Research Journal of Applied Sciences, Engineering and Technology 4(21): 4200-4205, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 18, 2011
Accepted: April 23, 2012
Published: November 01, 2012
Intuitionisitic Fuzzy B-algebras
Jiayin Peng
School of Mathematics and Information Science, Neijiang Normal University, Neijiang, CO
641112 China
Abstract: The notion of intuitionsitic fuzzy B-algebras is introduced and their some properties are investigated.
How to deal with the homomorphic image and inverse image of intuitionsitic fuzzy ideals are B-algebras. The
relations between a intuitionsitic fuzzy B-algebra and a intuitionsitic fuzzy B-algebra in the product B-algebras
are given.
Keywords: B-algebra, homomorphism, intuitionsitic fuzzy B-algebra, product B-algebra
INTRODUCTION
In 1966, (Imai and Iseki, 1966) defined a class of
algebras of type (2, 0) named BCK-algebras. At the same
time, (Iseki, 1966) generalized another class of algebras
of type (2, 0) called BCI-algebras. Hu and Li (1983)
introduced the concept of BCH-algebras and gave
examples of proper BCH-algebras (Hu and Li, 1985).
Neggers and Kim (2001) introduced and investigated a
class of algebras, viz., the class of B-algebras, which is
related to several classes of algebras of interest such as
BCH/BCI/BCK- algebras and which seems to have rather
nice properties without being excessively complicated
otherwise. Xi (1991) applied the notion of fuzzy sets
(Zadeh, 1966) to BCK/BCI-algebras. Jun (1993), Meng,
(1994) and Jun and Roh (1994) investigated fuzzy ideals
of BCK/BCI-algebras. Young et al. (2002) introduced
notions of fuzzy B-algebras and investigated its
properties.
In 1986, (Atanassov, 1986) introduced the concept of
intuitionistic fuzzy sets, which is a significant extension
of fuzzy set theory by Zadeh (1966).Li and Wang (2000)
studied intuitionistic fuzzy group and its homomorphisic
image and subsequently discussed the product and
extension principle of the intuitionistic fuzzy sets (Li and
Wang, 2002). In this study, we introduce notion of
intuitionistic fuzzy B-algebras and discuss their
properties. Some properties of the homomorphic
(isomorphic) image and inverse image of intuitionistic
fuzzy B-algebras will be studied, the notions of
intuitionistic fuzzy relation, strongest intuitionistic fuzzy
relation and Cartesian product of intuitionisitic fuzzy sets
will be given, some relations between intuitionistic fuzzy
B-algebras and its product B-algebras are exposed.
A collection of intuitionsitic fuzzy sets of X is
denoted by IFS(X). Let A, B, IFS(X). A = {< x, :A(x)
<A(x)>| x , X } and B = {< x, :B(x) <B(x)>| x , X
}.Consider the relation between A and B as follow:
C
C
AfB if and only if :A (x) < :B (x) and <A (x) $ <B (x)
for all x , X.
A = B if and only if :A (x) = :B (x) and <A (x) = <B(x)
for all x , X .
C
A ∩ B = {< x , µ A ( x ) ∧ µ B ( x )v A ( x ) > | x ∈ X }
C
A ∪ B = {< x , µ A ( x ) ∨ µ B ( x ), v A ( x ) ∧ v B ( x ) > | x ∈ X }
Otherwise,
C
IA
⎧
⎫
= ⎨ x , ∧ µ A j ( x ), ∨ v A j ( x ) x ∈ X ⎬
j ∈J
j ∈J
⎩
⎭
IA
⎫
⎧
= ⎨ x , ∨ µ A j ( x ), ∧ v A j ( x ) x ∈ X ⎬
j ∈J
j ∈J
⎭
⎩
j ∈J
C
j ∈J
i
i
C
~A = {< x, :A(x) 1! :A(x) >| x , X }
C
A = {< x, 1! <A(x), <A(x)>| x , X }
Definition 2: (Li and Wang, 2000) Let f be a mapping
from the set X to the set Y and B = {< x, :B(x), >*x , X }.
a intuitionsitic fuzzy set of Y. The inverse image of B,
denoted by f!1(B), is the intuitionsitic fuzzy set of X
defined by:
f!1(B)×{+x, :B(f(x)), vB(f(x)),|x0X}
PRELIMINARIES
Definition 1: Atanassov (1986) Let X be a nonempty set.
