BCI-Algebras 1 Introduction and

International Journal of Algebra, Vol. 4, 2010, no. 7, 341 - 352

Fuzzy Dot BCK/BCI-Algebras Arsham Borumand Saeid Department of Mathematics Shahid Bahonar University of Kerman, Kerman, Iran [email protected] Abstract In this paper the notion of fuzzy dot BCK-subalgebra is introduced. We state and prove some theorem in fuzzy dot BCK-subalgebra and level subalgebras.

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

1

Introduction and preliminaries

Processing of certain information especially inferences based on certain information, therefore is based on classical two-valued logic. Logic appears in a ‘scared’ form (resp., a ‘profane’) which is dominant in proof theory (resp., model theory). The role of logic in mathematics and computer science is two fold-as a tool for applications in both areas, and a technique for laying the foundations. Non-classical logic including many-valued logic, fuzzy logic, etc., takes the advantage of the classical logic to handle information with various facets of uncertainty, such as fuzziness, randomness, and so on. Non-classical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. Among all kinds of uncertainties, incomparability is an important one which can be encountered in our life. In recent years, motivated by both theory and application, the study of t-norm-based logic systems and the corresponding pseudo-logic systems has been become a greater focus in the field of logic. Here, t-norm-based algebraic investigations were first to the corresponding algebraic investigations, and in the case of pseudo-logic systems, algebraic development was first to the corresponding logical development. As it is well known, BCK/BCI-algebras are two classes of algebras of logic. They were introduced by Imai and Iseki

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[6, 7]. BCI-algebras are generalizations of BCK-algebras. Most of the algebras related to the t-norm based logic, such as MT L-algebras, BL-algebras [3, 4], hoop, MV -algebras and Boolean algebras et al., are extensions of BCKalgebras. The concept of fuzzy sets was first initiated by Zadeh [11]. Since then it has become a vigorous area of research in engineering, medical science, social science, physics, statistics, graph theory, etc. In the present paper, we introduced the concept of fuzzy dot BCK-subalgebras and fuzzy dot topological BCK-algebras and study this structure. We state and prove some theorem discussed in fuzzy dot BCK-subalgebras and level subalgebras and give the relationship between this notion and fuzzy BCKsubalgebras. Finally some of Fosters results on homomorphic images and inverse images in fuzzy dot topological BCK-algebras are studied. If μ is a fuzzy set in a BCK/BCI-algebra X. Then μ is called a fuzzy BCK/BCI-subalgebra (algebra) of X if μ(x ∗ y) ≥ min{μ(x), μ(y)} for all x, y ∈ X [9]. Definition 1.1. [8] A fuzzy topology on a set X is a family τ of fuzzy sets in X which satisfies the following condition: (i) For c ∈ [0, 1], kc ∈ τ , where kc has a constant membership function, (ii) If A, B ∈ τ , then A ∩ B ∈ τ , (iii) τ closed under arbitrary union, which means that if Aj ∈ τ for all Aj ∈ τ . j ∈ J, then j∈J

The pair (X, τ ) is called a fuzzy topological space and members of τ are called open fuzzy sets. Definition 1.2. [8] 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 ). Definition 1.3. [8] Let (X, τ ) and (Y, υ) be 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 ) be fuzzy subspace of fuzzy topological spaces (X, τ ) and (Y, υ) respectively, and let f be a mapping from (X, τ ) to (Y, υ).

Fuzzy dot BCK/BCI-algebras

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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 1.4. [2] 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 ).

