NOVEL CONCEPTS OF DOUBT BIPOLAR FUZZY H-IDEALS OF BCK/BCI-ALGEBRAS. Anas Al-Masarwah and Abd Ghafur Ahmad. Received February 2018; revised June 2018

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1 International Journal of Innovative Computing, Information Control ICIC International c 2018 ISS Volume 14, umber 6, December 2018 pp OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS OF BCK/BCI-LGEBRS nas l-masarwah bd Ghafur hmad School of Mathematical Sciences Faculty of Science Technology Universiti Kebangsaan Malaysia UKM Bangi, Selangor, Malaysia almasarwah85@gmail.com; ghafur@ukm.edu.my Received February 2018; revised June 2018 bstract. In this paper, we introduce the notion of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras investigate some interesting properties. We characterize strong doubt positive t-level cut set, strong doubt negative s-level cut set, homomorphism equivalence relation by considering doubt bipolar fuzzy H-ideals of BCK/BCI-algebras. Characterization theorem of characteristic doubt bipolar fuzzy H-ideals is also discussed. articularly, the notion of Cartesian product of two doubt bipolar fuzzy H-ideals by using max-min operations is introduced, some related properties are studied. Ordinary H-ideals are linked with doubt bipolar fuzzy H-ideals by means of doubt level cut set of Cartesian product of two bipolar-valued fuzzy sets. Keywords: BCK/BCI-algebra, Cartesian product, Doubt bipolar fuzzy H-ideal, Characteristic doubt bipolar fuzzy H-ideal 1. Introduction. The notion of fuzzy set fuzzy relation on a set was given by Zadeh [1, 2]. Since then, the concept of fuzzy set fuzzy relation provides a natural framework for generalizing some of the basic notions of algebra, for example, logic, set theory, group theory, semigroup theory, ring theory, semiring theory, hemiring theory, modules. lso, there are several applications of fuzzy set theory fuzzy relation in various fields, for example, decision making, artificial intelligence, expert system, computer science, operation research. Lee [3] introduced the notion of bipolar-valued fuzzy sets. Bipolarvalued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to the interval [ 1, 1]. In a bipolar-valued fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree 0, 1] of an element indicates that the element somewhat satisfies the property, the membership degree [ 1, 0 of an element indicates that the element somewhat satisfies the implicit counter-property. Imai Iséki [4] introduced two classes of abstract algebra, namely BCK-algebras BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. The study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy subgroups in 1971 by Rosenfeld [5]. fterwards many researchers had worked on the structures of fuzzy sets in BCK/BCI-algebras in other algebraic structures. Xi [6] applied the concept of fuzzy sets to BCK-algebras. lso, Jun [7] hmad [8] applied it to BCI-algebras. Khalid hmad [9] introduced fuzzy H- ideals in BCI-algebras studied their properties. Huang [10] gave another definition of fuzzy BCI-algebras some results about it. In 1994, Jun [11] established the definition DOI: /ijicic

2 2026. L-MSRWH D. G. HMD of a doubt fuzzy subalgebra a doubt fuzzy ideal in BCK/BCI-algebras to avoid the confusion created in Huang s definition of fuzzy BCI-algebras [10] gave some results about it. Zhan Tan [12, 13] introduced the notion of doubt fuzzy H-ideals in BCKalgebras studied their properties. Recently, based on the results of bipolar-valued fuzzy sets, more more researchers have devoted themselves to studying some bipolar fuzzy algebraic structures. Lee [14] applied the concept of bipolar-valued fuzzy set theory to BCK/BCI-algebras, introduced the notions of bipolar fuzzy subalgebras bipolar fuzzy ideals of BCK/BCIalgebras. Recently, the notion of bipolar-valued fuzzy set theory was applied to BCK/ BCI-algebras [15, 16] other algebraic structures such as semigroups [17], semirings [18], hemirings [19, 20], K-algebras [21], BF -algebras [22, 23], Lie algebras [24] Lie superalgebras [25]. For the general development of BCK/BCI-algebras, the H-ideal theory plays an important role. Thus, there are a number of works on BCK/BCI-algebras related algebraic systems but to the best of our knowledge no work is available on novel concepts of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras. For this reason we are motivated to develop these theories for BCK/BCI-algebras. In this paper, after introductory section, some basic notions are discussed in Section 2. In Section 3, we introduce the notion of doubt bipolar fuzzy H-ideals of BCK/BCIalgebras investigate some interesting properties. We characterize strong doubt positive t-level cut set, strong doubt negative s-level cut set homomorphism by considering doubt bipolar fuzzy H-ideals of BCK/BCI-algebras. In Section 4, the characterizations of the maps from the set of doubt bipolar fuzzy H-ideals to the set of H-ideals are investigated through the equivalence relation. In Section 5, we define Cartesian product of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras. We give some interesting results about Cartesian product of doubt bipolar fuzzy H-ideals. In addition, we prove that the Cartesian product of two bipolar-valued fuzzy sets becomes doubt bipolar fuzzy H-ideals if only if for any t, s [0, 1] [ 1, 0], doubt positive t-level cut set doubt negative s-level cut set are H-ideals of a BCK/BCI-algebra X Y. t last, conclusions are drawn in Section reliminaries. We first recall some elementary aspects which are used to present the paper. Throughout this paper, X always denotes a BCK/BCI-algebra without any specifications. BCK/BCI-algebra is an important class of logical algebras introduced by Imai Iséki [4, 26] was extensively investigated by several researchers. This algebra is defined as follows. By a BCI-algebra we mean an algebra X;, 0 of type 2, 0 satisfying the following axioms for all x, y, z X: 1 x y x z z y = 0, 2 x x y y = 0, 3 x x = 0, 4 x y = 0 y x = 0 imply x = y. If a BCI-algebra X satisfies 0 x = 0, then X is called a BCK-algebra. mapping f : X X of BCK/BCI-algebras is called a homomorphism if fx y = fx fy for all x, y X. If f is one to one onto, then f is called monomorphism epimorphism. partial ordering on a BCK/BCI-algebra X can be defined by x y if only if x y = 0. ny BCK/BCI-algebra X satisfies the following axioms for all x, y, z X: 1 x y z = x z y, 2 x y x, 3 x y z x z y z,

