Intuitionistic Fuzzy Ideals of Bg_Algebras
World Academy of Science, Engineering and Technology 5 2005
Intuitionistic Fuzzy Ideals of Bg_Algebras A.Zarandi ,A.Borumand Saeid
(i ) 0 ∈ I Abstract—We consider the intuitionistic fuzzification of the
(ii ) x ∗ y ∈ I and y ∈ I imply that x ∈ I
concept of subalgebras and ideals in BG-algebras, and investigate some of their properties.
for all x, y ∈ X .
Keywords—(Intuitionistic) fuzzy subalgebra, (Intuitionistic) fuzzy ideal, upper (respectively, lower) t-level cut, homomorphism.
Example 2.1. Let X = {0,1,2,3} be a set with the following table : TABLE I
I. INTRODUCTION
A
FTER the introduction of the concept of fuzzy sets by Zadeh several researches were conducted on the generalizations of the notion of fuzzy sets. The idea of “ Intuitionistic fuzzy set “ was first published by Atanassov, as a generalization of the notion of the fuzzy set . In this paper, using the Atanssov’s idea, we establish the intuitionistic fuzzification of the concept of subalgebras and ideals in BG-algebras, and investigate some of their properties
* 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0
Then ( X ,*,0 ) is a BG-algebra. By a fuzzy set
µ
in a non-empty set X we mean a
µ : X → [0,1] , and complement of ~ , is the fuzzy set in X given by denoted by µ µ~ ( x) = 1 − µ ( x) for all x, y ∈ X . A fuzzy set µ in a BG-algebra X is called a fuzzy function
II. PERLIMINARIES First we present the fundamental definitions. Definition 2.1. A BG-algebra is a non-empty set X with a constant 0 and a binary operating * satisfying the following axioms:
µ
,
subalgebra of X if
µ ( x ∗ y ) ≥ min{µ ( x), µ ( x)} for all x, y ∈ X . A fuzzy set µ in a BG-algebra X is called a fuzzy ideal of
(i) x ∗ x = 0,
X if : (i ) µ (0) ≥ µ ( x) for all x ∈ X ,
(ii ) x ∗ 0 = x, (iii )( x ∗ y ) ∗ (0 ∗ y ) = x, for all x, y ∈ X .
(ii ) µ ( x) ≥ min{µ ( x ∗ y ), µ ( y )} for all x, y ∈ X .
For brevity we also call X BG-algebra. A binary relation “ ≤ ” on X can be defined by
Definition 2.3. An intuitionistic fuzzy set (briefly IFS) A in a nonempty set X is an object having the form
x ≤ y if and only if x ∗ y = 0 .
A = {( x, α A ( x ), β A ( x)) x ∈ X }, where the functions α A : X → [0,1] and β A : X → [0,1] denoted the degree of membership and the degree of nonmembership, respectively , and
A non-empty set S of a BG-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S .
0 ≤ α A ( x) + β A ( x) ≤ 1 ∀x ∈ X .
Definition 2.2. A nonempty subset I of a BG-algebra X is called an ideal of X if
An intuitionistic fuzzy set
A = {( x, α A ( x), β A ( x)) x ∈ X }
In
X can be identified to an ordered pair (α A , β A ) in
I X × I X .For the sake of simplicity, we shall use the symbol A = (α A , β A ) for the IFS
187
World Academy of Science, Engineering and Technology 5 2005
α A (0) = α A (2) = (1), α A (1) = α A (3) = t ,
A = {( x, α A ( x), β A ( x)) x ∈ X } . and
III. INTUITIONSTIC FUZZY IDEALS
β A (0) = β A (2) = 0, β A (1) = β A (3) = s,
In what follows, let X is denoted a BG-algebra unless otherwise specified.
t , s ∈ [0,1] and t + s ≤ 1 By calculation we know A = (α A , β A ) is an intuitionstic fuzzy ideal of X . Where
Definition 3.1. An IFS A = (α A , β A ) in X is called an intuitionstic fuzzy subalgebra of X if it satisfies:
Lemma 3.1. Let A = (α A , β A ) in
( IS1) α A ( x ∗ y ) ≥ min{α A ( x), α A ( y )},
X be an intuitionstic
fuzzy ideal of X . If x ∗ y ≤ z then :
( IS 2) β A ( x ∗ y ) ≤ max{β A ( x), β A ( y )}, for all x, y ∈ X .