A set A = {< x, :A(x) <A(x)>| x , X } is said to a
intuitionsitic fuzzy set of X if mapping :A: ÷ [0, 1] and <A
: X÷ [0, 1] satisfy 0 # :A (x) + <A (x) # 1 for all x , X.
Conversely, let A = {< x, :A(x), <A (x), >| x , X }. be
a intuitionsitic fuzzy set of X. Then the image of A,
denoted by f (A), is the intuitionsitic fuzzy set of Y given
by:
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µ A (∏ x * x ) = µ A ( x ) and
f(A)×{+y, f(:A)(y), f$ (vA)(y) ,|y0Y}
n
where,
ν A (∏ x * x ) = ν A ( x ) Whenever n is even.
⎧⎪ f∨−1 µ A ( x ), f ( y ) ≠ ∅
f ( µ A )( y ) = ⎨ x ∈ ( y )
⎪⎩ 0,
f −1 ( y ) ≠ ∅
−1
n
Proof: Let x , X and assume that n is odd. Then n = 2k!1
for some positive interger k . Observe that :A(x*x) = :A
(0) $ :A (x) and <A (x*x) = <A (0) # <A(x). Suppose
that µ A ( ∏ x * x) ≥ µ A ( x) and ν A (∏ x * x) ≤ ν A ( x) for a positive
⎧⎪ f∨−1 ν A ( x ), f −1 ( y ) ≠ ∅
f (ν A )( y ) = ⎨ x ∈ ( y )
⎪⎩ 1,
f −1 ( y ) ≠ ∅
2 k −1
n
integer k . Then:
µ A ( ∏ x * x ) = µ A ( ∏ x * x )µ A ( ∏ x * ( x * ( x * x )))
2 ( k + 1)
Definition 3: Neggers and Kim (2001) An algebra of type
(X;*,0) is called a B-algebra if it satisfies the following
axioms:
2 k +1
2 k −1
= µ A (∏ x * x) ≥ µ A ( x)
2 k −1
ν A ( ∏ x * x ) = ν A ( ∏ x * x ) = ν A ( ∏ x * ( x * ( x * x )))
(B1) x * x = 0
2 ( k + 1)
2 k +1
2 k −1
= ν A (∏ x * x) ≤ ν A ( x)
(B2) x * 0 = x
2 k −1
which proves (1). Similarly we obtain the second part.
(BCH3) (x* y)* z = x* (z* (0 * y))
for all x ,y, z , X .
In Young et al. (2002), the concept of fuzzy Balgebras is introduced. A fuzzy set : in B-algebra X is
said to be an anti-fuzzy B-algebra of X if for all x, y , X ,
:(x*y) # :(x) ∨ :(y).
Proposition 3: Let A = {< x, :A(x), <A (x), >| x , X } be a
intuitionsitic fuzzy B-algebra, then for all x, y , X:
(IFB1) :A (0*x) $ :A(x) and) <A (0*x) # <A(x),
(IFB2) :A (x*(0*y)) $ :A(x) ∧ :A(y) and <A (x*(0*y))
# <A(x) ∨ <A(y)
INTUITIONSITIC FUZZY B-ALGEBRAS
Definition 1: Let (X; *, 0) be a B-algebra. A intuitionsitic
fuzzy set A = {< x, :A(x), <A (x), >| x , X } of X is said to
be a intuitionsitic fuzzy B-algebra if it satisfies :A(x*y) v
:A(x) :A(y) ∧ and <A(x*y) # <A(x)w<A(y) for all x, y , X:
Proof: For any x, y , X, we have :A (0*x) $ :A(0) ∧
:A(x), = :A(x) and :A (x*(0*y)) $ :A(x) ∧ :A(x) (0*y) $
:A(x) ∧ :A(y) .
Similarly we have that <A (0*x) # <A (x) and <A (x*
(0*y))# <A (x) ∨ <A (y).