3 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS x y x z y z, z y z x. Definition 2.1. [6] non-empty subset S of a BCK/BCI-algebra X is called an ideal of X if 1 0 S, 2 x y S y S then x S for all x, y X. Definition 2.2. [9] non-empty subset S of a BCK/BCI-algebra X is called an H-ideal of X if 1 0 S, 2 x y z S y S then x z S for all x, y, z X. We refer the reader to [27, 28, 29] for further information regarding BCK/BCIalgebras. Definition 2.3. [9] fuzzy set = x, x x X} in X is called a fuzzy H-ideal of X if 1 0 x, 2 x z min x y z, y} for all x, y, z X. Definition 2.4. [12] fuzzy set = x, x x X} in X is called a doubt fuzzy H-ideal of X if 1 0 x, 2 x z max x y z, y} for all x, y, z X. The proposed work is done on a bipolar-valued fuzzy set. The formal definition of a bipolar-valued fuzzy set is given below. Definition 2.5. [3] Let X be a non-empty set. bipolar-valued fuzzy set in X is an object having the form = x, x, x x X } where : X [0, 1] : X [ 1, 0] are mappings. We use the positive membership degree x to denote the satisfaction degree of an element x to the property corresponding to a bipolar-valued fuzzy set, the negative membership degree x to denote the satisfaction degree of an element x to some implicit counter-property corresponding to a bipolar-valued fuzzy set. If x 0 x = 0, it is the situation that x is regarded as having only positive satisfaction for. If x = 0 x 0, it is the situation that x does not satisfy the property of but somewhat satisfies the counter property of. It is possible for an element x to be such that x 0 x 0 when the membership function of the property overlaps that of its counter property over some portion of X. For the sake of simplicity, we shall use the symbol =, for the bipolar-valued fuzzy set = x, x, x x X }, use the notion of bipolar fuzzy sets instead of the notion of bipolar-valued fuzzy sets. 3. Doubt Bipolar Fuzzy H-Ideals. In this section, we introduce a new generalized doubt fuzzy H-ideal of a BCK/BCI-algebra called, doubt bipolar fuzzy H-ideal. We discuss the concepts of strong doubt positive t-level cut set strong doubt negative s-level cut set of a bipolar fuzzy set in BCK/BCI-algebras study some of their properties. lso, we investigate the homomorphism preimage doubt image of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras under some conditions.

4 2028. L-MSRWH D. G. HMD Definition 3.1. Let =, be a bipolar fuzzy subset of X. Then is called a doubt bipolar fuzzy H-ideal of X if it satisfies the following conditions for all x, y, z X: 1 0 x 0 x, 2 x z max x y z, y}, 3 x z min x y z, y}. Example 3.1. Let X = 0, a, b, c} be a BCK-algebra with the following Cayley 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 Let =, be a bipolar fuzzy set in X defined by: 0, if x = 0 0.1, if x = 0 x = 0.4, if x = a, c x = 0.2, if x = a, b, c. 0.3, if x = b, Then by routine calculation, we know that =, is a doubt bipolar fuzzy H-ideal of X. n interesting consequence of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras is the following. roposition 3.1. Let be a nonempty subset of X. bipolar fuzzy set =, is defined by n1, x = n 2, x, m1, x = m 2, x, where 0 n 1 n m 2 m 1 0 is a doubt bipolar fuzzy H-ideal of X if only if is an H-ideal of X. roof: The proof is obvious is omitted. The research about the relationships of doubt fuzzy subalgebras crisp subalgebras by level cut sets is usual but important, as it is a tie which can connect abstract algebraic structures fuzzy ones. However, now we encounter a significant challenge that the traditional level cut sets are not suitable for the framework of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras because of the characterization of bipolarity. s a consequence, we defined strong doubt positive t-level cut set strong doubt negative s-level cut set. Definition 3.2. Let =, be a bipolar fuzzy set on X. For any t, s [0, 1] [ 1, 0], the sets t = x X x t} s t = x X x < t} are called a doubt positive t-level cut set a strong doubt positive s-level cut set of =,, respectively. The sets s = x X x s} s s = x X x > s} are called a doubt negative t-level cut set a strong doubt negative s-level cut set of =,, respectively. The sets t,s = x X x t, x s} s t,s = x X x < t, x > s}

