α -ANTI FUZZY NEW IDEAL OF PUALGEBRA by IJFLS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

ANTI FUZZY NEW IDEAL OF PUALGEBRA

Samy M. Mostafa , Mokhtar A. Abdel Naby , Alaa Eldin I. Elkabany 1,2,3 Department of mathematics ,Ain Shams University, Roxy, Cairo

Abstract In this paper, the notion α -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The homomorphic images (pre images) of α -anti fuzzy new-ideal under homomorphism of a PU-algebras has been obtained. Some related result have been derived.

Keywords PU-algebra, α -anti fuzzy new-ideal, the homomorphic images (pre images) of α -anti fuzzy new-ideal . Mathematics Subject Classification: 03F55, 08A72

1. INTRODUCTION In 1966, Imai and Iseki [1, 2,3 ] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCIalgebras. In [4], Hu and Li introduced a wide class of abstract algebras: BCH-algebras. They are shown that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. In [9], Neggers and Kim introduced the notion of d-algebras, which is a generalization of BCK-algebras and investigated a relation between d-algebras and BCK-algebras. Neggers et al.[10] introduced the notion of Q-algebras , which is a generalization of BCH/BCI/BCK-algebras. Megalai and Tamilarasi [6 ] introduced the notion of a TM-algebra which is a generalization of BCK/BCI/BCH-algebras and several results are presented .Mostafa et al.[7] introduced a new algebraic structure called PU-algebra, which is a dual for TM-algebra and investigated severed basic properties. Moreover they derived new view of several ideals on PU-algebra and studied some properties of them .The concept of fuzzy sets was introduced by Zadeh [12]. In 1991, Xi [11] applied the concept of fuzzy sets to BCI, BCK, MV-algebras. Since its inception, the theory of fuzzy sets , ideal theory and its fuzzification has developed in many directions and is finding applications in a wide variety of fields. Jun [5 ] defined a doubt fuzzy sub-algebra , doubt fuzzy ideal in BCI-algebras and got some results about it. Mostafa et al [8] introduce the notion, α fuzzy new-ideal of a PU-algebra . They discussed the homomorphic image ( pre image) of α fuzzy new-ideal of a PU-algebra under homomorprhism of a PU-algebras. In this paper, we modify the ideas of Jun [5 ], to introduce the notion, α -anti fuzzy new-ideal of a PU-algebra . The homomorphic image ( pre image) of α -anti fuzzy new-ideal of a PU-algebra under homomorprhism of a PU-algebras are discussed . Many related results have been derived.

DOI : 10.5121/ijfls.2015.5301

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

2. PRELIMINARIES Now, we will recall some known concepts related to PU-algebra from the literature, which will be helpful in further study of this article

Definition 2.1 [7 ] A PU-algebra is a non-empty set X with a constant 0 ∈ X and a binary operation ∗ satisfying the following conditions: (I) 0 ∗ x = x, (II) (x ∗ z) ∗ (y ∗ z) = y ∗ x for any x, y, z ∈ X. On X we can define a binary relation " ≤ " by: x ≤ y if and only if y ∗ x = 0.

Example 2.2 [7 ] Let X = {0, 1, 2, 3, 4} in which ∗ is defined by ∗ 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 PU-algebra.

Proposition 2.2[7 ] In a PU-algebra (X, ∗, 0) the following hold for all x, y, z ∈ X: (a) x ∗ x = 0. (b) (x ∗ z) ∗ z = x. (c) x ∗ (y ∗ z) = y ∗ (x ∗ z). (d) x ∗ (y ∗ x) = y ∗ 0. (e) (x ∗ y) ∗ 0 = y ∗ x. (f) If x ≤ y, then x ∗ 0 = y ∗ 0. (g) (x ∗ y) ∗ 0 = (x ∗ z) ∗ (y ∗ z). (h) x ∗ y ≤ z if and only if z ∗ y ≤ x. (i) x ≤ y if and only if y ∗ z ≤ x ∗ z. (j) In a PU-algebra (X, ∗, 0) , the following are equivalent: (1) x = y, (2) x ∗ z = y ∗ z, (3) z ∗ x = z ∗ y. (k) The right and the left cancellation laws hold in X. (l) (z ∗ x) ∗ (z ∗ y) = x ∗ y, (m) (x ∗ y) ∗ z = (z ∗ y) ∗ x. (n) (x ∗ y) ∗ (z ∗ u) = (x ∗ z) ∗ (y ∗ u) for all x, y, z and u ∈ X.