Proposition 1: Let A = {< x, :A(x), <A (x), >| x , X } be a
intuitionsitic fuzzy B-algebra. Then , :A(0) $, :A(x) and <A
(0) # <A (x) ≤ , for all x , X.
Corollary 4: Let A = {< x, :A(x), <A (x), >| x , X }be a
intuitionsitic fuzzy B-algebra, then for all x , X, :A(0*x)
= :A(x) and <A(0*x) = <A(x).
Proof: Since x*x = 0 for all x , X , we havet hat :A(0) =
:A(x*x) $ :A(x)v:A(x) = :A(x) and <A (0) # <A (x)w<A (x)
= <A (x) <A(0) # .This completes the proof.
For any elements x and y of X, let us write ∏ x * y for
Proof: Since x = 0* (0*x), if A is a intuitionsitic fuzzy Balgebra, then <A(x) = <A(0*(0*x)) # <A(0) ∨ <A(0*x) =
<(0*x), i.e., <A(0*x) = <A (x). Similarly we can prove that
:A(0*x) = :A (x). for all x , X.
n
x*(…*(x*(x*y))) where, x occurs n times.
Proposition 2: Let A = {< x, :A(x), <A (x), >| x , X } be a
intuitionsitic fuzzy B-algebra and let n , N. Then for all:
µ A (∏ x * x ) ≥ µ A ( x ) and
n
Theorem 5: If a intuitionsitic fuzzy set A of X satisfies
(IFB1) and (IFB2), then A is a intuitionsitic fuzzy Balgebra.
Proof: Suppose that A satisfies (IFB1) and (IFB2) and let
x, y , X. Then,
ν A (∏ x * x ) ≤ ν A ( x ) Whenever n is odd
<A (x * y) = <A (x * (0 * (0 * y)))
n
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4200-4205, 2012
# <A (x) ∨ <A (0*y)
Case (2): If f!1(y1) = Ø and f!1(y2) = Ø, then let x10, x20 , X
such that :A(x10) = zefinf( y ) µ A ( z) , :A(x20) = zefinf( y ) µ( z) and:
−1
= <A (x)w<A (y)
−1
2
1
Similarly we have that :A(x * y) $ :A (x) v :A (y)
Hence A is a intuitionsitic fuzzy B-algebra.
µ ( z)
: A ( x 1 0 * x 2 0 ) zef inf
. Then f(:A)(y1*y2)
( y *y )
= sup µ A ( z) = µ A ( x10 * x20 ) ≥ µ A ( x10 ) ∧ µ A ( x20 )
−1
1
z∈ f
Theorem 6: If A and B are intuitionsitic fuzzy B-algebra
of X, then so is A 1B.
Proof: For all x, y , X, we get:
( y1 * y2 )
= ( inf
µ A ( z)) ∧ ( inf
−1
−1
z∈ f
( y1 )
z∈ f
( y2 )
µ A ( z)) = f ( µ A )( y1 ) ∧ f ( µ A )( y2 ) .
Similarly, we have that:
µ A∩ B ( x * y ) = µ A ( x * y ) ∧ µ B ( x * y )
≥ [ µ A ( x ) ∧ µ A ( y )] ∧ [ µ B ( x ) ∧ µ B ( y )]
f$ (ν A )( y1 * y2 ) ≤ f$
(v )( y ) ∨ f$ (v )( y )
A
1
A
2
Therefore, f(A) is a intuitionsitic fuzzy B-algebra of Y.
= [ µ A ( x ) ∧ µ B ( x )] ∧ [ µ A ( y ) ∧ µ B ( y )]
= µ A∩ B ( x ) ∧ µ A ∩ B ( y )
Theorem 10: Let f be a homomorphism from a B-algebra
X onto a B-algebra Y and A a intuitionsitic fuzzy Balgebra of Y. Then the preimage f!1(A) of A is a
intuitionsitic fuzzy B-algebra of X.
similarly, we have that:
ν A ∩ B ( x * y ) ≤ v A∩ B ( x ) ∨ v A∩ B ( y )
Proof: Let x,y , X. Since A is a intuitionsitic fuzzy Balgebra of Y , we have that:
and hence A1B is a intuitionsitic fuzzy B-algebra.