5 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS 2029 are called a doubt t, s-level cut set a strong doubt t, s-level cut set of =,, respectively. ote that t,s = t s s t,s = s t s s. For every γ [0, 1], the sets γ = γ γ s γ s γ are called a doubt γ-level cut set a strong doubt γ-level cut set of =,, respectively. roposition 3.2. Let =, be a doubt bipolar fuzzy H-ideal of X let x X. Then x = t, x = s if only if x t, x / u x s, x / v for all u < t, v > s. roof: Let =, be a doubt bipolar fuzzy H-ideal of X let x X. ssume x = t so that x t. If possible, let x u for u < t. Then x u < t. This contradicts the fact that x = t, concluding that x / u for all u < t. lso, assume x = s, then x s. If possible, let x v for v > s. Then x v > s. This contradicts the fact that x = s, concluding that x / v for all v > s. Conversely, let x t, x / u for all u < t. ow, x t x t. Since x / u for all u < t. Therefore, x = t. lso, let x s, x / v for all v > s. ow, x s x s. Since x / v for all v > s, x = s. From Definition 3.2, in the next two theorems, we can easily obtain the relation between doubt bipolar fuzzy H-ideals H-ideals of BCK/BCI-algebras. Theorem 3.1. Let =, be a bipolar fuzzy set over X let t, s [0, 1] [ 1, 0]. If =, is a doubt bipolar fuzzy H-ideal of X, then the nonempty strong doubt t, s-level cut set of is an H-ideal of X. roof: ssume that s t,s for t, s [0, 1] [ 1, 0]. Clearly, 0 s t,s. Let x, y, z X such that x y z s t,s y s t,s. Then x y < t, x y > s, y < t y > s. It follows from Definition 3.1 that x z max x y, y} < maxt, t} = t x z min x y, y} > mins, s} = s, so x z s t,s. Therefore, s t,s is an H-ideal of X. Theorem 3.2. Let =, be a bipolar fuzzy set over X assume that = s t = s s are H-ideals of X for all t, s [0, 1] [ 1, 0]. Then =, is a doubt bipolar fuzzy H-ideal of X. roof: ssume that s t s s are H-ideals of X for all t [0, 1] s [ 1, 0]. Suppose that there exists a X such that 0 > a 0 < a. Taking t o = a, s o = a, implies that a < t o < 0 a > s o > 0. This shows that 0 / s t 0 / s s, which leads to a contradiction. Therefore, 0 x 0 x for all x X. ow, suppose that there are a, b, c X such that a c > max a b c, b}. Then by taking t 1 = 1 2 a c + max a b c, b },

6 2030. L-MSRWH D. G. HMD we have max a b c, b} < t 1 < a c. Hence, a c / s t 1, a b c s t 1 b s t 1, that is s t 1 is not H-ideal of X, which is a contradiction. Therefore, x y max x y z, y} for all x, y, z X. Finally, assume that p, q, r X such that p r < min p q r, q}. Taking s 1 = 1 2 p r + min p q r, q }, then p r < s 1 < min p q r, q}. Therefore, p q r s s 1 q s s 1 but p r / s s 1. gain it is a contradiction. Thus, x z min x y z, y} for all x, y, z X. Hence, =, is a doubt bipolar fuzzy H-ideal of X. If =, is a bipolar fuzzy set in a BCK/BCI-algebra X if f is a self mapping of X, we define mappings [f] : X [0, 1] by [f]x = fx [f] : X [ 1, 0] by [f]x = fx for all x X, respectively. Theorem 3.3. If =, is a doubt bipolar fuzzy H-ideal of X if f is an epimorphism of X, then [f], [f] is a doubt bipolar fuzzy H-ideal of X. roof: Let [f]0 = f0 = 0 y [f]0 = f0 = 0 y for any y X. Since f is an epimorphism of X, then there exists x X such that fx = y. Thus, [f]0 fx = [f]x [f]0 fx = [f]x. s y is an arbitrary element of X, the above result is true for any x X. Moreover, for any x, y, z X, we have [f]x z = fx z = fx fz max fx fy fz, fy } = max fx fy z, fy } = max fx y z, fy } = max [f]x y z, [f]y } [f]x z = fx z = fx fz min fx fy fz, fy } = min fx fy z, fy } = min fx y z, fy } = min [f]x y z, [f]y }. Hence, [f], [f] is a doubt bipolar fuzzy H-ideal of X. Definition 3.3. n H-ideal C of a BCK/BCI-algebra X is said to be characteristic if fc = C for all f utx, where utx is the set of all automorphisms of a BCK/BCI-algebra X. doubt bipolar fuzzy H-ideal =, of X is called characteristic if fx = x fx = x for all x X f utx. For characteristic doubt bipolar fuzzy H-ideals in BCK/BCI-algebras, we have the following theorem. Theorem 3.4. doubt bipolar fuzzy H-ideal is characteristic if only if each of its doubt level cut set is a characteristic H-ideal.

