On fuzzy BF-algebras - m-hikari

International Mathematical Forum, 4, 2009, no. 1, 13 - 25

On Fuzzy BF -Algebras A. Borumand Saeid

1

and M. A. Rezvani

Dept. of Mathematics, Shahid Bahonar University of Kerman Kerman, Iran Abstract In this paper the notion of fuzzy BF -algebra and fuzzy topological BF -algebras are introduced. We stated and proved some theorem in fuzzy BF -algebras and level subalgebras. Finally the Foster’s results on homomorphic images and inverse images in fuzzy topological BF algebras are studied.

Mathematics Subject Classification: 06F35, 03G25, 94D05 Keywords: (fuzzy) BF -algebra, fuzzy BF -subalgebras, level subalgebras, fuzzy topological BF -algebras

1

Introduction

Y. Imai and K. Iseki [4] introduced two classes of abstract algebras: BCKalgebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In [8], J. Neggers and H. S. Kim introduced the notion of B-algebras, which is a generalization of BCK-algebra. In [7], Y. B. Jun , E. H. Roh , and H. S. Kim introduced BH-algebras, which are a generalization of BCK/BCI/B-algebras. Recently, Andrzej Walendziak defined a BF -algebra [10]. The concept of a fuzzy set, which was introduced in [11] by L. A. Zadeh Provides a natural framework for generalizing many of the concepts of general mathematics and topology. D. H. Foster (cf. [1]) combined the structure of a fuzzy topological spaces with that of a fuzzy group, introduced by A. Rosenfeld (cf. [9]), to formulated the elements of a theory of fuzzy topological groups. In the present paper, we introduced the concept of fuzzy BF -subalgebras and fuzzy topological BF -algebras and study this structure. We state and prove some theorem discussed in fuzzy BF -subalgebras and level subalgebras. Finally some of Fosters results on homomorphic images and inverse images in fuzzy topological BF -algebras are studied. 1

[email protected]

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2

A. Borumand Saeid and M. A. Rezvani

Preliminary

Definition 2.1. [10] A BF -algebra is a non-empty set X with a consonant 0 and a binary operation ∗ satisfying the following axioms: (I) x ∗ x = 0, (II) x ∗ 0 = x, (III) 0 ∗ (x ∗ y) = (y ∗ x), for all x, y ∈ X. Example 2.2. [10] (a) Let R be the set of real numbers and let A = (R; ∗, 0) be the algebra with the operation ∗ defined by ⎧ ⎪ ⎨

x x∗y =⎪ y ⎩ 0

if y = 0, if x = 0, otherwise

Then A is a BF -algebra. (b) Let A = [0; ∞). Define the binary operation ∗ on A as follows: x ∗ y = |x − y|, for all x, y ∈ A. Then (A; ∗, 0) is a BF -algebra. Proposition 2.3. [10] Let X be a BF -algebra. Then for any x and y in X, the following hold: (a) 0 ∗ (0 ∗ x) = x for all x ∈ A; (b) if 0 ∗ x = 0 ∗ y, then x = y for any x, y ∈ A; (c) if x ∗ y = 0, then y ∗ x = 0 for any x, y ∈ A. Definition 2.4. [10] A non-empty subset S of a BF -algebra X is called a subalgebra of X if x ∗ y ∈ S for any x, y ∈ S. A mapping f : X −→ Y of BF -algebras is called a BF -homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ X. We now review some fuzzy logic concept (see [11]). Let X be a set. A fuzzy set A in X is characterized by a membership function μA : X −→ [0, 1]. Let f be a mapping from the set X to the set Y and let B be a fuzzy set in Y with membership function μB . The inverse image of B, denoted f −1 (B), is the fuzzy set in X with membership function μf −1 (B) defined by μf −1 (B) (x) = μB (f (x)) for all x ∈ X. Conversely, let A be a fuzzy set in X with membership function μA Then the image of A, denoted by f (A), is the fuzzy set in Y such that:

On fuzzy BF -algebras

μf (A) (y) =

⎧ ⎨ ⎩

15

if f −1 (y) = {x : f (x) = y} = ∅,

sup μA (z)

z∈f −1 (y)

0

otherwise

A fuzzy set A in the BF -algebra X with the membership function μA is said to be have the sup property if for any subset T ⊆ X there exists x0 ∈ T such that μA (x0 ) = sup μA (t). t∈T