2

Main Results

From now on X is a BCK-algebra, unless otherwise is stated. Definition 2.1. Let μ be a fuzzy set in X. Then μ is called a fuzzy dot BCK-subalgebra (algebra) of X if μ(x ∗ y) ≥ μ(x).μ(y) for all x, y ∈ X. Example 2.2. Let X = {0, a, b, c} be a set with the following table: ∗ 0 a b c

0 0 a b c

a a 0 c b

b b c 0 a

c c b a 0

Then (X, ∗, 0) is a BCI-algebra. Define a fuzzy set μ : X → [0, 1] by μ(0) = 0.5, μ(x) = 0.7 for all x ∈ {a, b, c}. Then μ is a fuzzy dot BCI-subalgebra of X. Note that every fuzzy BCK-subalgebra of X is a fuzzy dot BCK-subalgebra of X, but the converse is not true. In fact, the fuzzy dot BCI-subalgebra in above example is not a fuzzy BCI-subalgebra, since μ(a ∗ a) = μ(0) = 0.5 < 0.7 = μ(a) = min{μ(a), μ(a)}. Lemma 2.3. If μ is a fuzzy dot BCK-subalgebra of X, then for all x ∈ X μ(0) ≥ (μ(x))2 .

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Proof. For all x ∈ X, we have x∗ x = 0. Hence μ(0) = μ(x ∗ x) ≥ μ(x).μ(x) = (μ(x))2 . Theorem 2.4. Let A be a fuzzy dot BCK-subalgebra of X. If there exists a sequence {xn } in X, such that lim (μ(xn ))2 = 1

n→∞

Then μ(0) = 1. Proof. By Lemma 2.3, we have μ(0) ≥ (μ(x))2 , for all x ∈ X, thus μ(0) ≥ (μ(xn ))2 , for every positive integer n. Consider 1 ≥ μ(0) ≥ lim (μ(xn ))2 = 1. n→∞

Hence μ(0) = 1. Theorem 2.5. Let μ and ν are fuzzy dot BCK-subalgebras of X. Then μ ∩ ν is a fuzzy dot BCK-subalgebras of X. Proof. Let x, y ∈ μ ∩ ν. Then x, y ∈ μ and ν, since μ and ν are fuzzy dot BCK-subalgebras of X by above theorem we have: (μ ∩ ν)(x ∗ y) = ≥ ≥ =

min{μ(x ∗ y), ν(x ∗ y)} min{μ(x).μ(y), ν(x).ν(y)} (min{μ(x), ν(x)}).(min{μ(y), ν(y)}) ((μ ∩ ν)(x)).((μ ∩ ν)(y))

Which proves the theorem. Corollary2.6. Let {μi |i ∈ Λ} be a family of fuzzy dot BCK-subalgebras of μi is also a fuzzy dot BCK-subalgebras of X. X. Then i∈Λ

Definition 2.7. Let μ be a fuzzy set in X and λ ∈ [0, 1]. Then the level BCK-subalgebra U(μ; λ) of μ and strong level BCK-subalgebra U(μ; >, λ) of μ are defined as following: U(μ; λ) := {x ∈ X | μ(x) ≥ λ}, U(μ; >, λ) := {x ∈ X | μ(x) > λ}.


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Remark. If μ is a fuzzy dot subalgebra of X, then U(μ; λ) or U(μ; >, λ) need not be a subalgebra of X. Since, in Example 2.2, define fuzzy subset μ by: μ(0) = 0.5, μ(a) = 0.6, μ(b) = 0.4 and μ(c) = 0.3, then μ is a fuzzy dot subalgebra of X. But U(μ; >, 0.35) = {0, a, b} = U(μ; 0.4) is not a subalgebra of X, since a, b ∈ U(μ; >, 0.35) and a ∗ b = c ∈ U(μ; >, 0.35). Theorem 2.8. Let μ be a fuzzy dot BCK-subalgebra of X. Then U(μ; 1) = {x ∈ X | μ(x) = 1} is either empty or is a subalgebra of X. Proof. If x, y ∈ U(μ; 1), then μ(x ∗ y) ≥ μ(x).μ(y) = 1. Hence μ(x ∗ y) = 1, which implies that x ∗ y ∈ U(μ; 1). Consequently, U(μ; 1) is a subalgebra of X. Proposition 2.9. Let f be a BCK-homomorphism from X into Y and μ be a fuzzy dot BCK-subalgebra of Y . Then the inverse image f −1 (μ) of μ is a fuzzy dot BCK-subalgebra of X. Proof. Let x, y ∈ X. Then f −1 (μ)(x ∗ y) = = ≥ =

μ(f (x ∗ y)) μ(f (x) ∗ f (y)) μ(f (x)).μ(f (y)) f −1 (μ)(x).f −1 (μ)(y).