7 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS 2031 roof: Let a doubt bipolar fuzzy H-ideal =, be characteristic, t Im, f utx let x t. Then fx = x t, which means that fx t. Thus, f t t. Since for each x t there exists y X such that fy = x, we have y = fy = x t, we conclude y t. Consequently, x = fy f t. Therefore, t f t ; hence, t = f t. Similarly s = f s. This proves that t s are characteristic. Conversely, if all doubt level cuts of =, are characteristic H-ideals of X, then for x X, f utx x = t > s = x, by roposition 3.2, we have x t, x / s x s, x / t. Thus, fx f t = t fx f s = s, i.e., fx t fx s. For fx = t 1 < t, fx = s 1 > s, we have fx t 1 = f t 1, fx s1 = f s 1, so x t1, x s 1. This is a contradiction. Thus, fx = x fx = x. Hence, =, is characteristic. ow, we discuss the properties of the preimage doubt image of doubt bipolar fuzzy H-ideals by homomorphism of BCK/BCI-algebras. Definition 3.4. If f is a self mapping of a BCK/BCI-algebra X =, is a bipolar fuzzy set in fx, then the bipolar fuzzy set f = f, f in X defined by f = f f = f i.e., f x = fx f x = fx for all x X is called the preimage of under f. Theorem 3.5. n onto homomorphic preimage of a doubt bipolar fuzzy H-ideal is a doubt bipolar fuzzy H-ideal. roof: Let f : X X be an onto homomorphism of BCK/BCI-algebras, =, be a doubt bipolar fuzzy H-ideal of X f = f, f be preimage of under f. For any x X there exists x X such that fx = x. We have f 0 = f0 = 0 x = fx = f x, f 0 = f0 = 0 x = fx = f x, where 0 is the zero element of X. So f 0 f x f 0 f x for all x X. Moreover, for any x, y, z X, we have f x z = fx z = fx fz max fx fy fz, fy } = max fx fy z, fy } = max fx y z, fy } } = max f x y z, y f x z = fx z = fx fz min fx fy fz, fy } = min fx fy z, fy } = min fx y z, fy } = min x y z, y }. Hence, f = f, f is a doubt bipolar fuzzy H-ideal of X.

8 2032. L-MSRWH D. G. HMD Definition 3.5. If =, is a bipolar fuzzy set in X f is a mapping on X, then the doubt image of under f, denoted by f, is a bipolar fuzzy set of fx defined by f = f inf, fsup, where f inf y = f sup y = inf x f 1 y sup x f 1 y x x, for each y fx. Definition 3.6. Let =, be a doubt bipolar fuzzy set in a BCK/BCI-algebra X, then we say that has inf property if for any subset S of X there exists s 1 S such that s 1 = inf r S r we say that has sup property if for any subset T of X there exists t 1 T such that t 1 = sup k. k T Theorem 3.6. n onto homomorphic doubt image of a doubt bipolar fuzzy H-ideal with has inf property has sup property is a doubt bipolar fuzzy H-ideal. roof: Let f : X X be an onto homomorphism of BCK/BCI-algebras, =, be a doubt bipolar fuzzy ideal of X with sup inf properties f be the doubt image of under f. Since =, is a doubt bipolar fuzzy H- ideal of X, we have 0 x 0 x for all x X. ote that 0 f 1 0 where 0 0 are the zeros of X X, respectively. Thus, f inf 0 = inf t = t f x f sup 0 = sup t = 0 x t f 1 0 for all x X. This implies that f inf 0 inf t = f inf x t f 1 x t = f sup x for any x X. f sup 0 sup t f 1 x Moreover, for any x, y, z X, let x o f 1 x, y o f 1 y z o f 1 z such that x o z o = inf t f 1 x z x o z o = sup t f 1 x z t, t, y o = inf t, t f 1 y y o = sup t, t f 1 y x o y o z o = f inf fxo y o z o = f inf x y z = inf x o y o z o x o y o z o f 1 x y z = inf t t f 1 x y z x o y o z o = f sup fxo y o z o = f sup x y z