α A ( x) ≥ min{α A ( y ), α A ( z )}, β A ( x) ≤ max{β A ( x), β A ( z )}.
Example 3.1. Consider BG-algebra X = {0,1,2,3} with table 1.1. in Example 2.1. Let A = (α A , β A ) be an IFS in
x, y, z ∈ X such that x ∗ y ≤ z .Then ( x ∗ y ) ∗ z = 0 ,and thus α A ( x) ≥ min{α A ( x ∗ y ), α A ( y )} Proof. Let
X defined by :
α A (0) = α A (1) = α A (3) = 0.7 > 0.3 = α A (2), β A (0) = β A (1) = β A (3) = 0.2 > 0.5 = β A (2). Then A = (α A , β A ) is an intuitionstic fuzzy subalgebra
≥ min{min{α A (( x ∗ y ) ∗ z ), α A ( z )}, α A ( y )} = min{min{α A (0), α A ( z )}, α A ( y )} = min{α A ( y ), α A ( z )}.
of X . Proposition 3.1. For every intuitionstic fuzzy subalgebra A = (α A , β A ) in X we have the following properties:
and similarity
β A ( x) ≤ max{β A ( x), β A ( z )}.
Lemma 3.2. Let A = (α A , β A ) in X be an
(i ) α A (0) ≥ α A ( x),
intuitionstic fuzzy ideal of
(ii ) β A (0) ≤ β A ( x) for all x ∈ X .
■
X . If x ≤ y then :
α A ( x) ≥ α A ( y ), β A ( x) ≤ β A ( y ). that is , α A is order –reserving and β A is order-preserving. ■ Proof. Let x, y, z ∈ X be such that x ∗ y = 0 and so α A ( x) ≥ min{α A ( x ∗ y ), α A ( y )} = min{α A (0), α A ( y )} = α A ( y ),
Proof. By definition 3.1. is clear. ■ Definition 3.2. An IFS A = (α A , β A ) in X is called an intuitionstic fuzzy ideal of X if it satisfies:
( IF1) α A (0) ≥ α A ( x) and β A (0) ≤ β A ( x) , ( IF 2) α A ( x) ≥ min{α A ( x ∗ y ), α A ( y )},
and
( IF 3) β A ( x) ≤ max{β A ( x ∗ y ), β A ( y )}, for all x, y ∈ X .
X = {0,1,2,3} with table 1.1. in Example 2.1. Let A = (α A , β A ) be an IFS in X as follows: Example 3.2. Consider BG-algebra
β A ( x) ≤ max{β A ( x ∗ y ), β A ( y )} = max{β A (0), β A ( y )} = β A ( y ). ■
f : X → Y of BG-algebras is called a homomorphism if f ( x ∗ y ) = f ( x ) ∗ f ( y ) for all x, y ∈ X .Note that if f : X → Y is a homomorphism of BG-algebras, then f (0) = 0 . Definition 3.3. A mapping
188
World Academy of Science, Engineering and Technology 5 2005
α A ( x ) = α A ( f (a )) = α Af ( f (a ))
f : X → Y be a homomorphism of BG-algebras for any IFS A = (α A , β A ) in Y , we define a new IFS
Let
≥ min{α Af (a ∗ b), α A ( f (b))} = min {α A ( f (a ∗ b)), α A f (b))}
A f = (α Af , β Af ) in X by :
α Af ( x) := α A ( f ( x)),
= min {α A ( f (a ) ∗ f (b)), α A f (b))}
β Af ( x) := β A ( f ( x)) ∀x ∈ X .
= min {α A ( x ∗ y ), α A ( y )},
f : X → Y be a homomorphism of BG-algebras . If an IFS A = (α A , β A ) , is an intuitionstic
and
Theorem 3.1. Let
fuzzy ideal of Y , then an IFS A an intuitionstic fuzzy ideal of
f
β A ( x) = β A ( f (a)) = β Af ( f (a ))
= (α Af , β Af ) in X is
≤ max{β Af (a ∗ b), β A ( f (b))} = max {β A ( f (a ∗ b)), β A f (b))}
X .
= max {β A ( f (a ) ∗ f (b)), β A f (b))}
Proof . We first have that
= max {β A ( x ∗ y ), β A ( y )}.