A fuzzy topology on a set X is a family τ of fuzzy sets in X which satisfies the following conditions: (i) For c ∈ [0, 1], kc ∈ τ , where kc has a constant membership function, (ii) If A, B ∈ τ , then A ∩ B ∈ τ , (iii) If Aj ∈ τ for all j ∈ J, then Aj ∈ τ . j∈J

The pair (X, τ ) is called a fuzzy topological space and members of τ are called open fuzzy sets. Let A be a fuzzy set in X and τ a fuzzy topology on X. Then the induced fuzzy topology on A is the family of fuzzy subsets of A which are the intersection with A of τ -open fuzzy sets in X. The induced fuzzy topology is denoted by τA , and the pair (A, τA ) is called a fuzzy subspace of (X, τ ). Let (X, τ ) and (Y, υ) are two fuzzy topological space. A mapping f of (X, τ ) into (Y, υ) is fuzzy continuous if for each open fuzzy set U in υ the inverse image f −1 (U) is in τ . Conversely, f is fuzzy open if for each fuzzy set V in τ , the image f (V ) is in υ. Let (A, τA ) and (B, υB ) are fuzzy subspace of fuzzy topological spaces (X, τ ) and (Y, υ) respectively, and let f be a mapping from (X, τ ) to (Y, υ). Then f is a mapping of (A, τA ) into (B, υB ) if f (A) ⊆ B. Furthermore f is relatively fuzzy continuous if for each open fuzzy set V in υB the intersection f −1 (V ) ∩ A is in τA . Conversely, f is relatively fuzzy open if for each open fuzzy set U , the image f (U ) is in υB . Lemma 2.5. [1] Let (A, τA ), (B, υB ) be fuzzy subspace of fuzzy topological space (X, τ ), (Y, υ) respectively, and let f be a fuzzy continuous mapping of (X, τ ) into (Y, υ) such that f (A) ⊂ B. Then f is a relatively fuzzy continuous mapping of (A, τA ) into (B, υB ).

3

Fuzzy BF -subalgebra

From now on X is a BF -algebra, unless otherwise is stated.

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Definition 3.1. Let μ be a fuzzy set in a BF -algebra. Then μ is called a fuzzy BF -subalgebra (algebra) of X if μ(x ∗ y) ≥ min{μ(x), μ(y)} for all x, y ∈ X. Example 3.2. Let X = {0, 1, 2} be a set with the following table: ∗ 0 1 2

0 0 1 2

1 1 0 0

2 2 0 0

Then (X, ∗, 0) is a BF -algebra, but is not a BCH/BCI/BCK-algebra. Define a fuzzy set μ : X → [0, 1] by μ(0) = 0.7, μ(1) = 0.1 and μ(2) = 0.3. Then μ is a fuzzy BF -subalgebra of X. Lemma 3.3. If A is a fuzzy BF -subalgebra of X, then for all x ∈ X μA (0) ≥ μA (x). Proof. For all x ∈ X, we have x ∗ x = 0 hence μA (0) = μA (x ∗ x) ≥ min{μA (x), μA (x)} = μA (x). Proposition 3.4. Let A be a fuzzy BF -subalgebra of X, and let n ∈ N . Then n (i) μA (

(ii) μA (

x ∗ x) ≥ μA (x), for any odd number n,

n

x ∗ x) = μA (x), for any even number n.

Proof. Let x ∈ X and assume that n is odd. Then n = 2k − 1 for some positive integer k. We prove by induction, definition and above lemma imply that μA (x ∗ x) = μA (0) ≥ μA (x). Now suppose that μA ( Then by assumption μA (

2(k+1)−1

x ∗ x) = μA ( = μA (

2k+1 2k−1 2k−1

2k−1

x ∗ x) ≥ μA (x).

x ∗ x) x ∗ (x ∗ (x ∗ x)))

x ∗ x) = μA ( ≥ μA (x).