µf
Corollary 7: Let Ai be intuitionsitic fuzzy B-algebra of X
for all i , J. Then I A is also a intuitionsitic fuzzy Bi∈ j
−1
2
−1
( A)
( x * y ) = µ A ( f ( x * y ))
= µ A ( f ( x ) * f ( y ))
i
algebra of X. Similarly, we have:
≥ µ A ( f ( x )) ∧ µ A ( f ( y ))
Theorem 8: Let A and B are intuitionsitic fuzzy Balgebras of X . Then ~ A and A are intuitionsitic fuzzy
B-algebras of X.
Definition 2: A intuitionsitic fuzzy set A of X has sup-inf
property if, for any TfX, there exist x0, y0, , T such that
:A(x0) = SUPz,T :A (z) and <A(y0) = inf z,T <A (z).
Theorem 9: Let f be a homomorphism from a B-algebra
X into a B-algebra Y and A a intuitionsitic fuzzy B-algebra
of X with sup-inf property. Then the image f(A ) of A is a
intuitionsitic fuzzy B-algebra of Y.
Proof: Let A = {< x, :A(x), <A (x), >| x , X } and let y1, y2
, Y. we consider the following two cases:
!1
!1
= µ f −1( A ) ( x ) ∧ µ f −1( A ) ( y )
Similarly,
ν f −1( A) ( x * y ) ≤ ν f −1( A) ( x ) ∨ ν f −1 ( A) ( y )
Hence f!1(A) is a intuitionsitic fuzzy B-algebra of X .
Corollary 11: Let f be a homomorphism from a B-algebra
X onto a B-algebra Y. Then the following conclusions
hold:
C
Case (1): If f (y1) = Ø or f (y2) = Ø, then f (y1* y2) = Ø.
And so f(:A)(y1* y2) = 0 and f$ (<A)(y1* y2) = 1. Thus
f(:A)(y1* y2) = 0 = f(:A)(y2) ∧ and:
If for all j , J, Aj, are intuitionsitic fuzzy B-algebras
A j ) is intuitionsitic fuzzy B-algebra
of X , then f ( I
j ∈J
!1
C
of Y.
If for all t , T, Bt are intuitionsitic fuzzy B-algebras
Bt ) is intuitionsitic fuzzy Bof Y, then f − 1(I
t ∈T
f$ (ν A )( y1 * y2 ) = 1 = f$ (ν A )( y1 ) ∨ f$ (ν A )( y2 )
algebras of X.
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4200-4205, 2012
C
C
If A is a intuitionsitic fuzzy B-algebra of X, then f(A)
and f(A) are intuitionsitic fuzzy B-algebra of Y.
If B is a intuitionsitic fuzzy B-algebra of Y, then
f!1(B) and f!1(B) are intuitionsitic fuzzy B-algebras
of X.
Theorem 12: Let f be an isomorphism from a B-algebra
X onto a B-algebra Y. If A is a intuitionsitic fuzzy Balgebra of X, then f!1( f(A)) = A.
where, :A×B = :A(x)v:B(y), <A×B = <A(x)w<B(y). Then A×B
is a binary intuitionsitic fuzzy relation on X .
Theorem 16: Let A = {< x, :A(x), <A (x),>|x , X } and
B = {< x, :B(x), <B (x),>|x , X } be intuitionsitic fuzzy Balgebra of X . Then A×B is a intuitionsitic fuzzy Balgebra of X ×X.