α Af ( x) = α A ( f ( x)) ≤ α A (0) = α A ( f (0)) = α Af (0),
■
β ( x) = β A ( f ( x)) ≤ β A (0) = β A ( f (0)) = β (0). f A
f A
REFERENCES
For all x ∈ X . Let x, y ∈ X . Then
[1] A.Zarandi, A.Borumand, On Fuzzy Topological BG[2] C.B.Kim, H.S. Kim, On BG-algebras (submitted).
min{α Af ( x * y ), α Af ( y )}
[3] K.T.Atanassov, intuitionstic fuzzy sets, fuzzy sets and systems (1986), no.1 87-96. [4] L.A.Zadeh,Fuzzy sets , information and control 8 (1965) 338-353. [5] S.S.Ahn and h.D.Lee, Fuzzy Subalgebras of BG-algebras, Commun. Korean Math Soc.19 (2004) 243-251.
= min{α A ( f ( x * y )), α A ( f ( y ))} = min {α A ( f ( x) * f ( y )), α A ( f ( y ))} ≤ α A ( f ( x)) = α Af ( x), and
max{β Af ( x * y ), β Af ( y )} = max{β A ( f ( x * y )), β A ( f ( y ))} = max {β A ( f ( x) * f ( y )), β A ( f ( y ))} ≤ β A ( f ( x )) = β Af ( x). Hence
A f = (α A , β A ) is an intuitionstic fuzzy ideal f
f
of X . Theorem 3.2. Left
f : X → Y be an epimorphism of
Algebras 1-5 .
■
BG-algebra and let A = (α A , β A ) be an IFS in Y . if
A = (α A , β A ) is an intuitionstic fuzzy ideal of Y . Proof . For any x ∈ X there exist a ∈ X such that f (a) = x . Then
α A ( x) = α A ( f ( a)) = α Af ( a) ≤ α Af (0) = α A ( f (0)) = α A (0), β A ( x) = β A ( f (a)) = β Af (a) ≥ β Af (0) = β A ( f (0)) = β A (0). Let x, y ∈ X . Then f ( a ) = x and f (b) = y for some
a, b ∈ X . Thus
189
Intuitionistic Fuzzy Ideals of Bg_Algebras A.Zarandi ,A.Borumand Saeid
(i ) 0 ∈ I Abstract—We consider the intuitionistic fuzzification of the
(ii ) x ∗ y ∈ I and y ∈ I imply that x ∈ I
concept of subalgebras and ideals in BG-algebras, and investigate some of their properties.
for all x, y ∈ X .
Keywords—(Intuitionistic) fuzzy subalgebra, (Intuitionistic) fuzzy ideal, upper (respectively, lower) t-level cut, homomorphism.
Example 2.1. Let X = {0,1,2,3} be a set with the following table : TABLE I
I. INTRODUCTION
A
FTER the introduction of the concept of fuzzy sets by Zadeh several researches were conducted on the generalizations of the notion of fuzzy sets. The idea of “ Intuitionistic fuzzy set “ was first published by Atanassov, as a generalization of the notion of the fuzzy set . In this paper, using the Atanssov’s idea, we establish the intuitionistic fuzzification of the concept of subalgebras and ideals in BG-algebras, and investigate some of their properties
* 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0
Then ( X ,*,0 ) is a BG-algebra. By a fuzzy set
µ
in a non-empty set X we mean a
µ : X → [0,1] , and complement of ~ , is the fuzzy set in X given by denoted by µ µ~ ( x) = 1 − µ ( x) for all x, y ∈ X . A fuzzy set µ in a BG-algebra X is called a fuzzy function
II. PERLIMINARIES First we present the fundamental definitions. Definition 2.1. A BG-algebra is a non-empty set X with a constant 0 and a binary operating * satisfying the following axioms:
µ
,
subalgebra of X if
µ ( x ∗ y ) ≥ min{µ ( x), µ ( x)} for all x, y ∈ X . A fuzzy set µ in a BG-algebra X is called a fuzzy ideal of
(i) x ∗ x = 0,
X if : (i ) µ (0) ≥ µ ( x) for all x ∈ X ,
(ii ) x ∗ 0 = x, (iii )( x ∗ y ) ∗ (0 ∗ y ) = x, for all x, y ∈ X .
