EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 652-670 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On some properties of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras Anas Al-Masarwah1,∗, Abd Ghafur Ahmad1 1 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia Abstract. In this research article, we study some properties of doubt bipolar fuzzy H-ideals in BCK/ BCI-algebras. Doubt bipolar fuzzy H-ideals are connected with doubt bipolar fuzzy sub- algebras and doubt bipolar fuzzy ideals. Moreover, doubt bipolar fuzzy H-ideals are characterized using doubt positive t-level cut set, doubt negative s-level cut set and H-Artin BCK/BCI-algebras. 2010 Mathematics Subject Classifications: 03G25, 06F35, 08A72 Key Words and Phrases: BCK/BCI-algebras, Doubt fuzzy ideals, Doubt fuzzy H-ideals, Doubt bipolar fuzzy subalgebras, Doubt bipolar fuzzy ideals, Doubt bipolar fuzzy H-ideals 1. Introduction In 1965, Zadeh [30] introduced the concept of fuzzy set to handle the uncertainties in our daily life. Fuzzy sets are extremely useful to solve many problems in applied mathematics, information sciences and decision making. After that many generalizations of fuzzy sets are presented, for example, interval valued fuzzy sets [31] and intuitionistic fuzzy sets [7]. Lee [22] introduced the notion of bipolar fuzzy sets which is an extension of fuzzy sets. Fuzzy sets give a degree of membership of an element in a given set, whereas bipolar fuzzy sets give both a positive membership degree belongs to the interval [0, 1] and a negative membership degree belongs to the interval [-1, 0]. In the case of bipolar fuzzy sets, the membership degrees range is enlarged from the interval [0, 1] to the interval [-1, 1]. Recently, the theory of bipolar fuzzy sets becomes a vigorous area of research in different domains such as group theory, semigroup theory, ring theory, semiring theory, graph theory, engineering, physics, statics, medical science, social science, artificial intelligent, computer networks, expert systems, decision making and so on. BCK-algebras introduced by Imai and Iséki [11] as a generalization of notion of the concept of set theoretic difference and propositional calculus and then Iséki [12] introduced ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3288 Email addresses: almasarwah85@gmail.com (A. Al-Masarwah), ghafur@ukm.edu.my (A. G. Ahmad) http://www.ejpam.com 652 c© 2018 EJPAM All rights reserved. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 653 the notion of BCI-algebras which is a generalization of BCK-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 [26] and later these ideas have been applied to other algebraic structures such as semigroups, rings, semirings, hemirings, ideals, modules and vector spaces. In 1991, Xi [29] applied the concept of fuzzy sets to BCK-algebras. After that, Jun [15] and Ahmad [1] applied the concept of fuzzy sets to BCI-algebras. Jun [14] provided characterizations of Noetherian BCK-algebras in terms of fuzzy ideals. Fuzzy H-ideals of BCI-algebras introduced in [20] by Khalid and Ahmad and the concept of H-Noetherian BCK-algebras was studied in [33] by Zhan and Tan. Huang [9] fuzzified BCI-algebras in little different ways. Jun [16] renamed Huanǵs definition as doubt fuzzy ideals in BCK/BCI-algebras and introduced the concepts of doubt fuzzy subalgebras and doubt fuzzy ideals in BCK/BCI-algebras. Zhan and Tan [32] introduced the concept of doubt fuzzy H-ideals and provided characterizations of H-Artin BCK-algebras in terms of doubt fuzzy H-ideals. Muhiuddin and Aldhafeeri [24] introduced the notions of uni- hesitant fuzzy algebras and uni-hesitant fuzzy (closed) ideals in BCK-algebras and BCI- algebras. Muhiuddin et al. [25] introduced hesitant fuzzy translations and