Then f −1 (μ) is a fuzzy dot BCK-subalgebra of X. Proposition 2.10. Let f be a BCK-homomorphism from X onto Y and μ be a fuzzy dot BCK-subalgebra of X with the sup property. Then the image f (μ) of μ is a fuzzy dot BCK-subalgebra of Y . Proof. Let x, y ∈ Y , A = f −1 (x), B = f −1 (y) and C = f −1 (x ∗ y). Consider A ∗ B = {t = a ∗ b | a ∈ A, b ∈ B}. It is clear that C ⊆ A ∗ B. We have f (μ)(x ∗ y) =

sup

t∈f −1 (x∗y)

μ(t)

= sup μ(t) t∈C

≥

sup μ(t)

t∈A∗B

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≥ ≥

sup μ(x ∗ y)

x∈A,y∈B

sup μ(x).μ(y).

x∈A,y∈B

We know that the operation . : [0, 1] × [0, 1] → [0, 1] is continuous, then for any > 0 there exists a δ > 0, such that if x ≥ sup μ(x)−δ and y ≥ sup μ(y)−δ. x∈A

y∈B

Then x ∗ y ≥ sup μ(x). sup μ(y) − . Choose a ∈ A and b ∈ B such that x∈A

y∈B

μ(a) ≥ sup μ(x) − δ and μ(b) ≥ sup μ(y) − δ. x∈A

y∈B

Thus μ(a).μ(b) ≥ sup μ(x). sup μ(y) − . Therefore x∈A

y∈B

f (μ)(x ∗ y) ≥

sup μ(x).μ(y)

x∈A,y∈B

≥ sup μ(x). sup μ(y) x∈A

y∈B

= f (μ)(x).f (μ)(y). Hence f (μ) is a fuzzy dot BCK-subalgebra of Y . Definition 2.11. Let ρ be a fuzzy subset of X. The strongest fuzzy ρrelation on X is a fuzzy subset μρ of X × X given by μρ (x, y) = ρ(x).ρ(y), for all x, y ∈ X. Theorem 2.12. Let μρ be the strongest fuzzy ρ-relation on X, where ρ is a fuzzy subset of X. If ρ is a fuzzy dot BCK-subalgebra of X, then μρ is a fuzzy dot BCK-subalgebra of X × X. Proof. Let ρ be a fuzzy dot BCK-subalgebra of X, x1 , x2 , y1 , y2 ∈ X. We have μρ ((x1 , y1 ) ∗ (x2 , y2)) = = ≥ = =

μρ (x1 ∗ y1 , x2 ∗ y2 ) ρ(x1 ∗ x2 ).ρ(y1 ∗ y2 ) (ρ(x1 ).ρ(x2 )).(ρ(y1 ).ρ(y2 )) (ρ(x1 ).ρ(y1 )).(ρ(x2 ).ρ(y2 )) μρ (x1 , y1 ).μρ (x2 , y2 ).

Therefore μρ is a fuzzy dot BCK-subalgebra of X × X.


