzzy subalgebras of Y, then

(

ˆ

)

1 T

f

−

µ

α an interval-valued fuzzy subalgebra of X.

Proof: Let

µ

ˆ

be an interval-valued fuzzy subalgebras of Y, therefore by Corollary 4.7

µ

ˆ

_αT_ˆ is also an interval-valued

fuzzy subalgebras of Y. Now

f

µ

T_αˆ 1

)

ˆ

(

−

is defined by

(

ˆ

)(

)

=

ˆ

(

))

.

1

X

x

f

x

(6)

Now

)}

(

{

ˆ

=

)

)(

ˆ

(

ˆ ˆ

1

y

x

f

y

x

f

−

µ

_αT

∗

µ

_αT

∗

=

ˆ

{

f

(

x

)

f

(

y

)}

rmin

{

ˆ

(

f

(

x

)),

ˆ

(

f

(

y

))}

T

T T

α α

α

µ

∗

≥

=

{

(

ˆ

)(

),

(

ˆ

)(

)}.

1 ˆ 1

y

f

x

f

rmin

T T

α

µ

−

Hence

(

ˆ

)

1 T

f

−

µ

α an interval-valued fuzzy subalgebras of X.

f

:

X

→

Y

be a homomorphism of BG-algebras. If

µ

ˆ

be an interval- valued fuzzy subalgebras of Y, then

f

−1

(

µ

ˆ

_βM_ˆ

)

an interval-valued fuzzy subalgebra of X.

Proof. Same as Theorem 5.2.

f

:

X

→

Y

be an onto homomorphism of BG-algebras. If

µ

ˆ

be an interval-valued fuzzy subset of Y such that

(

ˆ

)

1 T

f

−

µ

_α is an interval-valued fuzzy subalgebra of X. Then

µ

ˆ

is also an interval-valued fuzzy subalgebra of Y.

Proof: Since f is onto homomorphism therefore to each

x

′

,

y

′

∈

Y

,

there exists

x

,

y

∈

X

such that

f x

( ) =

x

′

,

( ) =

f y

y

′

and

f

(

x

∗

y

)

=

f

(

x

)

∗

f

(

y

)

=

x

′

∗

y

′

.Now since

(

ˆ

)

1 T

f

−

µ

_α is an interval-valued fuzzy subalgebras of X. Therefore

)}

)(

ˆ

(

),

)(

ˆ

(

{

)

)(

ˆ

(

ˆ 1 ˆ 1 ˆ 1

y

f

x

f

rmin

y

x

f

−

µ

αT

∗

≥

−

µ

αT −

µ

αT

)}

(

ˆ

),

(

ˆ

{

)

(

ˆ

_αT_ˆ

f

x

y

rmin

µ

_αT_ˆ

f

x

µ

_αT_ˆ

f

y

µ

∗

≥

⇒

)}

(

ˆ

),

(

ˆ

{

)}

(

)

(

{

ˆ

_αT_ˆ

f

x

f

y

rmin

µ

_αT_ˆ

f

x

µ

_αT_ˆ

f

y

µ

∗

≥

⇒

)}.

(

ˆ

),

(

ˆ

{

}

{(

ˆ

x

y

rmin

ˆ

x

ˆ

y

T T

T

′

_∗

′

_≥

′

⇒

µ

α

µ

α

µ

α

T

α

µ

ˆ

⇒

is an interval valued fuzzy subalgebra of Y. Hence By Corollary 4.7

µ

ˆ

is an interval valued fuzzy subalgebra of Y.

f

:

X

→

Y

be an onto homomorphism of BG-algebras. If

µ

ˆ

be an interval-valued fuzzy subset of Y such that

f

−1

(

µ

ˆ

_βM_ˆ

)

is an interval-valued fuzzy subalgebra of X. Then

µ

ˆ

is also an interval-valued fuzzy subalgebra of Y.

Proof: Same as Theorem 5.4.

f

:

X

→

Y

be an epimorphism, where X, Y are two two interval-valued fuzzy subalgebras and

]

ˆ

,

0ˆ

[

ˆ

∈

T

α

. Then the inverse image of

α

ˆ

translation of any fuzzy subalgebra

µ

ˆ

of Y is same as the

α

translation of the inverse image of fuzzy subalgebra

µ

ˆ

.

