International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 7 On Fuzzy b-Subimplicative Ideal Suad Abdulaali Neamah Department of Mathematics, Faculty of Education for Girls University of Kufa, Iraq E-mail: suada.shabib@uokufa.edu.iq or soshabib@yahoo.com Abstract: In this paper, we study a new notion of fuzzy subimplicative ideal of a BH-algebra, namely fuzzy subimplicative ideal with respect to an element in BH-algebra is introduced and some related properties are investigated. Keywords: BH-algebra, fuzzy subimplicative ideal, subimplicative ideal, fuzzy bsubimplicative ideal, Level subset, 1. PRELIMINARIES : In this section, issdevoted to some basic ordinarycconcepts of BH-algebra, fuzzy ideal, sub-implicative ideal in fuzzy and ordinary, level subset, image and preimage of fuzzy set and homomorphism in BH-algebra, we give some basic concepts about the image of function, the inverse image of a BH-algebra with some remarks in fuzzy sences. Definition (1.1): [ 9] A BH-algebra is a nonempty set X with a constant 0 and a binary operation * satisfying the following conditions: i. x * x = 0,  x X. ii. x * y = 0 and y * x = 0 imply x = y,  x, y X. iii. x *0 = x,  x X. Remark (1.2): [10] Let X and Y be BH-algebras. Aamapping f: XY is called ahhomomorphism if f(x*y) = f(x)*f(y), ∀ x, yX. A homomorphism f is called a monomorphism (resp., epimorphism) if it is injective (resp., surjective). A bijective homomorphism is called an isomorphism. Two BH-algebras X and Y are said to be isomorphic, written X  Y, if there exists an isomorphism f: XY. For any homomorphism f: X Y, the set {xX:f(x)=0'} is called the kernel of f, denoted by ker(f), and the set {f(x):xX} is called the image of f, denoted by Im(f). Notice that f(0)=0', ∀ homomorphism f. Definition (1.3): [1] if {A ,   } is a family of fuzzy sets in X, then :  I i A i (x)  inf { Ai (x), i I}, ∀ x  X.  Ii A  i (x)  sup { Ai (x), i I}, ∀ x  X. which are also fuzzy sets in X. Definition (1.4): [2] Let X and Y be any two sets, A be any fuzzy set in X and f: X→Y be any function. The set (y) = {x ∈ X | f(x) = y}, ∀ y ∈ Y. The fuzzy set B in Y defined by B(y) = ∈ , ∀ y ∈Y, is called the image of A under f and is denoted by f(A). International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 8 Definition (1.5): [2] Let X and Y be any two sets, f: X → Y be any function and B be any fuzzy set in f(A). The fuzzy set A in X defined by: A(x)=B(f(x)), ∀x∈X is called the preimage of B under f and is denoted by (B). Definition(1.6): [6] A fuzzy subset A of a BH-algebra X is said to be a fuzzy ideal if and only if: i. A(0) ≥ A(x), ∀ xX. ii. A(x) ≥ min{A(x*y), A(y)}, ∀ x, yX. Definition(1.7):[5] A nonempty subset I of a BH-algebra X is called subimplicative ideal of X if: i. 0I. ii. ((x*(x*y))*(y*x))*z  I and z  I imply y*(y*x) I, ∀x, y, z  X. Definition (1.8): [5] Let X be a BH-algebra and bX, a fuzzy subset A of X is called a sub-implicative ideal with respect to an element b (or briefly, b-subimplicative