extensions of subalgebras and ideals in BCK/BCI-algebras. Also, Jun et al. [17–19] studied the notions of subalgebras and ideals of BCK/BCI-algebras based on hesitant fuzzy soft sets, double-framed soft sets and cubic soft sets. In 2009, Lee [21] applied the concept of bipolar fuzzy set theory to BCK/BCI- algebras, and introduced the notions of bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras. Recently, the notion of bipolar fuzzy set theory was applied to BCK/BCI-algebras [4, 23] and other algebraic structures such that Lie algebras [2], Lie superalgebras [3], hemirings [8] and BF -algebras [27, 28], etc. Al-Masarwah and Ahmad [5] introduced the notions of doubt bipolar fuzzy subalgebras and doubt bipolar fuzzy ideals in BCK/BCI-algebras. Also, they introduced the concept of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras and investigated some interesting properties [6]. This paper is a continuation of the papers [5] and [6]. We study some properties of doubt bipolar fuzzy H-ideals in BCK/ BCI-algebras. We provide relations between a doubt bipolar fuzzy H-ideal and a doubt bipolar fuzzy ideal. We give conditions for a doubt bipolar fuzzy ideal to be a doubt bipolar fuzzy H-ideal. We investigate characterizations of doubt bipolar fuzzy H-ideals by means of doubt positive t-level cut set, doubt negative s-level cut set and H-Artin BCK/BCI-algebras. 2. Preliminaries We first recall some elementary aspects which are used to present the paper. A BCK/BCI-algebra is an important class of logical algebras introduced by Imai and Iséki [11, 12] and 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 : A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 654 (I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (II) (x ∗ (x ∗ y)) ∗ y = 0, (III) x ∗ x = 0, (IV) x ∗ y = 0 and y ∗ x = 0 imply x = y. If a BCI-algebra X satisfies 0 ∗ x = 0, then X is called a BCK-algebra. In a BCK/BCI-algebra, x∗0 = x hold. A BCI-algebra is said to be associative if (x∗y)∗z = x ∗ (y ∗ z) for all x, y, z ∈ X. A partial ordering ≤ on a BCK/BCI-algebra X can be defined by x ≤ y if and only if x∗y = 0. Any BCK/BCI-algebra X satisfies the following axioms for all x, y, z ∈ X: (1) x ∗ 0 = x, (2) (x ∗ y) ∗ z = (x ∗ z) ∗ y, (3) x ∗ y ≤ x, (4) (x ∗ y) ∗ z ≤ (x ∗ z) ∗ (y ∗ z), (5) x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x. Definition 1. [29] A non-empty subset S of a BCK/BCI-algebra X is called an ideal of X if (i) 0 ∈ S (ii) x ∗ y ∈ S and y ∈ S then x ∈ S, for all x, y ∈ X. Definition 2. [20] A non-empty subset S of a BCK/BCI-algebra X is called an H-ideal of X if (i) 0 ∈ S (ii) x ∗ (y ∗ z) ∈ S and y ∈ S then x ∗ z ∈ S, for all x, y, z ∈ X. We refer the reader to [10, 13] for further information regarding BCK/BCI-algebras. In what follows, we use (X; ∗, 0) to denote a BCK/BCI-algebra unless otherwise specified. For the sake of brevity, we call X a BCK/BCI-algebra. Definition 3. [16] A fuzzy set A = {(x, µA(x)) | x ∈ X} in X is called a doubt fuzzy ideal of X if (i) µA(0) ≤ µA(x), (ii) µA(x) ≤ max{µA(x ∗ y), µA(y)}, for all x, y ∈ X. Definition 4. [32] A fuzzy set A = {(x, µA(x)) | x ∈ X} in X is called a doubt fuzzy H-ideal of X if A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 655 (i) µA(0) ≤ µA(x), (ii) µA(x ∗ z) ≤ max{µA(x ∗ (y ∗ z)), µA(y)}, for all x, y, z ∈ X. The proposed work is done on a bipolar fuzzy set. The formal definition of a bipolar fuzzy set is given below: Definition 5. [22] Let X be a non-empty set. A bipolar fuzzy set A in X is an object having the form A = {(x, µPA(x), µNA (x))|x ∈ X} where µPA : X −→ [0, 1] and µNA : X −→ [−1, 0] are mappings. We use the positive membership degree µPA(x) to denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set A, and the negative membership degree µNA (x) to