Proof: Let

f

:

X

→

Y

be an epimorphism, where X, Y are two two interval-valued fuzzy subalgebras and

]

ˆ

,

0ˆ

[

ˆ

∈

T

α

. Let

µ

ˆ

be a fuzzy subalgebra of Y. Therefore by Theorem 4.6 The

α

ˆ

translation

µ

ˆ

_αT_ˆ is a fuzzy subalgebra of Y. Therefore by Corollary 4.7

(

ˆ

)

1 T

f

−

µ

_α is a fuzzy subalgebra of X.

Also

(

ˆ

)(

)

=

ˆ

(

))

=

ˆ

(

))

ˆ

=

(

ˆ

)(

)

ˆ

=

(

ˆ

))

ˆ

(

)

1 1 ˆ ˆ 1

x

f

x

f

x

f

x

f

x

f

−

µ

αT

µ

αT

µ

+

α

−

µ

+

α

−

µ

αT .

Hence

f

µ

_αT

f

µ

T_αˆ 1 ˆ 1

))

ˆ

(

=

)

ˆ

(

− −

f

:

X

→

Y

be an epimorphism, where X, Y are two two interval- valued fuzzy subalgebras and

1]

[0,

ˆ

_∈

_D

(7)

Proof: Same as Theorem 5.6.

6. INTERVAL-VALUED FUZZY SUBALGEBRA EXTENSION

µ

ˆ

₁ and

µ

ˆ

₂ be two interval-valued fuzzy subsets of X. If

µ

ˆ

₁

(

x

)

≤

µ

ˆ

₂

(

x

)

for all

x

∈

X

,

then we say that

µ

ˆ

₂ is a fuzzy (an interval-valued) S - extension of

µ

ˆ

₁

.

µ

ˆ

₁ and

µ

ˆ

₂ be two interval-valued fuzzy subsets of X such that

µ

ˆ

₂ is a fuzzy extension of

µ

ˆ

₁. If

1

ˆ

µ

is an interval-valued fuzzy sub algebra of X implies that

µ

ˆ

₂ is an interval-valued fuzzy sub algebra of X, then

µ

ˆ

₂ is called fuzzy S-extension of

µ

ˆ

₁

.

µ

ˆ

be an interval-valued fuzzy subalgebra of X and

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

, then the fuzzy

α

ˆ

translation

µ

ˆ

_αT_ˆ of

µ

ˆ

is a fuzzy S-extension of

µ

ˆ

.

Proof: Here

µ

ˆ

be an interval-valued fuzzy subalgebra of X, then by Corollary 4.7 the fuzzy

α

ˆ

translation

µ

ˆ

_αT_ˆ of

µ

ˆ

is also an interval-valued fuzzy subalgebra of X for all

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

. Also

µ

ˆ

αT_ˆ

(

x

)

=

µ

ˆ

(

x

)

+

α

ˆ

≥

µ

ˆ

(

x

)

∀

x

∈

X

.

Therefore

µ

ˆ

_αT_ˆ is a fuzzy S-extension of

µ

ˆ

.

µ

ˆ

be an interval-valued fuzzy subalgebra of X and

α

ˆ

,

β

ˆ

∈

[

0ˆ

,

T

ˆ

]

. If

α

ˆ

≥

β

ˆ

then the fuzzy

α

ˆ

translation

µ

ˆ

_αT_ˆ of

µ

ˆ

is a fuzzy S-extension of the fuzzy

β

ˆ

translation

µ

ˆ

T_β_ˆ of

µ

ˆ

.

Proof: Here

µ

ˆ

be an interval-valued fuzzy subalgebra of X, then by Corollary 4.7 the fuzzy

α

ˆ

translation

µ

ˆ

_αT_ˆ and

fuzzy

β

ˆ

translation

µ

ˆ

_βT_ˆ of

µ

ˆ

is also an interval-valued fuzzy subalgebra of X. Since Also

α

ˆ

≥

β

ˆ

therefore

X

x

+

α

≥

µ

+

β

∀

∈

µ

_ˆ

₍

₎

_ˆ

₍

₎

ˆ

_{. Therefore}

_ˆ

₍

₎

_ˆ

₍

₎

_.