ideal) of X if it satisfies: i. 0I. ii. (((x*(x*y))*(y*x))*z)*b  I and z  I imply y*(y*x) I, ∀x, y, z  X. Definition(1.9): [5] A fuzzy set A of a BH-algebra X is called a fuzzy subimplicative ideal of X if it satisfies: i. A(0) ≥ A(x) , ∀ xX. ii. A(y*(y*x))  min {A(((x *(x*y))*(y*x)) *z), A(z)}, ∀ x, y, z  X. Proposition(1.10):[5] Let X be a BH-algebra. Then every fuzzy sub-implicative ideal of X is fuzzy ideal of X. Definition(1.11): [7] Let μ be a fuzzy set in X, ∀  [0, 1], the set  { x  X, μ(x)  } is called a level subset of A. Note that, is a subset of X in the ordinary sense. Remark(1.12):[suad] Let A be a fuzzy subset of a BH-algebra X and w ∈ X. The set { x∈X | A(w) A(x)} is denoted by ↑A(w). 2. 3. THE FUZZY SUBIMPLICATIVE IDEAL WITH RESPECT TO AN ELEMENT OF A BH-ALGEBRA. We define the concept of a fuzzy sub-implicative ideal with respect to an element of a BH-algebra. We discuss some properties of this concept and link it with other types of fuzzy ideal of a BH-algebra. Definition (2.1): Let X be a BH-algebra and bX, a fuzzy subset A of X is called a fuzzy sub-implicative ideal with respect to an element b (or briefly, fuzzy b-subimplicative ideal) of X if it satisfies: i. i. A(0) ≥ A(x) , ∀ xX. ii. ii. A(y*(y*x))  min {A((((x *(x*y))*(y*x)) *z)*b), A(z)}, ∀ x, y, z  X. Example(2.2): Consider the BH-algebra X= with the following operation table: International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 9 * 0 1 2 0 0 1 2 1 1 0 2 2 2 2 0 The fuzzy subset A defined by { is a fuzzy 0-subimplicative ideal of X. Theorem (2.3): Let X be a BH-algebra. Then A is a fuzzy subimplicative ideal of X if and only if A is a fuzzy 0-subimplicative ideal of X. Proof : Let A be a fuzzy subimplicative ideal of X. Then i. A(0) ≥ A(x) , ∀ xX. [By definition (1.9)(i)] ii. Let x, y, z X. Then, we have A(y*(y*x)) min{A(((x*(x*y))*(y*x)))*z), A(z)}[By definition(1.5)(ii)] min{A((((x*(x*y))*(y*x)))*z)*0),A(z)}= min{A(((x*(x*y))*(y*x)))*z), A(z)} [Since X is a BH-algebra; x*0=x,  xX.]  A(y*(y*x))  min{A((((x*(x*y))*(y*x))*z)*0),A(z)} Therefore, A is a fuzzy 0-subimplicative ideal of X. Conversely, Let A be a fuzzy 0-subimplicative ideal of X. Then i. A(0) ≥ A(x) , ∀ xX. [By definition (2.1)(i)] ii. Let x, y, z X. Then A(y*(y*x))  min{A((((x*(x*y))*(y*x)))*z)*0),A(z)} [Since A is a fuzzy 0-subimplicative ideal of X. By definition (2.1)(ii)]  min{A(((x*(x*y))*(y*x))*z)*0), A(z)} = min{A(((x*(x*y))*(y*x))*z), A(z)} [Since X is a BH-algebra; x*0=x,  xX]  A(y*(y*x))  min {A(((x*(x*y))*(y*x))*z), A(z)}. Therefore, A is a fuzzy subimplicative ideal of X.