(ii ) µ ( x) ≥ min{µ ( x ∗ y ), µ ( y )} for all x, y ∈ X .
For brevity we also call X BG-algebra. A binary relation “ ≤ ” on X can be defined by
Definition 2.3. An intuitionistic fuzzy set (briefly IFS) A in a nonempty set X is an object having the form
x ≤ y if and only if x ∗ y = 0 .
A = {( x, α A ( x ), β A ( x)) x ∈ X }, where the functions α A : X → [0,1] and β A : X → [0,1] denoted the degree of membership and the degree of nonmembership, respectively , and
A non-empty set S of a BG-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S .
0 ≤ α A ( x) + β A ( x) ≤ 1 ∀x ∈ X .
Definition 2.2. A nonempty subset I of a BG-algebra X is called an ideal of X if
An intuitionistic fuzzy set
A = {( x, α A ( x), β A ( x)) x ∈ X }
In
X can be identified to an ordered pair (α A , β A ) in
I X × I X .For the sake of simplicity, we shall use the symbol A = (α A , β A ) for the IFS
187
World Academy of Science, Engineering and Technology 5 2005
α A (0) = α A (2) = (1), α A (1) = α A (3) = t ,
A = {( x, α A ( x), β A ( x)) x ∈ X } . and
III. INTUITIONSTIC FUZZY IDEALS
β A (0) = β A (2) = 0, β A (1) = β A (3) = s,
In what follows, let X is denoted a BG-algebra unless otherwise specified.
t , s ∈ [0,1] and t + s ≤ 1 By calculation we know A = (α A , β A ) is an intuitionstic fuzzy ideal of X . Where
Definition 3.1. An IFS A = (α A , β A ) in X is called an intuitionstic fuzzy subalgebra of X if it satisfies:
Lemma 3.1. Let A = (α A , β A ) in
( IS1) α A ( x ∗ y ) ≥ min{α A ( x), α A ( y )},
X be an intuitionstic
fuzzy ideal of X . If x ∗ y ≤ z then :
( IS 2) β A ( x ∗ y ) ≤ max{β A ( x), β A ( y )}, for all x, y ∈ X .
α A ( x) ≥ min{α A ( y ), α A ( z )}, β A ( x) ≤ max{β A ( x), β A ( z )}.
Example 3.1. Consider BG-algebra X = {0,1,2,3} with table 1.1. in Example 2.1. Let A = (α A , β A ) be an IFS in
x, y, z ∈ X such that x ∗ y ≤ z .Then ( x ∗ y ) ∗ z = 0 ,and thus α A ( x) ≥ min{α A ( x ∗ y ), α A ( y )} Proof. Let
X defined by :
α A (0) = α A (1) = α A (3) = 0.7 > 0.3 = α A (2), β A (0) = β A (1) = β A (3) = 0.2 > 0.5 = β A (2). Then A = (α A , β A ) is an intuitionstic fuzzy subalgebra
≥ min{min{α A (( x ∗ y ) ∗ z ), α A ( z )}, α A ( y )} = min{min{α A (0), α A ( z )}, α A ( y )} = min{α A ( y ), α A ( z )}.
of X . Proposition 3.1. For every intuitionstic fuzzy subalgebra A = (α A , β A ) in X we have the following properties:
and similarity
β A ( x) ≤ max{β A ( x), β A ( z )}.
Lemma 3.2. Let A = (α A , β A ) in X be an
(i ) α A (0) ≥ α A ( x),
intuitionstic fuzzy ideal of
(ii ) β A (0) ≤ β A ( x) for all x ∈ X .
■
X . If x ≤ y then :
α A ( x) ≥ α A ( y ), β A ( x) ≤ β A ( y ). that is , α A is order –reserving and β A is order-preserving. ■ Proof. Let x, y, z ∈ X be such that x ∗ y = 0 and so α A ( x) ≥ min{α A ( x ∗ y ), α A ( y )} = min{α A (0), α A ( y )} = α A ( y ),
Proof. By definition 3.1. is clear. ■ Definition 3.2. An IFS A = (α A , β A ) in X is called an intuitionstic fuzzy ideal of X if it satisfies:
( IF1) α A (0) ≥ α A ( x) and β A (0) ≤ β A ( x) , ( IF 2) α A ( x) ≥ min{α A ( x ∗ y ), α A ( y )},
and
( IF 3) β A ( x) ≤ max{β A ( x ∗ y ), β A ( y )}, for all x, y ∈ X .