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Which proves (i). Similarly we can prove (ii). Theorem 3.5. Let A be a fuzzy BF -subalgebra of X. If there exists a sequence {xn } in X, such that lim μA (xn ) = 1

n→∞

Then μA (0) = 1. Proof. By above lemma we have μA (0) ≥ μA (x), for all x ∈ X, thus μA (0) ≥ μA (xn ), for every positive integer n. Consider 1 ≥ μA (0) ≥ lim μA (xn ) = 1. n→∞

Hence μA (0) = 1. Theorem 3.6. Let A1 and A2 are fuzzy BF -subalgebras of X. Then A1 ∩ A2 is a fuzzy BF -subalgebras of X. Proof. Let x, y ∈ A1 ∩ A2 . Then x, y ∈ A1 and A2 , since A1 and A2 are fuzzy BF -subalgebras of X by above theorem we have: μA1 ∩A2 (x ∗ y) = min{μA1 (x ∗ y), μA2 (x ∗ y)} ≥ min{min(μA1 (x), μA1 (y)), min(μA2 (x), μA2 (y))} = min{μA1 ∩A2 (x), μA1 ∩A2 (y)}

Which Proves theorem. Corollary 3.7. Let {Ai |i ∈ Λ} be a family of fuzzy BF -subalgebras of X. Ai is also an fuzzy BF -subalgebras of X. Then i∈Λ

Definition 3.8. Let A be a fuzzy set in X and λ ∈ [0, 1]. Then the level BF -subalgebra U(A; λ) of A and strong level BF -subalgebra U(A; >, λ) of X are defined as following: U(A; λ) := {x ∈ X | μA (x) ≥ λ}, U(A; >, λ) := {x ∈ X | μA (x) > λ}. Theorem 3.9. Let A be a fuzzy BF -subalgebra of X with the least upper bound λ0 . Then the following condition are equivalent: (i) A is a fuzzy BF -subalgebra of X.

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(ii) For all λ ∈ Im(μA ), the nonempty level subset U(A; λ) of A is a BF subalgebra of X. (iii) For all λ ∈ Im(μA ) \ λ0 , the nonempty strong level subset U(A; >, λ) of A is a BF -subalgebra of X. (iv) For all λ ∈ [0, 1], the nonempty strong level subset U(A; >, λ) of A is a BF -subalgebra of X. (v) For all λ ∈ [0, 1], the nonempty level subset U(A; λ) of A is a BF subalgebra of X. Proof. (i −→ iv) Let A be a fuzzy BF -subalgebra of X, λ ∈ [0, 1] and x, y ∈ U(A; min{λ, λ} = λ thus x ∗ y ∈ U(A; >, λ). Hence U(A; >, λ) is a BF -subalgebra of X. (iv −→ iii) It is clear. (iii −→ ii) Let λ ∈ Im(μA ). Then U(A; λ) is a nonempty. Since U(A; λ) = U(A; >, λ), where β ∈ Im(μA )\λ0 . Then by (iii) and Corollary 3.7, U(A; λ)

λ>β

is a BF -subalgebra of X. (ii −→ v) Let λ ∈ [0, 1] and U(A; λ) be nonempty. Suppose x, y ∈ U(A; λ). Let α = min{μA (x), μA (y)}, it is clear that α = min{μA (x), μA (y)} ≥ {λ, λ} = λ . Thus x, y ∈ U(A; α) and α ∈ Im(μA ), by (ii) U(A; α) is a BF -subalgebra of X, hence x ∗ y ∈ U(A; α). Then we have μA (x ∗ y) ≥ min{μA (x), μA (y)} ≥ {α, α} = α ≥ λ. Therefore x ∗ y ∈ U(A; λ). Then U(A; λ) is a BF -subalgebra of X. (v −→ i) Assume that the nonempty set U(A; λ) is a BF -subalgebra of X, for every λ ∈ [0, 1]. In contrary, let x0 , y0 ∈ X be such that μA (x0 ∗ y0 ) < min{μA (x0 ), μA (y0 )}. Let μA (x0 ) = γ, μA (y0 ) = θ and μA (x0 ∗ y0 ) = λ. Then λ < min{γ, θ}. Consider

1 λ1 = (μA (x0 ∗ y0 ) + min{μA (x0 ), μA (y0 )}) 2

We get that λ1 =

1 (λ + min{γ, θ}) 2


Therefore

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1 γ > λ1 = (λ + min{γ, θ} > λ 2 1 θ > λ1 = (λ + min{γ, θ} > λ 2