9 Then, f inf x z = f sup x z = OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS 2033 = sup x o y o z o x o y o z o f 1 x y z = sup t. t f 1 x y z inf t = x o z o t f 1 x z max x o y o z o, y o } } = max inf t, inf t t f 1 x y z t f 1 y = max f inf x y z, f inf y } sup t = x o z o t f 1 x z min x o y o z o, y o } = min sup t, sup t t f 1 x y z t f 1 y = min f sup x y z, f sup y }. Hence, the image f of under f is a doubt bipolar fuzzy H-ideal of X. 4. Equivalence Relations on Doubt Bipolar Fuzzy H-Ideals. In this section, we continue to discuss the relation between doubt bipolar fuzzy H-ideals H-ideals of BCK/BCI-algebras by another means: equivalence relations. Let DBFHI X be the collection of all doubt bipolar fuzzy H-ideals of X. For any t, s [0, 1] [ 1, 0], define two binary relations t s on DBFHI X as follows:, B t t = B t, B s s = B s, for all =, DBFHI X, B = B, B DBFHI X. It is easy to know t s are equivalence relations on DBFHI X. For all =, DBFHI X, let [] t resp., [] s be the equivalence class of =, modular t resp., s. That is, DBFHI X/ t = [] t =, DBFHI X } resp., DBFHI X/ s = [] s =, DBFHI X }. Let HIX be the family of all H-ideals of X. Define maps g t : DBFHI X HIX ϕ}, t, h s : DBFHI X HIX ϕ}, s, for all =, DBFHI X. Then gt h s are clearly well-defined. In the light of the definition of equivalence relations on DBFHI X, we can obtain the following properties. Theorem 4.1. The maps g t h s are surjective for any t, s 0, 1 1, 0. roof: Clearly, a bipolar fuzzy set 1 = 1, 1 is a doubt bipolar fuzzy H-ideal of X, where 1 = 1 1 = 1 for all x X. Then we have g t 1 = 1 t = x X 1 x t } = ϕ g s 1 = 1 s = x X 1 x s } = ϕ. For any M in HIX, consider a bipolar fuzzy set M = M, M in X, where 0, x M M : X [0, 1], M x = 1, x M }

10 2034. L-MSRWH D. G. HMD M : X [ 1, 0], M x = 0, x M 1, x M. By roposition 3.1, M = M, M DBFHI X. ow, we get g t M = M = t x X M x t } = x X M x = 0 } = M h s M = M s = x X M x s } = x X M x = 0 } = M. Therefore, g t h s are surjective. Example 4.1. Consider a BCK-algebra X = 0, a, b, c} which is given in Example 3.1 we consider H-ideal M = 0} HIX. Define a bipolar fuzzy set M = M, M in X as follows: M x = 0, x M 1, x M, M x = 0, x M 1, x M. Clearly, M = M, M DBFHI X. We have gt M = M t = x X M x t } = 0} h s M = M s = x X M x s } = 0}. Thus, the maps g t h s are surjective for all t, s 0, 1 1, 0. ow, a natural question arises here: are there any relationships between the quotient sets the set of all H-ideals of X. In the following, we will concentrate on giving the answers. Theorem 4.2. The quotient sets DBFHI X/ t DBFHI X/ s are equipotent to HIX ϕ} for all t, s 0, 1 1, 0. roof: For all t, s [0, 1] [ 1, 0] =, DBFHI X. Let g t : DBFHI X/ t HIX ϕ}, t be defined by h s : DBFHI X/ s HIX ϕ}, s g t [] t = g t h s[] s = h s, respectively. For any =,, B = B, B DBFHI X, if t = B t s = B s, then, B t, B s, which means [] t = [B] t [] s = [B] s. Thus, g t h s are injective. For any nonempty M in HIX, consider the doubt bipolar fuzzy H-ideal M = M, M which is given in the proof in Theorem 4.1, we have g t [M ] t = g t M = M t = M h s [M ] s = h s M = M s = M. For any doubt bipolar fuzzy H-ideal 1 = 1, 1 of X, we have g t [1] t = gt 1 = 1 t = x X 1 x t } = ϕ hs [1] s = hs 1 = 1 s = x X 1 x s } = ϕ. Hence, gt h s are surjective. This completes the proof. For any 0 < γ < 1, we define another relation X γ on DBFHI X as follows:, B X γ γ = B γ, where γ = γ γ B γ = Bγ B γ. Then the relation X γ is also an equivalence relation on DBFHI X. Theorem 4.3. Let 0 < γ < 1. Then the map Φ γ : DBFHI X IX ϕ} defined by Φ γ = γ is surjective.