Definition 2.4[7] A non-empty subset I of a PU-algebra (X, ∗, 0) is called a sub-algebra of X if x ∗ y ∈ I whenever x, y ∈ I. Definition 2.5[7] A non-empty subset I of a PU-algebra (X, ∗, 0) is called a new-ideal of X if, (i) 0 ∈ I, (ii) (a * (b * x)) * x ∈ I, for all a, b ∈ I and x ∈ X. 2

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Example 2.6[7] Let X = {0, a, b, c} in which ∗ is defined by the following table: ∗

Then (X, ∗, 0) is a PU-algebra. It is easy to show that I1 = {0, a}, I2 = {0, b}, I3 = {0, c} are newideals of X.

Proposition 2.7[ 7 ] Let ( X ,∗,0) and ( X \ ,∗\ ,0 \ ) be PU-algebras and f : X → X \ be a homomorphism, then ker f is a new-ideal of X.

3. α -ANTI FUZZY NEW-IDEAL OF PU-ALGEBRA In this section, we will discuss and investigate a new notion called α -Anti fuzzy new-ideal of a PU-algebra and study several basic properties which related to α -Anti fuzzy new-ideal.

Definition 3.1. [12] Let X be a set, a fuzzy subset µ in X is a function µ : X → [0,1]. Definition 3.2. Let X be a PU-algebra. A fuzzy subset µ in X is called an anti fuzzy sub-algebra of X if µ ( x ∗ y ) ≤ max{µ ( x ), µ ( y )}, for all x, y ∈ X . Definition 3.3. Let µ be a fuzzy subset of a PU-algebra X. Let α ∈ [0,1]. Then the fuzzy set

µ α of X is called the α -anti fuzzy subset of X (w.r.t. fuzzy set µ ) and is defined as µ α ( x ) = max{µ ( x ), α } , for all x ∈ X . Remark 3.4. Clearly, µ 1 = 1 and µ 0 = µ . Lemma 3.5. If µ is an anti fuzzy sub-algebra of a PU-algebra X and α ∈ [0,1], then µ α ( x ∗ y ) ≤ max{µ α ( x ), µ α ( y )}, for all x, y ∈ X . Proof: Let X be a PU-algebra and α ∈ [0,1]. Then by Definition 3.3, we have that

µ α ( x ∗ y ) = max{µ ( x ∗ y ), α } ≤ max{max{µ ( x ), µ ( y )}, α } = max{max{µ ( x ), α }, max{µ ( y ), α }} = max{µ α ( x ), µ α ( y )}, for all x, y ∈ X .

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Definition 3.6. Let X be a PU-algebra. A fuzzy subset µ α in X is called an α -anti fuzzy subalgebra of X if µ α ( x ∗ y ) ≤ max{µ α ( x ), µ α ( y )}, for all x, y ∈ X . It is clear that an α -anti fuzzy sub-algebra of a PU-algebra X is a generalization of an anti fuzzy sub-algebra of X and an anti fuzzy sub-algebra of X is an α -anti fuzzy sub-algebra of X in case of

α = 0. Definition 3.7. Let (X, ∗, 0) be a PU-algebra, a fuzzy subset µ in X is called an anti fuzzy newideal of X if it satisfies the following conditions:

( F1 ) µ (0) ≤ µ ( x ), ( F2 ) µ (( x ∗ ( y ∗ z )) ∗ z ) ≤ max{µ ( x ), µ ( y )}, for all x, y , z ∈ X .