Hence min{γ, θ} > λ1 > λ = μA (x0 ∗ y0 ) so that x0 ∗ y0 ∈ U(A; λ) which is a contradiction, since μA (x0 ) = γ ≥ min{γ, θ} > λ1 μA (y0 ) = θ ≥ min{γ, θ} > λ1 imply that x0 , y0 ∈ U(A; λ). Thus μA (x ∗ y) ≥ min{μA (x), μA (y)} for all x, y ∈ X. Which completes the proof. Theorem 3.10. Each BF -subalgebra of X is a level BF -subalgebra of a fuzzy BF -subalgebra of X. Proof. Let Y be a BF -subalgebra of X, and A be an fuzzy set on X defined by α if x ∈ Y μA (x) = 0 otherwise where α ∈ [0, 1]. It is clear that U(A; α) = Y . Let x, y ∈ X. We consider the following cases: case 1) If x, y ∈ Y , then x ∗ y ∈ Y therefore μA (x ∗ y) = α = min{α, α} = min{μA (x), μA (y)}. case 2) If x, y ∈ Y , then μA (x) = 0 = μA (y) and so μA (x ∗ y) ≥ 0 = min{0, 0} = min{μA (x), μA (y)}. case 3) If x ∈ Y and y ∈ Y , then μA (x) = α and μA (y) = 0. Thus μA (x ∗ y) ≥ 0 = min{α, 0} = min{μA (x), μA (y)}. case 4) If y ∈ Y and x ∈ Y , then by the same argument as in case 3, we can conclude that μA (x ∗ y) ≥ min{μA (x), μA (y)}. Therefore A is a fuzzy BF -subalgebra of X. In the next theorem we generalize the above theorem.

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Theorem 3.11. Let X be a BF -algebra. Then for any chain of subalgebras A0 ⊂ A1 ⊂ . . . ⊂ Ar = X there exists a fuzzy subalgebra μ of X whose level subalgebras are exactly the subalgebras of this chain. Proof. Consider a set of numbers t0 > t1 > . . . > tr where each ti be in [0, 1]. Define μ : X → [0, 1] by μ(Ai \Ai−1 ) = ti for all 0 < i ≤ r, and μ(A0 ) = t0 . We prove that μ is a fuzzy subalgebra of X. Let x, y ∈ X, we consider the following cases: Case 1) Let x, y ∈ Ai \Ai−1 , then μ(x) = ti = μ(y). Since Ai is a subalgebra thus x ∗ y ∈ Ai , so x ∗ y ∈ Ai \Ai−1 or x ∗ y ∈ Ai−1 , and in each of them we have μ(x ∗ y) ≥ ti = min{μ(x), μ(y)}. Case 2) Let x ∈ Ai \Ai−1 , y ∈ Aj \Aj−1 , where i < j. Then μ(x) = ti μ(y) = tj , since A − j ⊆ Ai and Ai is a subalgebra of X, then x ∗ y ∈ Ai . Hence μ(x ∗ y) ≥ tj = min{μ(x), μ(y)}. It is clear that Im(μ) = {t0 , t1 , . . . , tr }, therefore the level subalgebras of μ are given by the chain of subalgebras μt0 ⊂ μt1 . . . ⊂ μtr = X. We have μt0 = {x ∈ X | μ(x) ≥ t0 } = A0 . It is clear that Ai ⊆ μti . Let x ∈ μti then μ(x) ≥ ti then x ∈ Aj for j > i. So μ(x) ∈ {t0 , t1 , . . . , ti }, thus x ∈ Ak for k ≤ i, since Ak ⊆ Ai we get that x ∈ Ai . Hence Ai = μti for 0 ≤ i ≤ r. Theorem 3.12. Let X be a BF -algebra. Then two level subalgebras μt1 , μt2 (where t1 < t2 ) of μ are equal if and only if there is no x ∈ X such that t1 ≤ μ(x) < t2 . Proof. In contrary let μt1 = μt2 where t1 < t2 and there exists x ∈ X such that t1 ≤ μ(x) < t2 . Then μt2 is a proper subset of μt1 , which is a contradiction. Conversely, suppose that there is no x ∈ X such that t1 ≤ μ(x) < t2 . Since t1 < t2 then μt1 ⊆ μt2 . If x ∈ μt1 , then μ(x) ≥ t1 by hypotheses we get that μ(x) ≥ t2 . Therefore x ∈ μt2 , then μt2 ⊆ μt1 . Hence μt1 = μt2 . Theorem 3.13. Let Y be a subset of X and A be a fuzzy set on X which is given in the proof of Theorem 3.10. If A is a fuzzy BF -subalgebra of X, then Y is a BF -subalgebra of X.