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Definition 2.13. Let ρ be a fuzzy subset of X. A fuzzy relation μ on X is called a fuzzy ρ-product relation if μ(x ∗ y) ≥ ρ(x).ρ(y), for all x, y ∈ X. Definition 2.14. Let ρ be a fuzzy subset of X. A fuzzy relation μ on X is called a left fuzzy relation on ρ if μ(x ∗ y) ≥ ρ(x), for all x, y ∈ X. Similarly we can define a right fuzzy relation on ρ. It is clear that a left (right) fuzzy relation on ρ is a fuzzy ρ-product relation. Theorem 2.15. Let μ be a left fuzzy relation on fuzzy relation ρ of X. If μ is a fuzzy dot BCK-subalgebra of X × X, then ρ is a fuzzy dot BCK-subalgebra of X. Proof. Let μ left fuzzy relation μ on ρ is a fuzzy dot BCK-subalgebra of X. Thus ρ(x1 ∗ x2 ) = = ≥ =

μ(x1 ∗ x2 , y1 ∗ y2 ) μ((x1 , y1 ) ∗ (x2 , y2)) μ(x1 , y1 ).μ(x2 , y2 ) ρ(x1 ).ρ(x2 ).

for all x1 , x2 , y1 , y2 ∈ X. Therefore ρ is a fuzzy dot subalgebra of X. Theorem 2.16. Let μ be a fuzzy relation on X satisfying the inequality μ(x, y) ≤ μ(x, 0), for all x, y ∈ X. Define ρt by ρt (x) = μ(x, t), for all x ∈ X and t ∈ X. If μ is a fuzzy dot BCK-subalgebra of X × X, then ρt is a fuzzy dot BCK-subalgebra of X, for all t ∈ X. Proof. Let x, y, t ∈ X. Then ρt (x ∗ y) = μ(x ∗ y, t) = μ(x ∗ y, t ∗ 0) = μ((x, t) ∗ (y, 0)) ≥ μ(x, t).μ(y, 0) ≥ μ(x, t).μ(y, t) = ρt (x).ρt (y). Therefore ρt is a fuzzy dot BCK-subalgebra of X. Theorem 2.17. Let μ be a fuzzy relation on X and ρμ be a fuzzy subset of X given by ρμ (x) = inf μ(x, y).μ(y, x), for all x ∈ X. If μ is a fuzzy dot y∈X

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BCK-subalgebra of X × X satisfying the equality μ(x, 0) = 1 = μ(0, x), for all x ∈ X. Then ρμ is a fuzzy dot BCK-subalgebra of X, for all t ∈ X. Proof. Let x, y, z ∈ X. Then μ(x ∗ y, z) = μ(x ∗ y, z ∗ 0) = μ((x, z) ∗ (y, 0)) ≥ μ(x, z).μ(y, 0) = μ(x, z). And μ(z, x ∗ y) = μ(z ∗ 0, x ∗ y) = μ((z, x) ∗ (0, y)) ≥ μ(z, x).μ(0, y) = μ(z, x). Therefore μ(x ∗ y, z).μ(z, x ∗ y) ≥ μ(x, z).μ(z, x) ≥ (μ(x, z).μ(z, x)).(μ(y, z).μ(z, y)). Now, consider ρμ (x ∗ y) = inf μ(x ∗ y, z).μ(z, x ∗ y) z∈X

≥ ( inf (μ(x, z).μ(z, x))).( inf (μ(y, z).μ(z, y))) z∈X

z∈X

= ρμ (x).ρμ (y). Therefore ρμ is a fuzzy dot BCK-subalgebra of X. Definition 2.18. A fuzzy map f from a set X to a set Y is an ordinary map from X to the set of all fuzzy subsets of Y satisfying the following conditions: (i) for all x ∈ X, there exists yx ∈ X such that (f (x))(yx ) = 1, (ii) for all x ∈ X, f (x)(y1 ) = f (x)(y2 ) implies y1 = y2 . We can see that a fuzzy map f from X to Y (i) gives rise to a unique ordinary map μf : X ×X → I, given by μf (x∗y) = f (x)(y). (ii) gives a unique ordinary map f1 : X → Y defined as f1 (x) = yx . Now we can generalize the notion of homomorphism to fuzzy homomorphism.