Lemma 3.8. If µ is an anti fuzzy new-ideal of a PU-algebra X and α ∈ [0,1], then µ α (0) ≤ µ α ( x ), ( F1α ) µ α (( x ∗ ( y ∗ z )) ∗ z ) ≤ max{µ α ( x ), µ α ( y )}, for all x, y , z ∈ X . ( F2α ) Proof: Let X be a PU-algebra and α ∈ [0,1]. Then by Definition 3.3.and Definition 3.7., we have that: µ α (0) = max{µ (0), α } ≤ max{µ ( x ), α } = µ α (x ), for all x ∈ X .

µ α (( x ∗ ( y ∗ z )) ∗ z ) = max{µ (( x ∗ ( y ∗ z )) ∗ z ), α } ≤ max{max{µ ( x ), µ ( y )}, α } = max{max{µ ( x ), α }, max{µ ( y ), α }} = max{µ α ( x ), µ α ( y )}, for all x, y , z ∈ X . Definition 3.9. Let (X, ∗, 0) be a PU-algebra, an α -anti fuzzy subset µ α in X is called an α -anti fuzzy new-ideal of X if it satisfies the following conditions:

( F1α ) µ α (0) ≤ µ α ( x ), ( F2α ) µ α (( x ∗ ( y ∗ z )) ∗ z ) ≤ max{µ α ( x ), µ α ( y )}, for all x, y , z ∈ X . It is clear that an α -anti fuzzy new-ideal of a PU-algebra X is a generalization of an anti fuzzy new-ideal of X and an anti fuzzy new-ideal of X is a special case, when α = 0.

Example 3.10. Let X = {0, 1, 2, 3} in which ∗ is defined by the following table: ∗

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Then (X, ∗, 0) is a PU-algebra. Define an α -fuzzy subset µ α : X → [0,1] by

max{α ,0.3} if x ∈ {0,1}  max{α ,0.7} otherwise

µ α ( x) = 

Routine calculation gives that µ α is an α -anti fuzzy new-ideal of X.

Lemma 3.11. Let µ α be an α -anti fuzzy new-ideal of a PU-algebra X. If the inequality x ∗ y ≤ z holds in X, then µ α ( y ) ≤ max{µ α ( x ), µ α ( z )}. Proof: Assume that the inequality x ∗ y ≤ z holds in X, then z ∗ ( x ∗ y ) = 0 and by 0 647 48 ( F2α ) µ α (( z ∗ ( x ∗ y )) ∗ y ) ≤ max{µ α ( x ), µ α ( z )} . Since µ α ( y ) = µ α (0 ∗ y ) , then we have

that µ α ( y ) ≤ max{µ α ( x ), µ α ( z )}.

Corollary 3.12. Let µ be an anti fuzzy new-ideal of a PU-algebra X. If the inequality x ∗ y ≤ z holds in X, then µ ( y ) ≤ max{µ ( x ), µ ( z )}. Lemma 3.13. If µ α is an α -anti fuzzy subset of a PU-algebra X and if x ≤ y , then µ α ( x ) = µ α ( y ). Proof: If x ≤ y , then y ∗ x = 0. Hence by the definition of PU-algebra and its properties we have

µ α ( x ) = max{µ ( x ), α } = max{µ (0 ∗ x ), α }

that

= max{µ (( y ∗ x ) ∗ x ), α } = max{µ ( y ), α } = µ α ( y ).