11 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS 2035 roof: Let 0 < γ < 1, for 1 = 1, 1 DBFHI X, we have Φγ 1 = 1 γ 1 γ = ϕ. For any nonempty M in DBFHI X, considering a doubt bipolar fuzzy H-ideal M = M, M which is given in the proof of Theorem 4.1, we obtain Φγ M = M γ = M γ M γ = x X M x γ } x X M x γ } = M. Therefore, Φ γ is surjective. Theorem 4.4. Let 0 < γ < 1. Then the quotient set DBFHI X/X γ is equipotent to IX ϕ}. roof: Suppose that 0 < γ < 1 Φ γ: DBFHI X/X γ IX ϕ} is a map defined by Φγ []X γ = Φγ for all [] X γ DBFHI X/X γ. Let Φ γ[] X γ = Φ γ[b] X γ for every [] X γ [B] X γ DBFHI X/X γ, then ϕ γ = ϕ γ B, i.e., γ = B γ. It implies that, B X γ. Thus, [] X γ = [B] X γ Φ γ is injective. Moreover, for any nonempty M in IX, we consider the doubt bipolar fuzzy H-ideal M = M, M which is given in the proof of Theorem 4.1, then we have Φ γ [M ] X γ = Φ γ M = M γ = x X M x γ } x X M x γ } = M. On the other h, for 1 = 1, 1 DBFHI X, we have Φ γ [1] X γ = ϕ γ 1 = 1 γ 1 γ = M. Therefore, Φ γ is surjective. This completes the proof. 5. Cartesian roduct of Doubt Bipolar Fuzzy H-Ideals. In this section, we give the definition of the Cartesian product of two doubt bipolar fuzzy H-ideals of two BCK/BCIalgebras X Y. lso, we provide some of their properties. In what follows, X Y are BCK/BCI-algebras, so we use X Y ;, 0, 0 to denote a BCK/BCI-algebra unless otherwise specified. For the sake of brevity, we call X Y a BCK/BCI-algebras. Definition 5.1. Let X, X, 0 X Y, Y, 0 Y be two BCK/BCI-algebras. The Cartesian product of X Y is defined to be the set X Y = x, y x X, y Y }. In X Y we define the product X Y as follows: x, y X Y u, v = x X u, y Y v for all x, y, u, v X Y. One can easily verify that the Cartesian product of two BCK/BCI-algebras is again a BCK/BCI-algebra. ow, we write the following definition. Definition 5.2. Let X be a BCK/BCI-algebra let =, B = B, B be two doubt bipolar fuzzy H-ideals of X. The Cartesian product of B is defined by B = B, B, where B : X X [0, 1] is given by Bx, y = max x, y } B : X X [ 1, 0] is given by Bx, y = min x, y } for all x, y X X. In the following, we extend the above definition to the Cartesian product of doubt bipolar fuzzy H-ideals of any BCK/BCI-algebras X Y. Definition 5.3. Let X Y be two BCK/BCI-algebras let =, B = B, B be two doubt bipolar fuzzy H-ideals of X Y, respectively. The Cartesian product of B is defined by B = B, B, where B : X Y [0, 1] is given by Bx, y = max x, y }

12 2036. L-MSRWH D. G. HMD B : X Y [ 1, 0] is given by Bx, y = min x, y } for all x, y X Y. Definition 5.4. bipolar fuzzy set B = B, B of a BCK/BCI-algebra X Y is called a doubt bipolar fuzzy H-ideal of X Y if it satisfies the following conditions for all x, y, u, v, w, z X Y : 1 B 0, 0 B x, y B 0, 0 B x, y, 2 B x, y w, z max B x, y u, v w, z, B u, v}, 3 B x, y w, z min B x, y u, v w, z, B u, v}. Example 5.1. Consider a BCK-algebra X = 0, a, b, c} a doubt bipolar fuzzy H-ideal =, of X which are given in Example 3.1. Define a doubt bipolar fuzzy H-ideal B = B, B in X as follows: 0.3, if x = 0 0.3, if x = 0 Bx = 0.5, if x = a B x = 0.4, if x = a, b, c. 0.6, if x = b, c, Let B = B, B, where B = max, } B = min, } are defined as: 0.3, if x, y = 0, 0, b, 0 0.4, if x, y = a, 0, c, 0 Bx = 0.5, if x, y = 0, a, a, a, b, a, c, a 0.6, otherwise, 0.3, if x, y = 0, 0, a, 0, b, 0, c, 0 Bx = 0.4, otherwise. By routine calculations, we know that B = B, B is a doubt bipolar fuzzy H-ideal of X X. Theorem 5.1. Let =, B = B, B be two doubt bipolar fuzzy H-ideals of BCK /BCI -algebras X Y, respectively. Then B = B, B is a doubt bipolar fuzzy H-ideal of X Y. roof: For any x, y X Y, we have B 0, 0=max 0, B 0} max x, B y} = B x, y B 0, 0 = min 0, B 0} min x, B y} = B x, y. ow, for any x, y, u, v, w, z X Y, we have Bx, y w, z = Bx w, y z = max x w, y z } max max x u, w, u }, max By v, z, Bv }} = max max x u, w, By v, z }, max u, Bv }} = max } B x u w, y v z, B u, v } = max Bx, y u, v w, z, Bu, v } Bx, y w, z = Bx w, y z = min x w, y z }