Proof: Since A, B are intuitionsitic fuzzy B-algebras of X,
we have:
Proof: For any x , X , let f(x) = y, since f is an
isomorphism, f!1(y) = {x}. Thus = f(:A)(f(x)) =
f(:A)(x)) = f(:A)(y)= ∨−1 µ A ( x ) = µ A ( x ) and:
x∈ f
f
−1
µ A× B (( x , y ) *( x ', y '))
= µ A× B ( x * x ' , y * y ' )
( y)
= µ A ( x * x ') ∧ µ B ( y * y ' )
( f$ (ν A ))( x ) f$ (ν A )( f ( x )) = f$ (ν A )( y )
∧ ν A ( x) = ν A ( x)
≥ [ µ A ( x ) ∧ µ A ( x ')] ∧ [ µ B ( y ) ∧ µ B ( y ')]
= x ∈ f −1
= [ µ A ( x ) ∧ µ B ( y )] ∧ [ µ A ( x ' ) ∧ µ B ( y ' )]
therefore f!1(f(A)) = (A).
Corollary 13: Let f be an isomorphism from a B-algebra
X onto a B-algebraY. If B is a intuitionsitic fuzzy Balgebra of Y, then f!1(f(B)) = (B).
= µ A× B ( x , y ) ∧ µ A× B ( x ' , y ' )
for all:
( x , y ),( x ' , y ' ) ∈ X × X
Corollary 14: Let f : X ÷ X be an automorphism. If A is
a intuitionsitic fuzzy B-algebra of X, then:
Similarly,
f ( A) = A ⇔ f
−1
( A) = A
ν A× B (( x , y ) * ( x ' , y ' )) ≤ ν A× B ( x , y ) ∨ ν A× B ( x ' , y ' )
A intuitionsitic fuzzy set: R = {+(x, y), :R (x, y), vR
(x, y),}|x0X, y0Y}0 IFS(X×Y) is called a binary
intuitionsitic fuzzy relation (Lei et al., 2005) from X into
Y. A binary intuitionsitic fuzzy relation from X into Y is
said to be a binary intuitionsitic fuzzy relation on X
if X = Y.
Definition 3: Let A = {< x, :A(x), <A (x),>|x , X } IFS[X]
A binary intuitionsitic fuzzy relation:
R=
{
( x , y ), µ R ( x , y ), ν R ( x , y ) x , y ∈ X
}
on X is called a intuitionsitic fuzzy relation on A if :R (x,
y) # :A (x) v:A(y) and <R(x,y) $ <A(x)w<A(y) for all x, y ,
X.
Lemma 15: Let A = {< x, :A(x), <A (x),>|x , X } and B =
{< x, :B(x), <B (x),>|x , X } be intuitionsitic fuzzy sets of
X. A Cartesian product of A and B defined by:
{
A × B ( x , y ), µ A× B ( x , y ), ν A× B ( x , y ) x , y ∈ X
}
for all (x, y), ( x ' , y ' ) X×X. Hence A×B is a intuitionsitic
fuzzy B-algebra of X×X.
Theorem 17: Let A = {< x, :A(x), <A (x),>|x , X }and B =
{< x, :B(x), <B (x),>|x , X } be intuitionsitic fuzzy sets of
a B-algebra X such that A×B is a intuitionsitic fuzzy Balgebra of X×X . Then,
C
C
C
C
C
C
C
C
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Either :A(x) # :A(0) or :B(x) # :B(0) for all x , X.
Either <A(x) # <A(0) or <B(x) $<B (0) for all x , X.
If :A(x) # :A(0) for all x , X, then :A (x) # :B(0) or
:B(x) :B(0) .
If <A(x) # <A(0) for all x , X, then <A(x) $ <B(0) or
<B(x) $ <B(0).
If :B(x) # :B(0) for all x , X, then :A (x) # :A(0) or :B
(x) # :A(0).
If <B(x) # <B (0) for all x , X, then <A(x) $ <A(0) or
<B(x) $ <A(0).
Either :A or :B is a fuzzy B-algebra (Young et al.,
2002) of X.
Either <A or <B is an anti-fuzzy B-algebra of X.’
Res. J. Appl. Sci. Eng. Technol., 4(21): 4200-4205, 2012
Proof: (i) Suppose that :A (x) # :A(0) and :B (y) > :B(0)
for some x, y , X. Then:
µ A× B ( x , y ) = µ A ( x ) ∧ µ B ( y )) > µ A (0) ∧ µ B (0) = µ A× B (0,0).
This is a contradiction. Hence (i) holds.
(ii) is by similar method to part (i).