■ Proposition (2.4): Let X be a BH-algebra, bX and A be a fuzzy b-subimplicative ideal of X, such that A(b)=A(0). Then A is a fuzzy ideal of X. Proof : Let A be a fuzzy b-subimplicative ideal of X. To prove A is a fuzzy ideal of X. i. A(0) ≥ A(x), ∀ xX. [Since A is a fuzzy b-sub-implicative ideal of X. ] ii. Let x, y , z X such that A(x*b)=A((x*0)*b [Since X is BH-algebra; x*0=x] = A((x*0)*0)*0)*b) [Since X is BH-algebra;x*x=0] International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 10 = A(((x*(x*x))*(x*x))*0)*b) [Since X is BH-algebra;x*x=0]  A(x*(x*x)) ≥ min{A(((x*(x*x))*(x*x))*0)*b), A(b)}=A(x*b) [Since A is a fuzzy b-sub-implicative of X. By definition (2.1)(ii)] Now, A(x*(x*x)) =A(x*0)=A(x) [Since X is BH-algebra ; x*x=0, x*0=x]  A(x) ≥ min{A(((x*(x*x))*(x*x))*0)*b) ,A(0)}=min{A(x*b) , A(0)}  A(x) ≥ min{A(x*b),A(b)}. [Since A(b)=A(0)] Therefore, A is a fuzzy ideal of X.■ Remark(2.5): The following example shows that converse of proposition (2.4) is not correct, bX. Example (2.6): Consider the BH-algebra X= {0, 1, 2, 3} with the binary operation '*' defined by the following table: * 0 1 2 3 0 0 0 0 0 1 1 0 0 2 2 2 2 0 1 3 3 3 3 0 The fuzzy subset A defined by { is a fuzzy ideal of X, but A is not a fuzzy 0-subimplicative ideal of X. Since if x=2, y=1, z=0, then A(1*(1*2))=A(1*0)=A(1)= 0.5 min{A((((2*(2*1))*(1*2))*0)*0) ,A(0)} = min{A((2*2))*(1*2)) ,A(0)} =min{A(0),A(0)}=A(0)=1. And A is not a fuzzy 1-subimplicative ideal of X. Since if x=2, y=1, z=0, then A(1*(1*2))=A(1*0)=A(1)= 0.5 min{A((((2*(2*1))*(1*2))*0)*1) ,A(0)} = min{A(((2*2))*(1*2))*1) ,A(0)} =min{A(0*1),A(0)}= min{A(0),A(0)}=A(0)=1. And A is not a fuzzy 2-subimplicative ideal of X. Since if x=2, y=1, z=0, then A(1*(1*2))=A(1*0)=A(1)= 0.5 min{A((((2*(2*1))*(1*2))*0)*2) ,A(0)} = min{A(((2*2))*(1*2))*2) ,A(0)} =min{A(0*2),A(0)}= min{A(0),A(0)}=A(0)=1. And A is not a fuzzy 3-subimplicative ideal of X. Since if x=2, y=1, z=0, then A(1*(1*2))=A(1*0)=A(1)= 0.5 min{A((((2*(2*1))*(1*2))*0)*3) ,A(0)} International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 11 = min{A(((2*2))*(1*2))*3) ,A(0)} =min{A(0*3),A(0)}= min{A(0),A(0)}=A(0)=1. Therefore, A is not a fuzzy b-subimplicative ideal. bX. Proposition (2.7): Let X be a BH-algebra. A be a fuzzy subimplicative ideal of X, bX such that A(b)=A(0). Then A is a fuzzy bsubimplicative ideal of X. Proof: Let A be a fuzzy subimplicative ideal of X. Then i. A(0)  A(x),  xX [By definition (1.9)(i)] ii. Let x, y, z X. Then A(y*(y*x))  min{A((((x*(x*y))*(y*x))*z),A(z)}  min{min{A((((x*(x*y))*(y*x))*z)*b),A(b)},A(z)} = min{A((((x*(x*y))*(y*x))*z)*b),A(z)} [Since A is a fuzzy ideal of X. By proposition (1.10) and A(b)=A(0)] Therefore, A is fuzzy b-subimplicative ideal of X.■ Theorem (2.8): Let X be a BH-algebra. Then a fuzzy ideal A of X satisfying the condition:  x, y ∈ X ; A(y*(y*x))  A((x *(x*y))*(y*x)) (b1) is a fuzzy b-subimplicative ideal of X, where bX and A(b)=A(0). Proof: Let A be a fuzzy ideal of X. Then, we have i. A(0)  A(x),  xX. [By definition (1.6)(i)] ii. Let x, y, zX. Then, we have A((x*(x*y))*(y*x)) ≥ min{A((x*(x*y))*(y*x))*z),A(z)} [Since A is a fuzzy ideal of X. By definition (1.6)(ii)] ≥ min{min{A((((x*(x*y))*(y*x))*z)*b),A(b)},A(z)}[Since A is a fuzzy ideal of X.] = min {A((((x*(x*y))*(y*x))*z)*b),A(z)} [Since A((((x*(x*y))*(y*x))*z)*b)=min{A((((x*(x*y))*(y*x))*z)*b),A(b)} and A(b)=A(0)]  A(y*(y*x)) ≥ min{A((((x *(x*y))*(y*x)) *z)*b), A(z)} [By the condition (b1)] Then A is a fuzzy b-subimplicative ideal of X, A(b)=A(0).