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Proof. Let A be a fuzzy BF -subalgebra of X, and x, y ∈ Y . Then μA (x) = α = μA (y), thus μA (x ∗ y) ≥ min{μA (x), μA (y)} = min{α, α} = α. which implies that x ∗ y ∈ Y . Theorem 3.14. If A is a fuzzy BF -subalgebra of X, then the set XμA := {x ∈ X | μA (x) = μA (0)} is a BF -subalgebra of X. Proof. Let x, y ∈ XμA . Then μA (x) = μA (0) = μA (y), and so μA (x ∗ y) ≥ min{μA (x), μA (y)} = min{μA (0), μA (0)} = μA (0). by Lemma 3.3, we get that μA (x ∗ y) = μA (0) which means that x ∗ y ∈ XμA . Theorem 3.15. Let M be a crisp subset of X. Suppose that N is a fuzzy set of X defined by μN as:

μN (x) =

α β

if x ∈ M otherwise

for all α, β ∈ [0, 1] with α ≥ β. Then N is a fuzzy BF -subalgebra if and only if M is a (crisp) BF -subalgebra of X. Moreover, in this case XμN = M. Proof. Let N be an fuzzy BF -subalgebra. Let x, y ∈ X be such that x, y ∈ M. Then μN (x ∗ y) ≥ min{μN (x), μN (y)} = min{α, α} = α and so x ∗ y ∈ M. Conversely, suppose that M is a BF -subalgebra of X, let x, y ∈ X. (i) If x, y ∈ M then x ∗ y ∈ M, thus μN (x ∗ y) = α = min{μN (x), μN (y)} (ii) If x ∈ M or y ∈ M , then μN (x ∗ y) ≥ β = min{μN (x), μN (y)} This shows that N is a fuzzy BF -subalgebra. Moreover, we have XμN := {x ∈ X | μN (x) = μN (0)} = {x ∈ X | μN (x) = α} = M.

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4


Fuzzy topological BF -algebra

Proposition 4.1. Let f be a BF -homomorphism from X into Y and G be a fuzzy BF -algebra of Y with the membership function μG . Then the inverse image f −1 (G) of G is a fuzzy BF -algebra of X. Proof. Let x, y ∈ X. Then μf −1 (G) (x ∗ y) = = ≥ =

μG (f (x ∗ y) μG (f (x) ∗ f (y)) min{μG (f (x)), μG (f (y))} min{μf −1 (G) (x), μf −1 (G) (y)}.

Proposition 4.2. Let f be a BF -homomorphism from X onto Y and D be a fuzzy BF -algebra of X with the sup property. Then the image f (D) of D is a fuzzy BF -algebra of Y . Proof. Let a, b ∈ Y , let x0 ∈ f −1 (a), y0 ∈ f −1 (b) such that μD (x0 ) = sup μD (t),

μD (y0 ) = sup μD (t).

t∈f −1 (a)

t∈f −1 (b)

Then by the definition of μf (D) , we have μf (D) (x ∗ y) =

sup

t∈f −1 (a∗b)

μD (t)

≥ μD (x0 ∗ y0 ) ≥ min{μD (x0 ), μD (y0 ) = min{ sup μD (t), sup μD (t)} t∈f −1 (a)

t∈f −1 (b)

= min{μf (D) (a), μf (D) (b)}. For any BF -algebra X and any element a ∈ X we denote by Ra the right translation of X defined by Ra (x) = x ∗ a for all x ∈ X. It is clear that R0 (x) = 0 = Rx (x) for all x ∈ X. Definition 4.3. Let τ be a fuzzy topology on X and D be a fuzzy BF algebra of X with induced topology τD . Then D is called a fuzzy topological BF -algebra of X if for each a ∈ X the mapping Ra : (D, τD ) → (D, τD ) is relatively fuzzy continuous.


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Theorem 4.4. Let X and Y be two BF -algebras, f : X → Y be a BF homomorphism. Let τ and υ be the fuzzy topologies on X and Y respectively, such that τ = f −1 (υ). Let G be a fuzzy topological BF -algebra of Y with membership function μG . Then f −1 (G) is a fuzzy topological BF -algebra of X with membership function μf −1 (G) . Proof. We must show that, for each a ∈ X, the mapping Ra : (f −1 (G), τf −1 (G) ) → (f −1 (G), τf −1 (G) ) is relatively fuzzy continuous. Let U be any open fuzzy set in τf −1 (G) on f −1 (G). Since f is a fuzzy continuous mapping from (X, τ ) into (Y, υ), from Lemma 1.4 follows that f is a relatively fuzzy continuous mapping of (f −1 (G), τf −1 (G) ) into (G, υG ). Note that there exists an open fuzzy set V ∈ υG such that f −1 (V ) = U. The membership function of Ra−1 (U) is given by (x) = μU (Ra (x)) = μU (x ∗ a) μR−1 a (U ) = μf −1 (V ) (x ∗ a) = μV (f (x ∗ a)) = μV (f (x) ∗ f (a)). Since G is a fuzzy topological BF -algebra of Y , the mapping Rb : (G, υG ) → (G, υG ) is relatively fuzzy continuous for each b ∈ Y . Hence (x) = μV (f (x) ∗ f (a)) = μV (Rf (a) (f (x)) μR−1 a (U ) = μR−1 (V ) (f (x)) = μf −1 (R−1 (V )) (x). f (a)