13 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS 2037 min min x u, w, u }, min B y v, z, B v }} = min min x u, w, B y v, z }, min u, B v }} = min } B x u w, y v z, B u, v } = min Bx, y u, v w, z, Bu, v }. Hence, B = B, B is a doubt bipolar fuzzy H-ideal of a BCK/BCI-algebra X Y. roposition 5.1. Let B = B, B be a doubt bipolar fuzzy H-ideal of X Y. If x, y u, v, then x, y u, v x, y u, v for all x, y, u, v X Y. roof: Let x, y, u, v X Y such that x, y u, v. Then x, y u, v = 0, 0. ow x, y = x, y 0, 0 max x, y u, v 0, 0, u, v } = max x, y u, v, u, v } = max 0, 0, u, v } = u, v. Therefore, x, y u, v for all x, y, u, v X Y. gain, x, y = x, y 0, 0 min x, y u, v 0, 0, u, v } = min x, y u, v, u, v } = min 0, 0, u, v } = u, v. Therefore, x, y u, v for all x, y, u, v X Y. roposition 5.2. Let B = B, B be a doubt bipolar fuzzy H-ideal of X Y such that Bx, y u, v Bu, v Bx, y u, v Bu, v for all x, y, u, v X Y, then B is constant. roof: ote that in a BCK/BCI-algebra X Y, x, y 0, 0 = x, y for all x, y X Y, by using assumption we have B x, y = B x, y 0, 0 B 0, 0 B x, y = B x, y 0, 0 B 0, 0. It follows from Definition 5.4, B x, y = B 0, 0 B x, y = B 0, 0 for all x, y, u, v X Y. Therefore, X Y is constant. roposition 5.3. Let =, B = B, B be two doubt bipolar fuzzy H- ideals of BCK/BCI-algebras X Y, respectively. If B is a doubt bipolar fuzzy H-ideal of X Y, the following are true: 1 0 B y B 0 x for all x X, y Y, 2 0 B y B 0 x for all x X, y Y. roof: 1 ssume that 0 > B y B 0 > x for some x X, y Y. Then Bx, y = max x, By } < max 0, B0 } = B0, 0,

14 2038. L-MSRWH D. G. HMD which is a contradiction. Therefore, 0 B y B 0 x for all x X, y Y. 2 ssume that 0 < B y B 0 < x for some x X, y Y. Then Bx, y = min x, B y } > min 0, B 0 } = B0, 0, which is also a contradiction. Therefore, 0 B y B 0 x for all x X, y Y. Theorem 5.2. Let =, B = B, B be two bipolar fuzzy subsets of X Y, respectively, such that B is a doubt bipolar fuzzy H-ideal of X Y. Then either is a doubt bipolar fuzzy H-ideal of X or B is a doubt bipolar fuzzy H-ideal of Y. roof: Since B is a doubt bipolar fuzzy H-ideal of X Y, then for all x, y, u, v, w, z X Y, we have B x, y w, z max B x, y u, v w, z, B u, v }. By putting y = z = v = 0, we have Bx, 0 w, 0 max Bx, 0 u, 0 w, 0, Bu, 0 }. 1 lso, we have Bx, 0 w, 0 = Bx w, 0 0 = max x w, B0 0 } = x w 2 Bx, 0 u, 0 w, 0 = Bx, 0 u w, 0 0 gain, by using roposition 5.3, we have = Bx u w, 0 0, 0 = max x u w, B0 0, 0 } = x u w. 3 Bu, 0 = max u, B0 } = u. 4 So, from 1, 2, 3 4 we get, x w max x u, w, u}. Similarly, we can prove x w min x u, w, u}. Hence, is a doubt bipolar fuzzy H-ideal of X. roposition 5.4. Let B = B, B be a doubt bipolar fuzzy H-ideal of a BCKalgebra X Y, then B 0, 0 0, 0 x, y B x, y B 0, 0 0, 0 x, y B x, y for all x, y X Y. roof: ote that B0, 0 0, 0 x, y max B0, 0 x, y 0, 0 x, y, Bx, y } = max B0, 0 x, y 0, 0, Bx, y } = max B0, 0 x, y, Bx, y } = max B0, 0, Bx, y } = Bx, y for all x, y X Y. Therefore, B 0, 0 0, 0 x, y B x, y for all x, y X Y. gain, B0, 0 0, 0 x, y min B0, 0 x, y 0, 0 x, y, Bx, y }