■ Theorem (2.9): If X is a BH-algebra of X satisfies the condition: ∀ x, yX ; y*(y*x) = (x *(x*y))*(y *x) (b2), then every fuzzy ideal of X is a fuzzy b-subimplicative ideal of X, where bX and A(b)=A(0). Proof: Let A be a fuzzy ideal of X. Then, we have i. A(0)  A(x),  xX. [By definition (1.6)(i)] ii. Let x, y, zX. Then A((x*(x*y))*(y*x)) ≥ min {A((x*(x*y))*(y*x))*z), A(z)} [Since A is a fuzzy ideal of X. By definition (1.6)(ii)] International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 12 Now, A(y*(y*x))=A((x*(x*y))*(y*x)) [ By (b2)]  A(y*(y*x)) ≥ min {A(((x *(x*y))*(y*x)) *z), A(z)}  A is a fuzzy subimplicative ideal of X. [By definition (1.9)] Therefore, A is a fuzzy b-subimplicative ideal of X. [By proposition (2.7)].■ Theorem (2.10) : Let X be a BH-algebra, b ∈ X and A be a fuzzy b-subimplicative ideal of X. Then the set XA is a bsubimplicative ideal of X. Proof: Let A be a fuzzy b-subimp1licative ideal of X. To prove XA is a b-subimplicative ideal of X. i. A(x) = A(0). If x=0, then 0 ∈ XA ii. Let x, y, z, b ∈ X such that (((x*(x*y))*(y* x))* z)*b ∈ XA and z ∈XA  A((((x*(x*y))*(y* x))* z)*b)=A(0) and A(z)= A(0)  by definition of fuzzy b-subimplicative ideal of X , we have A(y*(y*x)) ≥ min {A ((((x*(x*y))*(y* x))* z)*b), A(z)} = min {A(0) ,A(0)}=A(0) A(y*(y*x)) ≥ A(0). But A(0) ≥ A(x). [Since A is a fuzzy b-subimplicative ideal of X.]  A(y*(y*x))= A(0)  y*(y*x) ∈ XA. Therefore, XA is a b-subimplicative ideal of X. Proposition (2.11): Let {Aα |αλ} be a family of fuzzy b-subimplicative ideals of a BH-algebra X. Then   A α is a fuzzy bsubimplicative ideal of X. bX Proof: Let {Aα |αλ} be a family of fuzzy b-subimplicative ideals of X. i. Let xX. Then   A (0) = inf { Aα (0) |αλ }  inf { Aα(x) |αλ } =  A α(x) [Since Aα is a fuzzy b-subimplicative ideals of X, αλ. By definition (2.1)(i)]    A α (0)    A α (x) ii.Let x, y, z X. Then, we have   A α(y*(y*x)) = inf {Aα(y*(y*x))|αλ}  inf {min{Aα ((((x*(x*y))*(y*x))*z)*b),Aα (z) | αλ}} [Since Aα is a fuzzy b-subimplicative ideals of X, αλ. By definition (2.1)(ii)]] International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 13 = min {inf {A((((x *(x*y))*(y*x))*z)*b),Aα (z) | αλ}} = min {inf {Aα ((((x *(x*y))*(y*x))*z)*b) | αλ} , inf {Aα(z) | αλ}} = min {  A α((((x *(x*y))*(y*x)) *z)*b) ,   A α (z)}   A α(y*(y*x))  min{{  A α((((x*(x*y))*(y*x))*z)*b),  A α(z)} Therefore,   A α is a fuzzy b-subimplicative ideal of X. bX ■ Proposition (2.12): Let {Aα |αλ} be a chain of fuzzy b-subimplicative ideals of a BH-algebra X. Then   A α is a fuzzy bsubimplicative ideal of X. bX. Proof: Let {Aα |αλ} be a chain of fuzzy b-subimplicative ideal of X. i. Let xX. Then  A α(0)= sup{ Aα(0) |αλ })  sup{ Aα(x) |αλ } =  A α (x) [Since Aα is a fuzzy b-subimplicative ideal of X,  αλ. By definition(2.1)(i)]    A α (0)    A α (x) ii. Let x, y, z X. Then, we have   A α (y*(y*x)) = sup {Aα(y*(y*x)) |αλ}  sup {min{Aα ((((x *(x*y))*(y*x)) *z)*b),Aα(z) | αλ}} [Since Aα is a fuzzy b-subimplicative ideals of X, αλ. By definition (2.1)(ii)] = min{sup{Aα((((x*(x*y))*(y*x))*z)*b), Aα(z)| αλ}} [Since Aα is a chain] = min {sup {Aα ((((x *(x*y))*(y*x)) *z )*b) | αλ}, sup{A α(z) | αλ}} = min {  A α ((((x *(x*y))*(y*x)) *z)*b) ,  i A α(z) }    A α(y*(y*x))  min {{  A ((((x *(x*y))*(y*x)) *z)*b),  A α(z)} International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 14 Therefore,   A α is a fuzzy b-subimplicative ideal of X. bX ■ Proposition(2.13): Let f: (X,*, 0)→(Y,*', 0') be a BH-epimorphism. If is a fuzzy b-subimplicative ideal of X, then f( ) is a fuzzy b-subimplicative ideal of Y. bX Proof: Let A be a fuzzy b-subimplicative ideal of X. Then i. Let y ∈Y. Then there exists x ∈ X. (f(A))(0')=sup {A(x1) | x1 ∈ f -1(0')} =A(0) (x) | x ∈ X} sup {A(x1) | x = f -1(y)}=(f(A))(y) [Since A is a fuzzy b-subimplicative ideal of X. By definition (2.1)(i)]]  (f(A))(0') (f(A))(y), ∀ y ∈Y. iii. Let , ∈ Y. Then there exist f( )= , f( )= , f( )= , f(b)= such that ∈ X (f(A))(y2*'( y2*'y1))=sup{A( *( * ))| ∈ f -1(( y2*'( y2*'y1)))} sup{ (((( *( * ))*( * ))*z)*b),A(z)|((( *( * ))*( * ))*z ∈ f-1(((y1*'(y1*'y2))*'( y2*'y1))*'y3)*'y4)} and z∈ f -1( )}} [Since A is a fuzzy b-subimplicative ideal of X. By definition (2.1)(ii)] min {sup{ (((( *( * ))*( * ))*z)*b)|(( ) ) ∈ f-1(((y1*'(y1*'y2))*'( y2*'y1))*'y3)* )*'y4)} , sup {A(z) | z ∈ f -1( )}} = min {((f(A))((f((( *( * ))*( * ))*z)*b), (f(A))(f(z))} =min {((f(A))(((f( *'( * )))*'(f( * )))*'f(z))*'f(b), (f(A))(f(z))} [Since f is an epimorphism. By remark (1.2)] = min {(f(A))((((y1*'(y1*'y2)) *'(y2*' y1)))*' y3) )*'y4), (f(A))(y3)} f(A))(y2*'(y2*'y1)) min{(f(A))((((y1*'(y1*'y2))*'(y2*'y1)))*'y3) )*'y4),(f(A))(y3)} Therefore, f (A) is a fuzzy b-subimplicative ideal of Y.■ Proposition (2.14): Let X be a BH-algebra and A be a fuzzy subset of X. Then A is a fuzzy b-subimplicative ideal of X if and only if A#(x)=A(x)+1A(0) is a fuzzy b-subimplicative ideal of X, where bX. Proof: Let A be a fuzzy b-subimplicative ideal of X. Then International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 15 i. A# (0)=A(0)+1-A(0)  A#(0)=1. Then A#(0) ≥ A#(x), ∀xX ii. Let x, y, z X and bX. Then A# (y*(y*x))=A(y*(y*x))+1-A(0) ≥ min{A((((x*(x*y))*(y*x))*z)*b),A(z)}+1-A(0) [Since A is a fuzzy b-subimplicative ideal of X. By definition (2.1)(ii)] = min{A((((x*(x*y))*(y*x))*z)*b)+1-A(0) , A(z) +1-A(0)} ≥ min{A# ((((x*(x*y))*(y*x))*z)*b)) , A#(z)}  A#(y*(y*x)) ≥ min{ A# ((((x*(x*y))*(y*x))*z)*b), A# (z)}  A# is a fuzzy b-subimplicative ideal of X. Conversely, Let A# be a fuzzy b-subimplicative ideal of X. i. Let xX. Then we have A(0)=A# (0)-1+A(0) ≥ A# (0)-1+A(0)=A(x) [Since A# be a fuzzy b-subimplicative ideal of X. By definition (2.1)(i)]  A(0) ≥ A(x), ∀xX. ii. Let x, y, z X and bX. Then A(y*(y*x)) =A#(y*(y*x))-1+A(0) ≥ min{A# ((((x*(x*y))*(y*x))*z)*b), A#(z)}-1+A(0) [Since A# is a fuzzy b-subimplicative ideal of X. By definition (2.1)(ii)] = min { A# ((((x*(x*y))*(y*x))*z)*b)-1 +A(0) , A# (z) -1+A(0)} ≥ min { A((((x*(x*y))*(y*x))*z)*b)) , A(z)}  A(y*(y*x)) ≥ min { A ((((x*(x*y))*(y*x))*z)*b), A (z)} Then A is a fuzzy b-subimplicative ideal of X.