denote the satisfaction degree of an element x to some implicit counter-property corresponding to a bipolar fuzzy set A. If µPA(x) 6= 0 and µNA (x) = 0, it is the situation that x is regarded as having only positive satisfaction for A. If µPA(x) = 0 and µNA (x) 6= 0, it is the situation that x does not satisfy the property of A but somewhat satisfies the counter property of A. It is possible for an element x to be such that µPA(x) 6= 0 and µNA (x) 6= 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 A = (µPA, µ N A ) for the bipolar fuzzy set A = {(x, µPA(x), µNA (x))|x ∈ X}. Definition 6. [22] Let A = (µPA(x), µNA (x)) and B = (µPB(x), µNB (x)) be two bipolar fuzzy sets in X. Then A ⊆ B if and only if µPA(x) ≤ µPB(x) and µNA (x) ≥ µNB (x), for all x ∈ X. Doubt bipolar fuzzy subalgebras and doubt bipolar fuzzy ideals are extensions of doubt fuzzy subalgebras and doubt fuzzy ideals which are defined by Al-Masarwah and Ahmad [5] as follows: Definition 7. [5] A bipolar fuzzy set A = (µPA, µ N A ) in X is called a doubt bipolar fuzzy subalgebra of X if it satisfies: (i) µPA(x ∗ y) ≤ max{µPA(x), µPA(y)}, (ii) µNA (x ∗ y) ≥ min{µNA (x), µNA (y)}, for all x, y ∈ X. Definition 8. [5] A bipolar fuzzy set A = (µPA, µ N A ) in X is called a doubt bipolar fuzzy ideal of X if it satisfies: (i) µPA(0) ≤ µPA(x) and µNA (0) ≥ µNA (x), (ii) µPA(x) ≤ max{µPA(x ∗ y), µPA(y)}, (iii) µNA (x) ≥ min{µNA (x ∗ y), µNA (y)}, for all x, y ∈ X. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 656 3. Doubt bipolar fuzzy H-ideals In this section, the concepts of doubt bipolar fuzzy H-ideals were introduced by Al- Masarwah and Ahmad [6] will be used to study and investigate several properties of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras. Definition 9. [6] Let A = (µPA, µ N A ) be a bipolar fuzzy subset of X, then A is called a doubt bipolar fuzzy H-ideal of X if it satisfies: (i) µPA(0) ≤ µPA(x) and µNA (0) ≥ µNA (x), (ii) µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)}, (iii) µNA (x ∗ z) ≥ min{µNA (x ∗ (y ∗ z)), µNA (y)}, for all x, y, z ∈ X. Definition 10. Let M be a nonempty subset of X. A bipolar fuzzy set C̃M = (C̃PM , C̃ N M ) expressed by C̃PM (x) = { 0, x ∈M, 1, x 6∈M, and C̃PM (x) = { 0, x ∈M, −1, x 6∈M. is called a doubt bipolar fuzzy characteristic function. Lemma 1. Let M be a nonempty subset of X. Then the constant 0 of X is in M if and only if C̃PM (0) ≤ C̃PM (x) and C̃NM (0) ≥ C̃NM (x), for all x ∈ X. Proof. If 0 ∈ M, then C̃PM (0) = 0 and C̃NM (0) = 0. Thus, C̃PM (0) = 0 ≤ C̃PM (x) and C̃NM (0) = 0 ≥ C̃NM (x), for all x ∈ X. Conversely, assume that C̃PM (0) ≤ C̃PM (x) and C̃NM (0) ≥ C̃NM (x), for all x ∈ X. Since M is a nonempty subset of X, we have m ∈M for some m ∈ X. Then C̃PM (0) ≤ C̃PM (m) = 0 and C̃NM (0) ≥ C̃PM (m) = 0. Thus, C̃PM (0) = 0 and C̃NM (0) = 0. So 0 ∈M. Theorem 1. Let M be a nonempty subset of X. Then M is an H-ideal of X if and only if the doubt bipolar fuzzy characteristic function C̃M = (C̃PM , C̃ N M ) is a doubt bipolar fuzzy H-ideal of X. Proof. Assume that M is an H-ideal of X. Since 0 ∈M, it follows from Lemma 1 that C̃PM (0) ≤ C̃PM (x) and C̃NM (0) ≥ C̃NM (x), for all x ∈ X. Next, let x, y, z ∈ X. Then we have the following cases: Case(1). Suppose that x ∗ (y ∗ z) ∈ M and y ∈ M, then C̃PM (x ∗ (y ∗ z)) = 0, C̃PM (y) = 0, C̃NM (x ∗ (y ∗ z)) = 0, and C̃NM (y) = 0. Therefore, max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)} = max{0, 0} = 0, and min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)} = min{0, 0} = 0. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 657 Since x ∗ (y ∗ z) ∈M and y ∈M, we have x ∗ z ∈M. So C̃PM (x ∗ z) = 0 and C̃NM (x ∗ z) = 0. Therefore, C̃PM (x ∗ z) = 0 ≤ 0 = max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)}, and C̃NM (x ∗ z) = 0 ≥ 0 = min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)}. Case(2). Suppose that x ∗ (y ∗ z) 6∈ M and y 6∈ M, then C̃PM (x ∗ (y ∗ z)) = 1, C̃PM (y) = 1, C̃NM (x ∗ (y ∗ z)) = −1, and C̃NM (y) = −1. So, max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)} = max{1, 1} = 1, and min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)} = min{−1,−1} = −1. Therefore, C̃PM (x ∗ z) ≤ 1 = max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)}, and C̃NM (x ∗ z) ≥ −1 = min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)}. Case(3). Suppose that x ∗ (y ∗ z) ∈M or y ∈M. Then we have two subcases: Subcase (3a). If x ∗ (y ∗ z) ∈ M and y 6∈ M, then C̃PM (x ∗ (y ∗ z)) = 0, C̃PM (y) = 1, C̃NM (x ∗ (y ∗ z)) = 0, and C̃NM (y) = −1. So, max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)} = max{0, 1} = 1, and min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)} = min{0,−1} = −1. Therefore, C̃PM (x ∗ z) ≤ 1 = max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)}, and C̃NM (x ∗ z) ≥ −1 = min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)}. Subcase (3b). If x ∗ (y ∗ z) 6∈ M and y ∈ M, then C̃PM (x ∗ (y ∗ z)) = 1, C̃PM (y) = 0, C̃NM (x ∗ (y ∗ z)) = −1, and C̃NM (y) = 0. So, max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)} = max{1, 0} = 1, and min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)} = min{−1, 0} = −1. Therefore, C̃PM (x ∗ z) ≤ 1 = max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)}, and C̃NM (x ∗ z) ≥ −1 = min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)}. Hence, C̃M = (C̃PM , C̃ N M ) is a doubt bipolar fuzzy H-ideal of X. Conversely, assume that C̃M = (C̃PM , C̃ N M ) is a doubt bipolar fuzzy H-ideal of X. Since C̃PM (0) ≤ C̃PM (x) and C̃NM (0) ≥ C̃NM (x), for all x ∈ X. It follows that from Lemma 1 that A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 658 0 ∈M. Next, let x, y, z ∈ X such that x ∗ (y ∗ z) ∈M and y ∈M. To show that x ∗ z ∈M, assume that x ∗ z 6∈M. Then C̃PM (x ∗ z) = 1 and C̃NM (x ∗ z) = −1. So 1 = C̃PM (x ∗ z) ≤ max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)}, and −1 = C̃NM (x ∗ z) ≥ min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)}. Thus, max{C̃PM (x ∗ (y ∗ z)), C̃PM (y)} = 1, and min{C̃NM (x ∗ (y ∗ z)), C̃NM (y)} = −1. This implies that C̃PM (x∗(y∗z)) = 1 or C̃PM (y) = 1 and C̃NM (x∗(y∗z)) = −1 or C̃NM (y) = −1. So, x ∗ (y ∗ z) 6∈M or y 6∈M, a contradiction. Hence, x ∗ z ∈M, and thus M is an H-ideal of X. Theorem 2. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of associative BCK/BCI- algebras X. If the inequality x ∗ y ≤ z holds in X, then µPA(x ∗ y) ≤ µPA(z) and µNA (x ∗ y) ≥ µNA (z) for all x, y, z ∈ X. Proof. Let x, y, z ∈ X such that x ∗ y ≤ z. Then (x ∗ y) ∗ z = 0 and since A is a doubt bipolar fuzzy H-ideal of X, so µPA(x ∗ y) ≤ max{µPA(x ∗ (z ∗ y)), µPA(z)} = max{µPA((x ∗ z) ∗ y), µPA(z)} = max{µPA((x ∗ y) ∗ z), µPA(z)} = max{µPA(0), µPA(z)} = µPA(z). Therefore, µPA(x ∗ y) ≤ µPA(z) for all x, y, z ∈ X. Again, µNA (x ∗ y) ≥ min{µNA (x ∗ (z ∗ y)), µNA (z)} = min{µNA ((x ∗ z) ∗ y), µNA (z)} = min{µNA ((x ∗ y) ∗ z), µNA (z)} = min{µNA (0), µNA (z)} = µNA (z). Therefore, µNA (x ∗ y) ≥ µNA (z) for all x, y, z ∈ X. Proposition 1. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of X. If the inequality x ≤ y holds in X, then µPA(x) ≤ µPA(y) and µNA (x) ≥ µNA (y) for all x, y ∈ X. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 659 Proof. Let x, y ∈ X such that x ≤ y. Then x ∗ y = 0. Now µPA(x) = µPA(x ∗ 0) ≤ max{µPA(x ∗ (y ∗ 0)), µPA(y)} = max{µPA(x ∗ y), µPA(y)} = max{µPA(0), µPA(y)} = µPA(y). Therefore, µPA(x) ≤ µPA(y) for all x, y ∈ X. Again, µNA (x) = µNA (x ∗ 0) ≥ min{µNA (x ∗ (y ∗ 0)), µNA (y)} = min{µNA (x ∗ y), µNA (y)} = min{µNA (0), µNA (y)} = µNA (y). Therefore, µNA (x) ≥ µNA (y) for all x, y ∈ X. Proposition 2. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of a BCK-algebra X, then µPA(0 ∗ (0 ∗ x)) ≤ µPA(x) and µNA (0 ∗ (0 ∗ x)) ≥ µNA (x) for all x ∈ X. Proof. Note that µPA(0 ∗ (0 ∗ x)) ≤ max{µPA(0 ∗ (x ∗ (0 ∗ x))), µPA(x)} = max{µPA(0 ∗ (x ∗ 0)), µPA(x)} = max{µPA(0 ∗ x), µPA(x)} = max{µPA(0), µPA(x)} = µPA(x), for all x ∈ X. Therefore, µPA(0 ∗ (0 ∗ x)) ≤ µPA(x) for all x ∈ X. Again, µNA (0 ∗ (0 ∗ x)) ≥ min{µNA (0 ∗ (x ∗ (0 ∗ x))), µNA (x)} = min{µNA (0 ∗ (x ∗ 0)), µNA (x)} = min{µNA (0 ∗ x), µNA (x)} = min{µNA (0), µNA (x)} = µNA (x), for all x ∈ X. Therefore, µNA (0 ∗ (0 ∗ x)) ≥ µNA (x) for all x ∈ X. Theorem 3. Every doubt bipolar fuzzy H-ideal of X is both a doubt bipolar fuzzy subalgebra of X and a doubt bipolar fuzzy ideal of X. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 660 Proof. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of X, then for any x, y ∈ X, we have µPA(x ∗ y) ≤ max{µPA(x ∗ (y ∗ y)), µPA(y)} = max{µPA(x ∗ 0), µPA(y)} = max{µPA(x), µPA(y)}, and µNA (x ∗ y) ≥ min{µNA (x ∗ (y ∗ y)), µNA (y)} = min{µNA (x ∗ 0), µNA (y)} = min{µNA (x), µNA (y)}. Hence, A = (µPA, µ N A ) is a doubt bipolar fuzzy subalgebra of X. Also, since A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. Then µPA(0) ≤ µPA(x) and µNA (0) ≥ µNA (x). Now, since x ∗ 0 = x for all x ∈ X, we obtain µPA(x) = µPA(x ∗ 0) ≤ max{µPA(x ∗ (y ∗ 0)), µPA(y)} = max{µPA(x ∗ y), µPA(y)}, and µNA (x) = µNA (x ∗ 0) ≥ min{µNA (x ∗ (y ∗ 0)), µNA (y)} = min{µNA (x ∗ y), µNA (y)}. Therefore, A = (µPA, µ N A ) is a doubt bipolar fuzzy ideal of X. The converse of Theorem 3 is not true. That is every doubt bipolar fuzzy subalgebra of X and doubt bipolar fuzzy ideal of X is not necessarily to be a doubt bipolar fuzzy H-ideal of X. It can be verified by the following example: Example 1. Let X = {0, a, b} be a BCI-algebra with the Cayley table which is appeared in Table 1. Table 1: Cayley table for the ∗-operation. ∗ 0 a b 0 0 b a a a 0 b b b a 0 Define a bipolar fuzzy set A = (µPA, µ N A ) in X as follows: µPA(x) = { 0, if x = 0 0.8, if x = a, b, and µNA (x) = { −0.2, if x = 0 −0.4, if x = a, b. Then, A = (µPA, µ N A ) is a doubt bipolar fuzzy subalgebra of X and a doubt bipolar fuzzy ideal of X. But A = (µPA, µ N A ) is not a doubt bipolar fuzzy H-ideal of X, since µPA(a ∗ b) = 0.8 max{µPA(a ∗ (0 ∗ b)), µPA(0)} = µPA(0) = 0. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 661 In the following example, we have a doubt bipolar fuzzy subalgebra of X but it is neither a doubt bipolar fuzzy ideal of X nor a doubt bipolar fuzzy H-ideal of X. Example 2. Let X = {0, a, b, c} be a BCK-algebra with the Cayley table which is appeared in Table 2. Table 2: Cayley table for the ∗-operation. ∗ 0 a b c 0 0 0 0 0 a a 0 0 a b b a 0 b c c c c 0 Define a bipolar fuzzy set A = (µPA, µ N A ) in X as follows: µPA(x) = { 0.5, if x = 0, a, c 0.6, if x = b, and µNA (0) = µNA (a) = µNA (b) = µNA (c) = −0.5. Then by routine calculation we know that A = (µPA, µ N A ) is a doubt bipolar fuzzy subalgebra of X. But, it is not a doubt bipolar fuzzy ideal of X, since µPA(b) = 0.6, µPA(b) = 0.6 � 0.5 = max{µPA(b ∗a), µPA(a)}, and hence it is not a doubt bipolar fuzzy H- ideal of X, since µPA(b ∗ c) = µPA(b) = 0.6, µPA(b) = 0.6 � 0.5 = max{µPA(b ∗ (a ∗ c)), µPA(a)}. Now, we give a condition for the bipolar fuzzy set A = (µPA, µ N A ), which is a doubt bipolar fuzzy ideal of X to be a doubt bipolar fuzzy H-ideal of X. Theorem 4. In associative BCK/BCI-algebras X, every doubt bipolar fuzzy ideal is a doubt bipolar fuzzy H-ideal of X. Proof. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy ideal of X. Then µPA(0) ≤ µPA(x) and µNA (0) ≥ µNA (x), for all x ∈ X. Now, since X is an associative, then x ∗ (y ∗ z) = (x ∗ y) ∗ z, for x, y, z ∈ X. Now, max{µPA(x ∗ (y ∗ z)), µPA(y)} = max{µPA((x ∗ y) ∗ z), µPA(y)} = max{µPA((x ∗ z) ∗ y), µPA(y)} ≥ µPA(x ∗ z). Therefore, µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} for all x, y, z ∈ X. Again, min{µNA (x ∗ (y ∗ z)), µNA (y)} = min{µNA ((x ∗ y) ∗ z), µNA (y)} = min{µNA ((x ∗ z) ∗ y), µNA (y)} ≤ µNA (x ∗ z). A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 662 Therefore, µNA (x∗z) ≥ min{µNA (x∗(y∗z)), µNA (y)} for all x, y, z ∈ X. Hence, A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. Example 3. Let X = {0, a, b, c, d} be a BCK-algebra with the Cayley table which is appeared in Table 3. Table 3: Cayley table for the ∗-operation. ∗ 0 a b c d 0 0 0 0 0 0 a a 0 a a a b b b 0 b b c c c c 0 c d d d d d 0 Here, X is an associative BCK-algebra. Define a bipolar fuzzy set A = (µPA, µ N A ) in X as follows: µPA(x) =  0, if x = 0 0.6, if x = a 0.4, if x = b 0.8, if x = c 0.9, if x = d, and µNA (0) = µNA (a) = µNA (b) = µNA (c) = µNA (d) = r, where r ∈ [−1, 0]. Hence, A = (µPA, µ N A ) is a doubt bipolar fuzzy ideal as well as a doubt bipolar fuzzy H-ideal of X. Corollary 1. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of X. Then the sets DµPA = {x ∈ X : µPA(x) = µPA(0)} and DµNA = {x ∈ X : µNA (x) = µPN (0)} are H-ideals of X. Proof. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of X. Obviously, 0 ∈ DµPA and 0 ∈ DµNA . Now, let x, y, z ∈ DµPA such that x ∗ (y ∗ z), y ∈ DµPA . Then µPA(x ∗ (y ∗ z)) = µPA(0) = µPA(y). Now, µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} = µPA(0). Again, since A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X,µPA(0) ≤ µPA(x ∗ z). Therefore, µPA(0) = µPA(x ∗ z). It follows that x ∗ z ∈ DµPA , for all x, y, z ∈ X. Therefore, DµPA is an H-ideal of X. Also, let x, y, z ∈ DµNA such that x ∗ (y ∗ z), y ∈ DµNA . Then µNA (x ∗ (y ∗ z) = µNA (0) = µNA (y). Now, µNA (x ∗ z) ≥ min{µNA (x ∗ (y ∗ z)), µNA (y)} = µNA (0). Again, since A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X,µNA (0) ≥ µNA (x ∗ z). Therefore, µNA (0) = µNA (x ∗ z). It follows that x ∗ z ∈ DµNA , for all x, y, z ∈ X. Therefore, DµNA is an H-ideal of X. Lemma 2. Let µ be a fuzzy set in X. Then the following statements holds, for all x, y ∈ X, A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 663 (1) 1−max{µ(x), µ(y)} = min{1− µ(x), 1− µ(y)}, (2) 1−min{µ(x), µ(y)} = max{1− µ(x), 1− µ(y)}. Proof. (1) If max{µ(x), µ(y)} = µ(x), then µ(y) ≤ µ(x). Thus, 1−µ(y) ≥ 1−µ(x), so min{1−µ(x), 1−µ(y)} = 1−µ(x) = 1−max{µ(x), µ(y)}. Similarly, if max{µ(x), µ(y)} = µ(y), then min{1− µ(x), 1− µ(y)} = 1− µ(y) = 1−max{µ(x), µ(y)}. (2) If min{µ(x), µ(y)} = µ(x), then µ(x) ≤ µ(y). Thus, 1− µ(x) ≥ 1− µ(y), so max{1− µ(x), 1 − µ(y)} = 1 − µ(x) = 1 − min{µ(x), µ(y)}. Similarly, if min{µ(x), µ(y)} = µ(y), then max{1− µ(x), 1− µ(y)} = 1− µ(y) = 1−min{µ(x), µ(y)}. Remark 1. A = (µPA, µ N A ) is a bipolar fuzzy set defined on any universe set X if and only if µPA and −µNA are fuzzy subsets of X. Theorem 5. A bipolar fuzzy set A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X if and only if the fuzzy subsets µPA and −µNA are doubt fuzzy H-ideals of X. Proof. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of X. Then clearly µPA is a doubt fuzzy H-ideal ofX.Also, µNA (0) ≥ µNA (x) and µNA (x∗z) ≥ min{µNA (x∗(y∗z)), µNA (y)}, implies that, −µNA (0) ≤ −µNA (x) and −µNA (x ∗ z) ≤ −min{µNA (x ∗ (y ∗ z)), µNA (y)} = max{−µNA (x ∗ (y ∗ z)),−µNA (y)}. Therefore, −µNA is a doubt fuzzy H-ideal of X. Conversely, assume that µPA and −µNA are doubt fuzzy H-ideals of X. So that µPA(0) ≤ µPA(x) and µPA(x∗z) ≤ max{µPA(x∗(y∗z)), µPA(y)}, for all x, y, z ∈ X. Now, we prove that µNA (0) ≥ µNA (x) and µNA (x ∗ z) ≥ min{µNA (x ∗ (y ∗ z)), µNA (y)} for all x, y, z ∈ X. Since −µNA is a doubt fuzzy H-ideal of X, so that −µNA (0) ≤ −µNA (x) and −µNA (x ∗ z) ≤ max{−µNA (x ∗ (y ∗ z)),−µNA (y)} = −min{µNA (x ∗ (y ∗ z)), µNA (y)}, implies that, µNA (0) ≥ µNA (x) and µNA (x∗z) ≥ min{µNA (x∗(y∗z)), µNA (y)} for all x, y, z ∈ X. Therefore, A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. Theorem 6. A bipolar fuzzy set A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X if and only if 4A = (µPA,−µPA) and 5A = (−µNA , µNA ) are also doubt bipolar fuzzy H-ideals of X. Proof. A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X if and only if the fuzzy subsets µPA and −µNA are doubt fuzzy H-ideals of X by Theorem 5. That is, if and only if 4A = (µPA,−µPA) and 5A = (−µNA , µNA ) are also doubt bipolar fuzzy H-ideals of X by definition of 4A and 5A. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 664 4. Characterizations of doubt bipolar fuzzy H-ideals In this section, we define a doubt positive t-level cut set and a doubt negative s-level cut set of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras. We investigate characteri- zations of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras by means of doubt positive t-level cut set, doubt negative s-level cut set and H-Artin BCK/BCI-algebras. Definition 11. Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of a BCK/BCI-algebra X, and (s, t) ∈ [−1, 0]× [0, 1]. Then the doubt positive t-level cut set and the doubt negative s-level cut set of A are as follows: APt = {x ∈ X : µPA(x) ≤ t} and ANt = {x ∈ X : µNA (x) ≥ s}. The set S(s,t) = {x ∈ X : µPA(x) ≤ t and µNA (x) ≥ s} is called a doubt (s, t)-level cut set of A. For every γ ∈ [0, 1], the set APγ ∩AN−γ is called a doubt γ-level cut set of A. From Definition 11, we can easily obtained the relation between a doubt bipolar fuzzy H-ideal and H-ideal in BCK/BCI-algebras. Theorem 7. For a bipolar fuzzy set A = (µPA, µ N A ) in X, the following are equivalent: 1. A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. 2. A = (µPA, µ N A ) satisfies the following assertions: i. (∀t ∈ [0, 1])(APt 6= ∅ ⇒ APt = {x ∈ X : µPA(x) ≤ t} is an H-ideal of X). ii. (∀s ∈ [−1, 0])(ANt 6= ∅ ⇒ ANs = {x ∈ X : µNA (x) ≥ s} is an H-ideal of X). Proof. (1⇒ 2) Let A = (µPA, µ N A ) be a doubt bipolar fuzzy H-ideal of X. Let t ∈ [0, 1] and s ∈ [−1, 0] such that APt 6= ∅ and ANs 6= ∅. Then there exists a ∈ APt and b ∈ ANs , that is µPA(a) ≤ t and µNA (b) ≥ s. Since A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X, we have µPA(0) ≤ µPA(x) and µNA (0) ≥ µNA (x), for all x ∈ X. Thus, µPA(0) ≤ µPA(a) ≤ t and µNA (0) ≥ µNA (b) ≥ s, so 0 ∈ APt and 0 ∈ ANs . Let x, y, z ∈ X such that x ∗ (y ∗ z) ∈ APt and y ∈ APt . Then µPA(x ∗ (y ∗ z)) ≤ t and µPA(y) ≤ t. Using Definition 9, we have µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} ≤ max{t, t} = t, so x∗z ∈ APt . Hence, APt is an H-ideal ofX. Finally, Let x, y, z ∈ X such that x∗(y∗z) ∈ ANs and y ∈ ANs . Then µNA (x ∗ (y ∗ z) ≥ s and µNA (y)) ≥ s. It follows that µNA (x ∗ z) ≥ min{µNA (x ∗ (y ∗ z)), µNA (y)} ≥ min{s, s} = s, so x ∗ z ∈ ANs . Hence, ANs is an H-ideal of X. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 665 (2 ⇒ 1) Suppose that APt 6= ∅ and ANt 6= ∅ are H-ideals of X for all t ∈ [0, 1] and s ∈ [−1, 0]. Assume that there exists a ∈ X such that µPA(0) > µPA(a) and µNA (0) < µNA (a). Taking to = 1 2 (µPA(0) + µPA(a)), so = 1 2 (µNA (0) + µNA (a)), implies that µPA(a) < to < µPA(0) and µNA (a) > so > µNA (0). This shows that 0 /∈ APt and 0 /∈ ANs , which leads to a contradiction. Therefore, µPA(0) ≤ µPA(x) and µNA (0) ≥ µNA (x) for all x ∈ X. Now, suppose that there are a, b, c ∈ X such that µPA(a ∗ c) > max{µPA(a ∗ (b ∗ c)), µPA(b)}. Then by taking t1 = 1 2 (µPA(a ∗ c) + max{µPA(a ∗ (b ∗ c)), µPA(b)}), we have max{µPA(a∗ (b∗ c), µPA(b)} < t1 < µPA(a∗ c). Hence a∗ c /∈ APt1 , a∗ (b∗ c)) ∈ APt1 and b ∈ APt1 , that