Corollary 3.14. If µ is an anti fuzzy subset of a PU-algebra X and if x ≤ y , then µ ( x ) = µ ( y ). Definition 3.15. Let µ α be an α -anti fuzzy new-ideal of a PU-algebra X and let x be an element of X. We define ( ∪ µiα )( x ) = sup( µiα ( x )) i∈I . i∈I

Proposition 3.16. The union of any set of α -anti fuzzy new-ideals of a PU-algebra X is also an

α -anti fuzzy new-ideal of X. Proof: Let {µiα }i∈I be a family of α -anti fuzzy new-ideals of a PU-algebra X, then for any

x, y , z ∈ X ,

( ∪ µiα )(0) = sup( µiα (0)) i∈I ≤ sup( µiα ( x )) i∈I = ( ∪ µiα )( x ) i∈I

i∈I

and

( ∪ µiα )(( x ∗ ( y ∗ z )) ∗ z ) = sup( µiα (( x ∗ ( y ∗ z )) ∗ z ) i∈I i∈I

≤ sup(max{µiα ( x ), µiα ( y )}) i∈I

= max{sup(µiα ( x )) i∈I , sup( µiα ( y )) i∈I } = max{( ∪ µ iα )( x ), ( ∪ µ iα )( y )}. This completes the proof. i∈I

i∈I

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Theorem 3.17. Let µ α be an α -anti fuzzy subset of a PU-algebra X. Then µ α is an α -anti fuzzy new-ideal of X if and only if it satisfies:

(∀ε ∈ [0,1]) ( L( µ α ; ε ) ≠ φ ⇒ L( µ α ; ε ) is a new-ideal of X), where L( µ α ; ε ) = {x ∈ X : µ α ( x ) ≤ ε }. Proof: Assume that µ α is an α -anti fuzzy new-ideal of X. Let ε ∈ [0,1] be such that

L( µ α ; ε ) ≠ φ . Let x ∈ L( µ α ; ε ), then µ α ( x ) ≤ ε . Since µ α ( 0) ≤ µ α ( x ) for all x ∈ X , then

µ α (0) ≤ ε . Thus 0 ∈ L( µ α ; ε ). Let x ∈ X and a , b ∈ L( µ α ; ε ), then µ α ( a ) ≤ ε and µ α (b) ≤ ε . It follows by the definition of α -anti fuzzy new-ideal that µ α (( a ∗ (b ∗ x )) ∗ x ) ≤ max{µ α ( a ), µ α (b)} ≤ ε , so that ( a ∗ (b ∗ x )) ∗ x ∈ L( µ α ; ε ). Hence L( µ α ; ε ) is a new-ideal of X. Conversely, suppose that (∀ε ∈ [0,1]) ( L( µ α ; ε ) ≠ φ ⇒ L( µ α ; ε ) is a new-ideal of X), where

L( µ α ; ε ) = {x ∈ X : µ α ( x ) ≤ ε }.

for some then µ α ( 0) > µ α ( x ) x∈ X, α α α α α µ (0) > ε 0 > µ ( x ) by taking ε 0 = ( µ (0) + µ ( x )) 2 . Hence 0 ∉ L( µ ; ε 0 ), which is a If

contradiction.

µ α (( a ∗ (b ∗ c )) ∗ c ) > max{µ α (a ), µ α (b)}. Taking we have and ε 1 = ( µ α (a ∗ (b ∗ c ) ∗ c ) + max{µ α (a ), µ α (b)}) 2 , ε 1 ∈ [0,1] α α α α µ (( a ∗ (b ∗ c )) ∗ c ) > ε 1 > max{µ ( a ), µ (b)}. It follows that a , b ∈ L( µ ; ε 1 ) and ( a ∗ (b ∗ c )) ∗ c ∉ L( µ α ; ε 1 ). This is a contradiction, and therefore µ α is an α -anti fuzzy new-

Let

a, b, c ∈ X be

such

that

ideal of X.

Corollary 3.18. Let µ be a fuzzy subset of a PU-algebra X. Then µ is an anti fuzzy new-ideal of X if and only if it satisfies:

(∀ε ∈ [0,1]) ( L( µ ; ε ) ≠ φ ⇒ L( µ ; ε ) is a new-ideal of X), where L( µ ; ε ) = {x ∈ X : µ ( x ) ≤ ε }.