15 OVEL COCETS OF DOUBT BIOLR FUZZY H-IDELS 2039 = min B0, 0 x, y 0, 0, Bx, y } = min B0, 0 x, y, Bx, y } = min B0, 0, Bx, y } = Bx, y for all x, y X Y. Therefore, B 0, 0 0, 0 x, y B x, y for all x, y X Y. Corollary 5.1. Let B = B, B be a doubt bipolar fuzzy H-ideal of X Y. Then the sets D B = x, y X Y B x, y = B 0, 0} D B = x, y X Y B x, y = B 0, 0} are H-ideals of X Y. roof: Let B = B, B be a doubt bipolar fuzzy H-ideal of X Y. Obviously, 0, 0 D B 0, 0 D B. ow, let x, y, u, v, w, z D B such that x, y u, v w, z, u, v D B. Then B x, y u, v w, z = B 0, 0 = B u, v. ow B x, y w, z max B x, y u, v w, z, u, v} = 0, 0. gain, since B = B, B is a doubt bipolar fuzzy H-ideal of X Y, so B 0, 0 x, y w, z. Therefore, B 0, 0 = B x, y w, z. It follows that x, y w, z D B for all x, y, u, v, w, z X Y. Therefore, D B is an H-ideal of X Y. Similarly, we can prove that D B is an H-ideal of X Y. Definition 5.5. Let B = B, B be a doubt bipolar fuzzy H-ideal of X Y, t, s [0, 1] [ 1, 0]. Then the doubt positive t-level cut set the doubt negative s-level cut set of B are as follows: B t = x, y X Y Bx, y t } B s = x, y X Y Bx, y s }. Theorem 5.3. Let B = B, B be a bipolar fuzzy set of X Y. Then B = B, B is a doubt bipolar fuzzy H-ideal of X Y if only if B t B s are H-ideals of X Y for all t, s [0, 1] [ 1, 0]. roof: ssume that B = B, B is a doubt bipolar fuzzy H-ideal of X Y t, s [0, 1] [ 1, 0] such that B t B s. Let a, b B t c, d B s. Then we have B a, b t B c, d s. So we deduce that B 0, 0 B a, b t B 0, 0 B c, d s. This shows that 0, 0 B t 0, 0 B s. Let x, y, u, v, w, z X Y such that x, y u, v w, z B t u, v B t, let x, y, u, v, w, z X Y such that x, y u, v w, z B s u, v B s. Then B x, y u, v w, z t, B u, v t, B x, y u, v w, z s B u, v s. Since B is a doubt bipolar fuzzy H-ideal of X Y, it follows that Bx, y w, z max Bx, y u, v w, z, Bu, v } maxt, t} = t Bx, y w, z min Bx, y u, v w, z, Bu, v } mins, s} = s,

16 2040. L-MSRWH D. G. HMD so x, y w, z B t x, y w, z B s. Therefore, B t B s are H-ideals of X Y. Conversely, assume that B t B t are H-ideals of X Y for all t, s [0, 1] [ 1, 0]. Let x, y X Y such that B 0, 0 > B x, y B 0, 0 < B x, y. By taking t o = 1 [ 2 B 0, 0 + Bx, y ], s o = 1 [ 2 B 0, 0 + Bx, y ], we get B 0, 0 > t o > B x, y B 0, 0 < s o < B x, y. Therefore, x, y B t o, 0, 0 B t o, x, y B s o 0, 0 B s o. This is a contradiction. Hence, B 0, 0 B x, y B 0, 0 B x, y for all x, y X Y. gain, we assume that x, y, u, v, w, z X Y such that B x, y w, z > max B x, y u, v w, z, B u, v} B x, y w, z < min B x, y u, v w, z, B u, v}. Then by taking t 1 = 1 [ 2 B x, y w, z + max Bx, y u, v w, z, Bu, v }], s 1 = 1 2 we have [ B x, y w, z + max Bx, y u, v w, z, Bu, v }], Bx, y w, z > t 1 > max Bx, y u, v w, z, Bu, v }, Bx, y w, z < s 1 < min Bx, y u, v w, z, Bu, v }. This shows that, x, y u, v w, z B t 1, u, v B t 1, but x, y w, z / B t 1, which is a contradiction. lso, x, y u, v w, z B s 1, u, v B s 1, but x, y w, z / B s 1, again this is a contradiction. Therefore, Bx, y w, z max Bx, y u, v w, z, Bu, v } Bx, y w, z min Bx, y u, v w, z, Bu, v } for all x, y, u, v, w, z X Y. Hence, B = B, B is a doubt bipolar fuzzy H-ideal of X Y. 6. Conclusions. To investigate the structure of an algebraic system, it is clear that doubt bipolar fuzzy H-ideals with special properties play an important role. In this paper, we introduced the notion of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras investigated related properties. We characterized strong doubt positive t-level cut set, strong doubt negative s-level cut set, homomorphism equivalence relation by considering doubt bipolar fuzzy H-ideals of BCK/BCI-algebras. lso, we considered the structure of BCK/BCI-algebras defined Cartesian product of two doubt bipolar fuzzy H-ideals. We presented some interesting results about Cartesian product of two doubt bipolar fuzzy H-ideals of BCK/BCI-algebras. In addition, we proved that the Cartesian product of two bipolar fuzzy sets becomes doubt bipolar fuzzy H-ideals if only if for any t, s [0, 1] [ 1, 0], doubt positive t-level cut set doubt negative s- level cut set are H-ideals of a BCK/BCI-algebra X Y. In our opinion, these definitions main results can be similarly extended to some other fuzzy algebraic systems such as groups, semigroups, rings, nearrings, semirings hemirings, lattices Lie algebras.

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