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Definition 2.19. Let X, Y be BCK/BCI-algebras. A fuzzy map f from a set X to a set Y is called a fuzzy dot homomorphism if μf (x1 ∗ x2 , y) = sup μf (x1 , y1 ).μf (x2 , y2 ), y=y1 ∗y2

for all x1 , x2 ∈ X and y ∈ Y . Proposition 2.20. Let f : X → Y be a fuzzy homomorphism of BCK/BCIalgebras. Then (i) μf (x1 ∗ x2 , y1 ∗ y2) ≥ μf (x1 , y1 ).μf (x2 , y2 ), for all x1 , x2 ∈ X and y1 , y2 ∈ Y. (ii) μf (0, 0) = 1. (iii) μf (0 ∗ x, 0 ∗ y) ≥ μf (x, y), for all x ∈ X and y ∈ Y . Proof. (i) For any x1 , x2 ∈ X and y1 , y2 ∈ Y , we have μf (x1 ∗ x2 , y1 ∗ y2 ) =

sup

y1 ∗y2 =y1 ∗y2

μf (x1 , y1 ).μf (x2 , y2 )

≥ μf (x1 , y1 ).μf (x2 , y2). (ii) Let s ∈ X. Since f is a fuzzy homomorphism then there exists a yx ∈ Y such that μf (x, yx ) = 1, then we get that μf (0, 0) = μf (x ∗ x, yx ∗ yx ) ≥ μf (x, yx ).μf (x, yx ) = 1. (iii) By (i) and (ii) is clear. For any BCK/BCI-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 2.21. Let τ be a fuzzy topology on X and D be a fuzzy dot BCK-subalgebra of X with induced topology τD . Then D is called a fuzzy dot topological BCK/BCI-algebra of X if for each a ∈ X the mapping Ra : (D, τD ) → (D, τD ) is relatively fuzzy continuous. Theorem 2.22. Let X and Y be two BCK/BCI-algebras, f : X → Y be a BCK/BCI-homomorphism. Let τ and υ be the fuzzy dot topologies on X and Y respectively, such that τ = f −1 (υ). Let G be a fuzzy dot topological BCK-subalgebra of Y with membership function μG . Then f −1 (G) is a fuzzy dot topological BCK/BCI-algebra of X with membership function μf −1 (G) . Proof. We must show that, for each a ∈ X, the mapping

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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 dot topological BCK/BCI-algebra of Y , the mapping Rb : (G, υG ) → (G, υG ) is relatively fuzzy continuous for each b ∈ Y . Hence μR−1 (x) = μV (f (x) ∗ f (a)) = μV (Rf (a) (f (x)) 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 dot topology on f −1 (G). Theorem 2.23. Given BCK/BCI-algebras X and Y and a BCK/BCIhomomorphism f from X onto Y , let τ be the fuzzy dot topology on X and υ be the fuzzy dot topology on Y such that f (τ ) = υ. Let D be a fuzzy dot topological BCK/BCI-algebra of X. If the membership function μD of D is a f -invariant, then f (D) is a fuzzy dot topological BCK/BCI-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) .


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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) .

3

Conclusion

In the present paper, we have introduced the concept of fuzzy dot subalgebras of BCK/BCI-algebras and investigated some of their useful properties. In our opinion, these definitions and main results can be similarly extended to some other fuzzy algebraic systems such as groups, semigroups, rings, nearrings, semirings (hemirings), lattices and Lie algebras. It is our hope that this work would other foundations for further study of the theory of BCK/BCI-algebras. Our obtained results can be perhaps applied in engineering, soft computing or even in medical diagnosis. In our future study of fuzzy structure of BCK/BCI-algebras, may be the following topics should be considered: (1) To establish a fuzzy dot ideals of BCK/BCI-algebras; (2) To consider the structure of quotient BCK/BCI-algebras by using these fuzzy dot ideals; (3) To get more results in fuzzy dot BCK/BCI-algebras and application. ACKNOWLEDGEMENTS

The author wishes to express his gratitude to the referee for his/her suggestions to improve the readability of this paper. The author has been supported

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by Mahani Mathematical research center of Shahid Bahonar University of Kerman, Kerman, Iran.

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