Definition 3.19. Let f be a mapping from X to Y. If µ α is an α -anti fuzzy subset of X, then the

α -anti fuzzy subset β α of Y defined by  inf µ α ( x ) if f −1 ( y ) ≠ φ f ( µ α )( y ) = β α ( y ) =  x∈ f −1 ( y )  1 otherwise is said to be the image of µ α under f. Similarly if β α is an α -anti fuzzy subset of Y, then the α -anti fuzzy subset µ α = ( β α o f ) of X (i.e. the α -anti fuzzy subset defined by µ α ( x ) = β α ( f ( x )) for all x ∈ X ) is called the preimage of β α under f. 6

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Theorem 3.20. Let ( X ,∗,0) and ( X \ ,∗\ ,0 \ ) be PU-algebras and

f : X → X \ be a

homomorphism. If β α is an α -anti fuzzy new-ideal of X \ and µ α is the pre-image of β α under f, then µ α is an α -anti fuzzy new-ideal of X. Proof: Since µ α is the pre-image of β α under f, then µ α ( x ) = β α ( f ( x )) for all x ∈ X . Let then µ α (0) = β α ( f ( 0)) ≤ β α ( f ( x )) = µ α ( x ). Now

x∈X , α

let x, y , z ∈ X ,

then

µ (( x ∗ ( y ∗ z )) ∗ z ) = β ( f (( x ∗ ( y ∗ z )) ∗ z )) = β α ( f ( x ∗ ( y ∗ z )) ∗\ f ( z )) = β α (( f ( x ) ∗\ f ( y ∗ z )) ∗\ f ( z )) = β α (( f ( x ) ∗\ ( f ( y ) ∗\ f ( z ))) ∗\ f ( z )) The proof is completed. Theorem 3.21. Let ( X ,∗,0) and (Y ,∗\ ,0 \ ) be PU-algebras, f : X → Y be a homomorphism, µ α be an α -anti fuzzy subset of X, β α be the image of µ α under f and µ α ( x ) = β α ( f ( x )) for all x ∈ X . If µ α is an α -anti fuzzy new-ideal of X, then β α is an α -anti fuzzy new-ideal of Y. Proof: Since 0 ∈ f

−1

(0 \ ), then f −1 (0 \ ) ≠ φ . It follows that

β α (0 \ ) = inf µ α (t ) = µ α (0) ≤ µ α (x ), for all x ∈ X . t∈ f −1 ( 0 \ )

Thus β ( 0 \ ) = inf µ α (t ) for all x \ ∈ Y . Hence β α (0 \ ) ≤ β α ( x \ ) for all x \ ∈ Y . −1 \ t∈ f

(x )

For any x \ , y \ , z \ ∈ Y , If f

−1

( x \ ) = φ or f −1 ( y \ ) = φ ,

then β α ( x \ ) = 1 or β α ( y \ ) = 1 ,it follows that max{β α ( x \ ), β α ( y \ )} = 1 and hence

β α (( x \ ∗\ ( y \ ∗\ z \ )) ∗\ z \ ) ≤ max{β α ( x \ ), β α ( y \ )}. If f

−1

( x \ ) ≠ φ and f −1 ( y \ ) ≠ φ , let x0 ∈ f −1 ( x \ ), y 0 ∈ f −1 ( y \ ) be such that

µ α ( x0 ) = inf µ α (t ) and µ α ( y 0 ) = inf µ α (t ) . t∈ f −1 ( x \ )

t∈ f −1 ( y \ )