(iii) Assume that there exist x, y , X. such that :A(x)
>:B(0) and :A(y)>:B(0).
Then
µ A× B ( x , y ) = µ A ( x ) ∧ µ B ( y ) > µ B (0) = µ B (0)) = µ A× B (0,0) , which
is a contradiction. Hence (iii) holds.
(vi), (v) and (vi) are by similar method to part (iii).
(vii) Since by (i) either :A(x) # :A(0) or :B(x) > :B(0)
for all x , X , without loss of generality we may
assume that :B(x) # :B(0) for all x , X . From (v), it
follows that :A(x) # :A(0) or :B(x) > :A(0). If :B(x) >
:A(0), then :A×B(0, x) = :A(0)v:B(x) = :B(x). Let (x1,
x2), (y1, y2), , X×X . Since A×B is a intuitionsitic
fuzzy B-algebra of , X×X we have:
From the proof of Theorem 17 (vii) and Theorem 17
(viii), the following results hold up.
Theorem 18: Let A = {< x, :A(x), <A (x),>|x , X } and B
= {< x, :B(x), <B (x),>|x , X }be intuitionsitic fuzzy sets of
a B-algebra X such that A×B is a intuitionsitic fuzzy Balgebra of X ×X. Then:
C
C
If :A(x) # :B(0) v:B(0) and <A(x) # <B(0)w<B(0) for
all x , X, then A is a intuitionsitic fuzzy B-algebra of
X.
If :B(x) # :A(0) v:B(0) and wfor all x , X, then B is a
intuitionsitic fuzzy B-algebra of X.
Definition 4: Let A = {< x, :A(x), <A (x),>|x , X } ,
IFS(X). A intuitionsitic fuzzy relation:
R=
{ ( x, y), µ ( x, y),ν
R
R
( x, y) x, y ∈ X
}
on X is called a strongest intuitionsitic fuzzy relation on A
if:
µ A × B (( x1 , x2 ) * ( y1 , y2 )) ≥ µ A × B
(IFB3) (( x , x ) ∧ µ
1 2
A × B ( y1 , y2 ))
:B(x ,y) = :A(x)v:A(y) and <R(x, y) = <A(x)w<A(y)
= [ µ A ( x1 ) ∧ µ B ( x 2 )] ∧ [ µ A ( y1 ) ∧ µ B ( y 2 )]
for all x, y , X.
If we take x1 = y2 = 0, then
Proposition 19: For a given intuitionsitic fuzzy set A = {<
x, :A(x), <A(x),>|x , X } of a B-algebra X, let R be a
:B(x2* y2) = :A×B (0, x2* y2)
strongest intuitionsitic fuzzy relation on A . If R is a
intuitionsitic fuzzy B-algebra of X × X , then :A(x) $ :A(0)
and <A(x) $ <A(0) for all x , X.
$[ :A(0)v:B(x2)]v[ :A(0)v:B(y2)]
= :B (x2)v:B(y2)
This proves that :B is a fuzzy B-algebra of X. Now
we consider the case :A(y) # :A(0) for all x , X.
Suppose that :B(y) > :A(0) for some y , X. Then
:A(0) < :B(y) # :B(0). Since :A(x) # :A (0) for all x ,
X , we have :B(0) > :A(x). for all x , X. Hence
:A×B(x,0) = :A(x)v:B(0) = :A(x). Taking x2 , y2 = 0 in
(IFB3), then:
Proof: Since R is a intuitionsitic fuzzy B-algebra, we
have :R(x, x) # :R(0,0) and <R (x ,x) $ <R (0,0) for all x ,
X, that is, v and <A(x) v<A(x) $ <A(0 w<A(0) . So :A(x) #
:A(0) and <A(x) $ <A (0) for all x , X.
Theorem 20: Let A = {< x, :A(x), <A (x),>|x , X } Let be
a intuitionsitic fuzzy set of B-algebra X and:
R=
:A×B(x1*y1) = :A×B (x1*y1, 0)
:A×B ((x1,0)*( y1,0))
{ ( x, y), µ ( x, y),ν
R
R
( x, y) x, y ∈ X
}
strongest intuitionsitic fuzzy relation on A. Then A is a
intuitionsitic fuzzy B-algebra of X if and only if R is a
intuitionsitic fuzzy B-algebra of X × X.