■ Proposition(2.15) Let X be a BH-algebra and let w, b ∈ X. If A is a fuzzy b-sub-implicative ideal of X, then ↑A(w) is a bsubimplicative ideal of X . Proof: Let A be a fuzzy b-subimplicative ideal of X. Then i. A(0) A(x), ∀ x∈X. [Since A is a fuzzy b-subimplicative ideal of X. By definition (2.1)(i)]  A(0) A((w)  0 ∈↑A(w) ii. Let x, y, z ∈ X such that (((x *(x*y)*(y*x))*z)*b∈ ↑A(w) and z ∈ ↑A(w)  A(w) A((((x *(x*y))*(y*x))*z)*b) and A(w) A(z)  A(w) min{A((((x *(x*y))*(y*x))*z)*b) , A(z)} But A(y*(y*x)) ≥ min {A((x *(x*y))*(y*x))*z)*b) , A(z)} [Since A is a fuzzy b-subimplicative ideal of X .By definition (2.1)(ii)]  A(w) A(y*(y*x))  y*(y*x) ∈↑A(w). International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 7-17 www.ijeais.org 16 Therefore, ↑A(w) is a b-subimplicative ideal of X . ■ Theorem (2.16): Let X be a BH-algebra, A be a fuzzy ideal of X A(b) =A(0). Then A is a fuzzy b-subimplicative ideal of X if and only if Aα is a b-subimplicative ideal of X,  α[0, A(0)]. Proof: Let A be a fuzzy b-subimplicative ideal of X. To prove Aα is a b-subimplicative ideal of X. i. Let x Aα. Then A(x)  α [By definition (1.11) of Aα] But A(0)  A(x). [Since A is a fuzzy b-subimplicative ideal of X. By definition(2.1)(i)]  A(0)  α.  0 Aα. ii. Let x, y, z, b X such that (((x*(x*y))*(y*x))*z)*b Aα and z Aα.  A((((x*(x*y))*(y*x))*z)*b)  α and A(z)  α [By definition(1.11) of Aα ] min { A((((x*(x*y))*(y*x))*z)*b) , A(z) }  α But A(y*(y*x))  min{ A((((x*(x*y))*(y*x))*z)*b) , A(z) } [Since A is a fuzzy b-subimplicative ideal of X. By definition (2.1)(ii)]  A(y*(y*x))  α  y*(y*x)  Aα [By definition (1.11) of Aα] Therefore, Aα is a b-subimplicative ideal of X,  b Aα. Conversely, To prove A is a fuzzy b-subimplicative ideal of X. i. 0 Aα . [By definition (1.8)(i)]. Then A(0)  α =A(x),  xX. ii. Let x, y, z  X such that α = min{ A((((x*(x*y))*(y*x))*z)*b) , A(z)}  A((((x*(x*y))*(y*x))*z)*b)  α and A(z)  α  (((x*(x*y))*(y*x))*z)*b  Aα and z Aα .  y*(y*x) Aα [Since Aα is a b-subimplicative ideal of X. By definition (1.8)(ii)]  A(y*(y*x))  α  A(y*(y*x))  min { A((((x*(x*y))*(y*x))*z)*b) , A(z) } Therefore, A is a fuzzy b-subimplicative ideal of X, A(b) =A(0). ■ References 1. D. Dubois and H. Prade, "Fuzzy Sets and Systems", Academic Press. INC. (London) LTD, fifth Avenue, New Yurk, 1980. 2. E. M.kim and S.S. Ahn," On Fuzzy n-fold Strong Ideals of BH-algebras", J. Appl. Math. & Informatics. 30,(No.3) – 4,pp. 665 676,2012 3. H. H. Abbass and H. A. Dahham,"Some Types of Fuzzy Ideals with Respect to an Element of a BG-algebra", M.S.c thesis, Kufa University, 2012. 4. H. H. Abbass and H. M. A. Saeed, "The Fuzzy Closed BCH-algebra with Respect to an Element of a BH-algebra", M.S.c thesis, Kufa University, 2011. 5. H. H. Abbass and S. A. Neamah. 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