It follows by given and properties of PU-algebra that

β α (( x \ ∗\ ( y \ ∗\ z \ )) ∗\ z \ ) = β α (( z \ ∗\ ( y \ ∗\ z \ )) ∗\ x \ ) = β α (( y \ ∗\ ( z \ ∗\ z \ )) ∗\ x \ ) = β α (( y \ ∗\ 0 \ ) ∗\ x \ ) = β α (( f ( y 0 ) ∗\ f (0)) ∗\ f ( x0 )) = β α ( f (( y 0 ∗ 0) ∗ x0 )) = µ α (( y0 ∗ 0) ∗ x0 ) = µ α (( y0 ∗ ( z0 ∗ z0 )) ∗ x0 ) = µ α (( z0 ∗ ( y 0 ∗ z0 )) ∗ x0 ) = µ α (( x0 ∗ ( y0 ∗ z0 )) ∗ z0 ) ≤ max{µ α ( x0 ), µ α ( y 0 )} = max{ inf µ α (t ), inf µ α (t )} = t∈ f −1 ( x \ )

t∈ f −1 ( y \ )

max{β ( x ), β ( y )}. Hence β α is an α -anti fuzzy new-ideal of Y.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Corollary 3.22. Let ( X ,∗,0) and (Y ,∗\ ,0 \ ) be PU-algebras, f : X → Y be a homomorphism,

µ be an anti fuzzy subset of X, β be an anti fuzzy subset of Y defined by  inf µ ( x ) if f −1 ( y ) ≠ φ −1 ( y)  1 otherwise

β ( y ) =  x∈ f

and µ ( x ) = β ( f ( x )) for all x ∈ X . If µ is an anti fuzzy new-ideal of X, then β is an anti fuzzy new-ideal of Y.

4. CARTESIAN PRODUCT OF α - ANTI FUZZY NEW-IDEALS OF PU-ALGEBRAS In this section, we introduce the concept of Cartesian product of an α -anti fuzzy new-ideal of a PU-algebra.

Definition 4.1. An α -anti fuzzy relation on any set S is an α -anti fuzzy subset µ α : S × S → [0,1]. Definition 4.2. If µ α is an α -anti fuzzy relation on a set S and β α is an α -anti fuzzy subset of S, then µ α is an α -anti fuzzy relation on β α if µ α ( x, y ) ≥ max{β α ( x ), β α ( y )}, for all x, y ∈ S . Definition 4.3. If β α is an α -anti fuzzy subset of a set S, the strongest α -anti fuzzy relation on S that is an α -anti fuzzy relation on β α is µ βαα given by µ βαα ( x, y ) = max{β α ( x ), β α ( y )} for all x, y ∈ S .

Definition 4.4. We define the binary operation * on the Cartesian product X × X as follows: ( x1 , x 2 ) ∗ ( y1 , y 2 ) = ( x1 ∗ y1 , x2 ∗ y 2 ) for all ( x1 , x 2 ), ( y1 , y 2 ) ∈ X × X . Lemma 4.5. If (X, ∗, 0) is a PU-algebra, then ( X × X ,∗, (0,0)) is a PU-algebra, where ( x1 , x 2 ) ∗ ( y1 , y 2 ) = ( x1 ∗ y1 , x2 ∗ y 2 ) for all ( x1 , x 2 ), ( y1 , y 2 ) ∈ X × X . Proof: Clear.

Theorem 4.6. Let β α be an α -anti fuzzy subset of a PU-algebra X and µ βαα be the strongest α anti fuzzy relation on X, then β α is an α -anti fuzzy new-ideal of X if and only if µ βαα is an α anti fuzzy new-ideal of X × X . Proof: ( ⇒) : Assume that β α is an α -anti fuzzy new-ideal of X, we note from ( F1α ) that: µ βαα (0,0) = max{β α ( 0), β α ( 0)} ≤ max{β α ( x ), β α ( y )} = µ βαα ( x, y ) for all x, y ∈ X . Now, for any ( x1 , x 2 ), ( y1 , y 2 ), ( z1 , z 2 ) ∈ X × X , we have from ( F2α ) :