f (a)

which implies that Ra−1 (U) = f −1 (Rf−1 (a) (V )) therefore −1 Ra−1 (U) ∩ f −1 (G) = f −1 (Rf−1 (G) (a) (V )) ∩ f

is a open in the relative fuzzy topology on f −1 (G). The membership function μG of a fuzzy BF -algebra G of X is said to be f -invariant [9] if f (x) = f (y) implies μG (x) = μG (y), for all x, y ∈ X. Theorem 4.5. Given BF -algebras X and Y and a BF -homomorphism f from X onto Y , let τ be the fuzzy topology on X and υ be the fuzzy topology on Y such that f (τ ) = υ. Let D be a fuzzy topological BF -algebra of X.

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If the membership function μD of D is a f -invariant, then f (D) is a fuzzy topological BF -algebra of Y . Proof. It is enough to show that the mapping Rb : (f (D), υf (D) ) → (f (D), υf (D)) is relatively fuzzy continuous, for all b ∈ Y . It is clear that f is a relatively fuzzy open mapping, since for U ∈ τD there exists U ∈ τ such that U = U ∩D, by f -invariance of μD we have f (U) = f (U) ∩ f (D) ∈ υf (D) .

Let V be an open fuzzy set in υf (D) . For any b ∈ Y by hypothesis there exists a ∈ X such that b = f (a). Thus μf −1 (R−1 (V ) (x) = μf −1 (R−1 b

f (a)

(V )) (x)

= μR−1

f (a)

(V ) (f (x))

= μV (Rf (a) (f (x)) = μV (f (x) ∗ (f (a)) = μV (f (x ∗ a)) = μf −1 (V ) (x ∗ a) = μf −1 (V ) (Ra (x)) = μR−1 −1 (V )) (x). a (f

which implies that f −1 (Rb−1 (V )) = Ra−1 (f −1 (V )). By hypothesis, Ra is a relatively fuzzy continuous mapping from (D, τD ) to (D, τD ) and f is a relatively fuzzy continuous mapping from (D, τD ) to (f (D), υf (D) ). Therefore

f −1 (Rb−1 (V )) ∩ G = Ra−1 (f −1 (V )) ∩ D is open in τD . Since f is relatively fuzzy open, then

f (f −1 (Rb−1 (V )) ∩ D) = Rb−1 (V ) ∩ f (D) is open in υf (D) . ACKNOWLEDGEMENTS

This research has been supported by Mahani Mathematical research center and Fuzzy sets and its application center of Exellence, Kerman, Iran.

References [1] D. H. Foster, Fuzzy topological groups, J. Math. Anal. Appl., 67 (1979), 549-564.


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[2] Q. P. Hu and X. Li, On BCH-algebras, Mathematics Seminar Notes, 11 (1983), 313-320. [3] Q. P. Hu and X. Li, On proper BCH-algebras, Math. Japanica, 30 (1985), 659-661. [4] Y. Imai and K. Iseki, On axiom systems of propositional calculi, XIV Proc. Japan Academy, 42 (1966), 19-22. [5] K. Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japanica, 23 (1978), 1-26. [6] K. Iseki, On BCI-algebras, Mathematics Seminar Notes, 8 (1980), 125130. [7] Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Sci. Math. Jpn. 1 (1998), 347354. [8] J. Neggers and H. S. Kim, On B-algebras, Math. Vensik 54 (2002), 21-29. [9] A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl., 35 (1971), 512-517. [10] Andrzej Walendziak, On BF -algebras, Math. Slovaca 57 (2007), No. 2, 119-128. [11] L. A. Zadeh, Fuzzy Sets, Inform. Control, 8 (1965), 338-353. Received: October 1, 2008