ˆ

x

X

T

≥

∀

∈

β

α

µ

Therefore

µ

ˆ

_αT_ˆ is a fuzzy S-extension of

.

ˆ

T_ˆ

β

µ

Theorem 6.5: Intersection of any two interval-valued fuzzy S-extensions of an interval-valued fuzzy subalgebra

µ

ˆ

is a

fuzzy S-extensions of

µ

ˆ

.

Proof: Let

µ

ˆ

₁ and

µ

ˆ

₂ be two interval-valued fuzzy S-extensions of a fuzzy subalgebra

µ

ˆ

of X. Then

)

(

ˆ

)

(

ˆ

₁

x

µ

x

µ

≥

and

µ

ˆ

₂

(

x

)

≥

µ

ˆ

(

x

)

∀

x

∈

X

.

Now By Remark 3.5

µ

ˆ

₁

∩

µ

ˆ

₂ is an interval-valued fuzzy subalgebra of X. Now

1 2 1 2

ˆ

(

µ

∩

µ

)( ) =

x

rmin

{ ( ),

µ

x

µ

( )}

x

≥

rmin

{ ( ), ( )} =

µ

x

µ

x

µ

( )

x

.

Hence

(

µ

ˆ

₁

∩

µ

ˆ

₂

)

is fuzzy S-extensions of

µ

ˆ

.

µ

ˆ

be an interval-valued fuzzy set of X and

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

and

β

ˆ

∈

D

[0

1]

.If

µ

ˆ

_βM_ˆ is a fuzzy

subalgebra of X then the fuzzy

α

ˆ

translation

µ

ˆ

_αT_ˆ of

µ

ˆ

is a fuzzy S-extension of

ˆ

M_ˆ

.

β

µ

Proof: Let

α

ˆ

∈

[

0ˆ

,

T

ˆ

]

and

β

ˆ

∈

D

[0

1]

and if fuzzy

β

ˆ

multiplication

µ

ˆ

_βM_ˆ is an interval-valued fuzzy subalgebra

of X then by Corollary 4.7 the interval-valued fuzzy set

µ

ˆ

and the fuzzy

α

ˆ

translation

µ

ˆ

_αT_ˆ of

µ

ˆ

is an

interval-valued fuzzy subalgebra of X. Now

µ

ˆ

_αTˆ

(

x

)

=

µ

ˆ

(

x

)

+

α

ˆ

≥

µ

ˆ

(

x

)

≥

µ

ˆ

(

x

).

β

ˆ

=

µ

ˆ

_βMˆ . Hence T

α

µ

ˆ

ˆ is a fuzzy

S-extension of

ˆ

M_ˆ

.

β

(8)

REFERENCES

1. S. S. Ahn and H. D. Lee, Fuzzy subalgebras of BG-algebras, Commun Korean Math. Soc 19(2) (2004) 243-251. 2. K. T. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems,31(1) (1989),

343-349.

3. K. T. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 64 (1994), 159-174.

4. Y. Imai and K. Iseki, On Axiom systems of Propositional calculi XIV, Proc, Japan Academy, 42 (1966)19-22. 5. K. Iseki, On some ideals in BCK-algebras, Math.Seminar Notes, 3 (1975), 65-70.

6. C. B. Kim and H. S. Kim, on BG-algebras, Demonstratio Mathematica, 41(3) (2008) 497-505.

7. K. J. Lee,Y. B. Jun, M. I. Doh , Fuzzy translation and Fuzzy multiplications of BCK/BCI -algebras, Commun Korean Math.Soc 24(3) (2009) 353-360.

8. S. K. Majumder, S. K. Sardar, Fuzzy Magnified translation on groups, Journal of Mathematics, North Bengal University, 1(2) (2008) 117-124.

9. J. Neggers and H. S. Kim, On B-algebras, Matematicki Vesnik, 54,No.1-2, (2002), 21-29.

10. A. B. Saeid, Interval-valued fuzzy BG-algebras, Kangweon-Kyungki Math. Jour, 14(2) (2006), 203-215. 11. Y. B. Jun, translation of fuzzy ideals in BCK/BCI-Algebras, Hacettepe Journal of Mathematics and Statistics,

Volume 40(3) (2011),349-358.

12. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353.

13. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Information Science 8 (1975), 199-249.

Source of support: Nil, Conflict of interest: None Declared