µ βαα ((( x1 , x 2 ) ∗ (( y1 , y 2 ) ∗ ( z1 , z2 ))) ∗ ( z1 , z 2 )) = 8

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

µ βαα ((( x1 , x2 ) ∗ ( y1 ∗ z1 , y 2 ∗ z 2 )) ∗ ( z1 , z 2 )) = µ βαα (( x1 ∗ ( y1 ∗ z1 ), x 2 ∗ ( y 2 ∗ z 2 )) ∗ ( z1 , z 2 )) = µ βαα (( x1 ∗ ( y1 ∗ z1 )) ∗ z1 , ( x 2 ∗ ( y 2 ∗ z 2 )) ∗ z 2 ) = max{β α (( x1 ∗ ( y1 ∗ z1 )) ∗ z1 ), β α (( x 2 ∗ ( y 2 ∗ z 2 )) ∗ z 2 )} ≤ max{max{β α ( x1 ), β α ( y1 )}, max{β α ( x 2 ), β α ( y 2 )}} = max{max{β α ( x1 ), β α ( x2 )}, max{β α ( y1 ), β α ( y 2 )}} = max{µ βαα ( x1 , x2 ), µ βαα ( y1 , y 2 )}. Hence µ βαα is an α -anti fuzzy new-ideal of X × X .

( ⇐) : For all ( x, x ) ∈ X × X , we have µ βαα (0,0) = max{β α ( 0), β α ( 0)} ≤ µ βαα ( x, x ) . Then β α (0) = max{β α ( 0), β α ( 0)} ≤ max{β α ( x ), β α ( x )} = β α (x ) for all x ∈ X . Now, for all x, y , z ∈ X , we have

β α (( x ∗ ( y ∗ z )) ∗ z ) = max{β α (( x ∗ ( y ∗ z )) ∗ z ), β α (( x ∗ ( y ∗ z )) ∗ z )} = µ βαα (( x ∗ ( y ∗ z )) ∗ z , ( x ∗ ( y ∗ z )) ∗ z ) = µ βαα (( x ∗ ( y ∗ z ), x ∗ ( y ∗ z )) ∗ ( z, z )) = µ βαα ((( x, x ) ∗ (( y ∗ z ), ( y ∗ z ))) ∗ ( z, z )) = µ βαα ((( x, x ) ∗ (( y , y ) ∗ ( z, z ))) ∗ ( z, z )) ≤ max{µ βαα ( x, x ), µ βαα ( y , y )} = max{max{β α ( x ), β α ( x )}, max{β α ( y ), β α ( y )}} = max{β α ( x ), β α ( y )}. Hence β α is an α -anti fuzzy new-ideal of X.

Definition 4.7. Let µ and be the anti fuzzy subsets in X. The Cartesian product µ × : X × X [0,1] is defined by (µ × ) ( x, y) = max {(x),(y)}, for all x, y X.

Definition 4.8. Let µ α and δ α be the α - anti fuzzy subsets in X. The Cartesian product µ α × δ α : X × X → [0,1] is defined by ( µ α × δ α )( x, y ) = max{µ α ( x ), δ α ( y )}, for all x, y ∈ X . Theorem 4.9. If µ α and δ α are α -anti fuzzy new-ideals in a PU-algebra X, then µ α × δ α is an α -anti fuzzy new-ideal in X × X. Proof:

( µ α × δ α )( 0,0) = max{µ α ( 0), δ α (0)} ≤ max{µ α ( x1 ), δ α ( x 2 )} = ( µ α × δ α )( x1 , x 2 ) for all (x1, x2) X × X. Let (x1, x2), (y1, y2), (z1, z2) X × X. Then we have that

( µ α × δ α )((( x1 , x 2 ) ∗ (( y1 , y 2 ) ∗ ( z1 , z 2 ))) ∗ ( z1 , z 2 )) = ( µ α × δ α )((( x1 , x 2 ) ∗ ( y1 ∗ z1 , y 2 ∗ z 2 )) ∗ ( z1 , z 2 )) = ( µ α × δ α )(( x1 ∗ ( y1 ∗ z1 ), x 2 ∗ ( y 2 ∗ z 2 )) ∗ ( z1 , z 2 )) = ( µ α × δ α )(( x1 ∗ ( y1 ∗ z1 )) ∗ z1 , ( x2 ∗ ( y 2 ∗ z 2 )) ∗ z 2 ) = max{µ α ( x1 ∗ ( y1 ∗ z1 )) ∗ z1 ), δ α ( x 2 ∗ ( y 2 ∗ z 2 )) ∗ z 2 )} ≤ 9