X = {0,1,2,3} with table 1.1. in Example 2.1. Let A = (α A , β A ) be an IFS in X as follows: Example 3.2. Consider BG-algebra
β A ( x) ≤ max{β A ( x ∗ y ), β A ( y )} = max{β A (0), β A ( y )} = β A ( y ). ■
f : X → Y of BG-algebras is called a homomorphism if f ( x ∗ y ) = f ( x ) ∗ f ( y ) for all x, y ∈ X .Note that if f : X → Y is a homomorphism of BG-algebras, then f (0) = 0 . Definition 3.3. A mapping
188
World Academy of Science, Engineering and Technology 5 2005
α A ( x ) = α A ( f (a )) = α Af ( f (a ))
f : X → Y be a homomorphism of BG-algebras for any IFS A = (α A , β A ) in Y , we define a new IFS
Let
≥ min{α Af (a ∗ b), α A ( f (b))} = min {α A ( f (a ∗ b)), α A f (b))}
A f = (α Af , β Af ) in X by :
α Af ( x) := α A ( f ( x)),
= min {α A ( f (a ) ∗ f (b)), α A f (b))}
β Af ( x) := β A ( f ( x)) ∀x ∈ X .
= min {α A ( x ∗ y ), α A ( y )},
f : X → Y be a homomorphism of BG-algebras . If an IFS A = (α A , β A ) , is an intuitionstic
and
Theorem 3.1. Let
fuzzy ideal of Y , then an IFS A an intuitionstic fuzzy ideal of
f
β A ( x) = β A ( f (a)) = β Af ( f (a ))
= (α Af , β Af ) in X is
≤ max{β Af (a ∗ b), β A ( f (b))} = max {β A ( f (a ∗ b)), β A f (b))}
X .
= max {β A ( f (a ) ∗ f (b)), β A f (b))}
Proof . We first have that
= max {β A ( x ∗ y ), β A ( y )}.
α Af ( x) = α A ( f ( x)) ≤ α A (0) = α A ( f (0)) = α Af (0),
■
β ( x) = β A ( f ( x)) ≤ β A (0) = β A ( f (0)) = β (0). f A
f A
REFERENCES
For all x ∈ X . Let x, y ∈ X . Then
[1] A.Zarandi, A.Borumand, On Fuzzy Topological BG[2] C.B.Kim, H.S. Kim, On BG-algebras (submitted).
min{α Af ( x * y ), α Af ( y )}
[3] K.T.Atanassov, intuitionstic fuzzy sets, fuzzy sets and systems (1986), no.1 87-96. [4] L.A.Zadeh,Fuzzy sets , information and control 8 (1965) 338-353. [5] S.S.Ahn and h.D.Lee, Fuzzy Subalgebras of BG-algebras, Commun. Korean Math Soc.19 (2004) 243-251.
= min{α A ( f ( x * y )), α A ( f ( y ))} = min {α A ( f ( x) * f ( y )), α A ( f ( y ))} ≤ α A ( f ( x)) = α Af ( x), and
max{β Af ( x * y ), β Af ( y )} = max{β A ( f ( x * y )), β A ( f ( y ))} = max {β A ( f ( x) * f ( y )), β A ( f ( y ))} ≤ β A ( f ( x )) = β Af ( x). Hence
A f = (α A , β A ) is an intuitionstic fuzzy ideal f
f
of X . Theorem 3.2. Left
f : X → Y be an epimorphism of
Algebras 1-5 .
■
BG-algebra and let A = (α A , β A ) be an IFS in Y . if
A = (α A , β A ) is an intuitionstic fuzzy ideal of Y . Proof . For any x ∈ X there exist a ∈ X such that f (a) = x . Then
α A ( x) = α A ( f ( a)) = α Af ( a) ≤ α Af (0) = α A ( f (0)) = α A (0), β A ( x) = β A ( f (a)) = β Af (a) ≥ β Af (0) = β A ( f (0)) = β A (0). Let x, y ∈ X . Then f ( a ) = x and f (b) = y for some
a, b ∈ X . Thus
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