≥ [ µ A ( x1 ) ∧ µ B (0)] ∧ [ µ A ( y1 ) ∧ µ B (0)]
= µ A ( x1 ) ∧ µ A ( y1 ))
Therefore :A is a fuzzy B-algebra of X.
(viii) is by similar method to part (vii) and the proof
is completed.
Proof: Assume that A is a intuitionsitic fuzzy B-algebra
of X. Then:
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µ R (( x1 , x2 ) * ( y1 , y2 )) = µ R ( x1 * y1, x2 * y2 )
Res. J. Appl. Sci. Eng. Technol., 4(21): 4200-4205, 2012
= µ A ( x1 * y1 ) ∧ µ A ( x 2 * y 2 )
≥ [ µ A ( x1 ) ∧ µ A ( y1 )] ∧ [ µ A ( x 2 ) ∧ µ A ( y 2 )]
= [ µ A ( x1 ) ∧ µ A ( x 2 )] ∧ [ µ A ( y1 ) ∧ µ A ( y 2 )]
= µ R ( x1 , x 2 ) ∧ µ R ( y1 , y 2 )
for all (x1, x2), (y1, y2) , X × X. Similarly, we have that <R
(x1, x2),*(y1, y2)) # <R (x1, x2) w<R (y1, y2) for all (x1, x2), (y1,
y2) , X × X. Hence R is a intuitionsitic fuzzy B-algebra of
X × X.
Conversely, suppose that R is a intuitionsitic fuzzy Balgebra of X × X and let x, y , X, Then,
:A (x*y) = :R (x*y, x*y)
= :R((x, x)*, (y, y))
$ :R(x, x)* v = :R(
∧
y, y))
=:R(x) v:R(y)
Similarly, we have <A(x*y) # <A(x) w<A(y). And
completes the proof.
REFERENCES
Atanassov, K., 1986. Intuitionistic fuzzy sets. Fuzzy Sets
Syst., 20(1): 87-96.
Hu, Q.O. and X. Li, 1983. On BCH-algebras. Math. Sem.
Notes, 11: 13-320.
Hu, Q.O. and X. Li, 1985. On proper BCH-algebras.
Math. Japan., 30: 659-661.
Imai, Y. and K. Iseki, 1966. On axiom systems of
propositional calculi XIV. Proc. Japan Acad.,
41: 19-22.
Iseki, K., 1966. An algebra related with a propositional
calculus XIV. Proc. Japan Acad., 42: 26-29.
Jun, Y.B., 1993. Characterization of fuzzy ideals by their
level ideals in BCK (BCI)-algebras. Math. Japan., 38:
67-71.
Jun, Y.B. and E.H. Roh, 1994. Fuzzy commutative ideals
of BCK-algebras. Fuzzy Sets Syst., 64: 401-405.
Li, X.P. and G.J. Wang, 2000. Intuitionistic fuzzy group
and its homomorphisic image. Fuzzy Syst. Math.,
14(1): 45-50.
Li, X.P. and G.J. Wang, 2002. The extension operations
of the intuitionistic fuzzy sets. Fuzzy Syst. Math.,
16(1): 40-46.
Lei, Y.J., B.S. Wang and Q.G. Miao, 2005. On the
intuitionsitic fuzzy relations with compositional
operations. Syst. Eng. Theor. Pract., 2: 113-133.
Neggers, J. and H.S. Kim, 2001. On B-algebras. Int. J.
Math. Sci., 27: 749-757.
Meng, J., 1994. Fuzzy ideals of BCI-algebras. SEA Bull.
Math., 18: 65-68.
Xi, O.G., 1991. Fuzzy BCK-algebras. Math. Japan., 36:
935-942.
Young, B.J., H.R. Eun and J. Chin, 2002. On fuzzy Balgebras. Czechoslovak Math. J., 52(127): 375-384.
Zadeh, L.A., 1966. Fuzzy sets. Inform. Contr., 8:
338-352.
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