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

max{max{µ α ( x1 ), µ α ( y1 )}, max{δ α ( x 2 ), δ α ( y 2 )}} = max{( µ α × δ α )( x1 , x2 ), ( µ α × δ α )( y1 , y 2 )}. Therefore µ α × δ α is an α -anti fuzzy new-ideal in X × X. This completes the proof.

5. CONCLUSIONS In the present paper, we have introduced the concept of α -anti fuzzy new-ideal of PU -algebras and investigated some of their useful properties. We believe that these results are very useful in developing algebraic structures also these definitions and main results can be similarly extended to some other algebraic structure such as PS -algebras , Q- algebras, SU- algebras , ISalgebras, β algebras and semirings . It is our hope that this work would other foundations for further study of the theory of BCI-algebras. In our future study of anti fuzzy structure of PUalgebras, may be the following topics should be considered: (1) To establish the intuitionistic α -anti fuzzy new-ideal in PU-algebras. (4) To get more results in τ~ - cubic α - fuzzy new-ideal of PU -algebras and it’s application.

Acknowledgements The authors would like to express sincere appreciation to the referees for their valuable suggestions and comments helpful in improving this paper.

Algorithms for PU-algebra Input (X: set with 0 element,* : Binary operation) Output ("X is a PU-algebra or not") If X = φ then; Go to (1.) End if If 0 ∉ X then go to (1.); End If Stop: = false i = 1; While i ≤ X and not (Stop) do If 0 * xi ≠ xi, then Stop: = true End if j = 1; While j ≤ X , and not (Stop) do k = 1; While k ≤ X and not (stop) do If (xi * xk)* (xj * xk) ≠ xj * xi, then Stop: = true End if End while End if End while 10

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

If stop then Output ("X is a PU-algebra") Else (1.) Output ("X is not a PU-algebra") End if End.

Algorithms for PU-ideal in PU-algebra Input (X: PU-algebra, I: subset of X) Output ("I is a PU-ideal of X or not") If I = φ then Go to (1.); End if If 0 ∉ I then Go to (1.); End if Stop: = false i = 1; While i ≤ X and not (stop) do j=1 While j ≤ X and not (stop) do k=1 While k ≤ X and not (stop) do If xj * xi ∈ I, and xi * xk ∈ I then If xj* xk ∉ I then Stop: = false End if End while End while End while If stop then Output ("I is a PU-ideal of X") Else (1.) Output ("I is not ("I is a PU-ideal of X") End if End.

Algorithm for anti fuzzy subsets Input ( X : PU-algebra, A : X → [0,1] ); Output (“ A is an anti fuzzy subset of X or not”) Begin Stop: =false; i := 1 ; While i ≤ X and not (Stop) do If ( A( xi ) < 0 ) or ( A( xi ) > 1 ) then Stop: = true; End If 11

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

End While If Stop then Output (“ A is an anti fuzzy subset of X ”) Else Output (“ A is not an anti fuzzy subset of X ”) End If End.

Algorithm for new-ideal in PU-algebra Input ( X : PU-algebra, I : subset of X ); Output (“ I is an new-ideal of X or not”); Begin If I = φ then go to (1.); End If If 0 ∉ I then go to (1.); End If Stop: =false; i := 1 ; While i ≤ X and not (Stop) do

j := 1 While j ≤ X and not (Stop) do

k := 1 While k ≤ X and not (Stop) do If xi , x j ∈ I and x k ∈ X ,then If (x i * (x j * x k )) * x k ∉ I then Stop: = true; End If End If End While End While End While If Stop then Output (“ I is is new-ideal of X ”) Else (1.) Output (“ I is not is new-ideal of X ”) End If End.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

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