is APt1 is not H- ideal of X, which a contradiction. Therefore, µPA(x ∗ y) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} for all x, y, z ∈ X. Finally, assume that p, q, r ∈ X such that µNA (p ∗ r) < min{µNA (p ∗ (q ∗ r)), µNA (q)}. Taking s1 = 1 2 (µNA (p ∗ r) + min{µNA (p ∗ (q ∗ r)), µNA (q)}), then µNA (p ∗ r) < s1 < min{µNA (p ∗ (q ∗ r)), µNA (q)}. Therefore, p ∗ (q ∗ r) ∈ ANs1 and q ∈ ANt1 but p ∗ r /∈ ANt1 . Again a contradiction. Thus, µNA (x ∗ z) ≥ min{µNA (x ∗ (y ∗ z)), µNA (y)} for all x, y, z ∈ X. Hence, A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. Example 4. Let X = {0, a, b, c} be a BCK-algebra with the Cayley table which is appeared in Table 4. Table 4: Cayley table for the ∗-operation. ∗ 0 a b c 0 0 0 0 0 a a 0 a a b b a 0 0 c c a c 0 Define a bipolar fuzzy set A = (µPA, µ N A ) in X as follows: µPA(x) =  0.2, if x = 0 0.6, if x = a 0.8, if x = b 0.7, if x = c, and µNA (x) = { −0.3, if x = 0, a, c −0.5, if x = b, A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 666 which is not a doubt bipolar fuzzy H- ideal of X, since µPA(b ∗ 0) = µPA(b) = 0.8 max{µPA((b ∗ (a ∗ 0))), µPA(a)} = max{µPA(a), µPA(a)} = 0.6. Now, for t = 0.75 and s = −0.45, we get APt = ANs = {0, a, c} which are not H-ideals of X, since a ∈ {0, a, c} and b ∗ (a ∗ 0) = b ∗ a = a ∈ {0, a, c}, but b ∗ 0 = b 6∈ {0, a, c}. Corollary 2. If A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X, then the doubt γ-level cut set of A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X, for all γ ∈ [0, 1]. Corollary 3. If A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. Then S(s,t) is an H-ideal of X for all (s, t) ∈ [−1, 0] × [0, 1]. In particular, the nonempty doubt γ-level cut set of A = (µPA, µ N A ) is an H-ideal of X for all γ ∈ [0, 1]. Theorem 8. If A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X and µPA(z)+µNA (z) ≤ 0 for all z ∈ X, then APγ ∪AN−γ is an H- ideal of X for all γ ∈ [0, 1]. Proof. Given that A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X and µPA(z) + µNA (z) ≤ 0 for all z ∈ X. Assume that APγ and AN−γ are nonempty for all γ ∈ [0, 1]. Then by Theorem 7, APγ and AN−γ are H-ideals of X. Let x, y, z ∈ X such that x ∗ (y ∗ z) ∈ APγ ∪AN−γ and y ∈ APγ ∪AN−γ . Here we have four cases to prove the theorem: (i) x ∗ (y ∗ z) ∈ APγ and y ∈ APγ , (ii) x ∗ (y ∗ z) ∈ APγ and y ∈ AN−γ , (iii) x ∗ (y ∗ z) ∈ AN−γ and y ∈ APγ , (iv) x ∗ (y ∗ z) ∈ AN−γ and y ∈ AN−γ . Case(i). If x ∗ (y ∗ z) ∈ APγ and y ∈ APγ , implies that µPA(x ∗ (y ∗ z)) ≤ γ and µPA(y) ≤ γ. Since A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X, it follows that µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} ≤ γ. Therefore, x ∗ z ∈ APγ ⊆ APγ ∪AN−γ . Case(ii). If x∗(y∗z) ∈ APγ and y ∈ AN−γ , implies that µPA(x∗(y∗z)) ≤ γ and µNA (y) ≥ −γ. Since µPA(y) + µNA (y) ≤ 0, so µPA(y) ≤ −µNA (y) ≤ γ, it follows that µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} ≤ max{µPA(x ∗ (y ∗ z)),−µNA (y)} ≤ γ. Therefore, x ∗ z ∈ APγ ⊆ APγ ∪AN−γ . Case(iii). If x∗(y∗z) ∈ AN−γ and y ∈ APγ , implies that µNA (x∗(y∗z)) ≥ −γ and µPA(y) ≤ γ. Since µPA(x ∗ (y ∗ z)) + µNA (x ∗ (y ∗ z)) ≤ 0, so µPA(x ∗ (y ∗ z)) ≤ −µNA (x ∗ (y ∗ z)) ≤ γ, it follows that µPA(x ∗ z) ≤ max{µPA(x ∗ (y ∗ z)), µPA(y)} A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 667 ≤ max{−µNA (x ∗ (y ∗ z)), µPA(y)} ≤ γ. Therefore, x ∗ z ∈ APγ ⊆ APγ ∪AN−γ . Case(iv). If x ∗ (y ∗ z) ∈ AN−γ and y ∈ AN−γ , implies that µNA (x ∗ (y ∗ z)) ≥ −γ and µNA (y) ≥ −γ. Since A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X, it follows that µNA (x ∗ z) ≥ min{µNA (x ∗ (y ∗ z)), µNA (y)} ≥ −γ. Therefore, x ∗ z ∈ AN−γ ⊆ APγ ∪AN−γ . Hence, APγ ∪AN−γ is an H-ideal of X. Definition 12. [33] A BCK/BCI-algebra X is said to satisfy the H-ascending (resp. H-descending) chain condition (briefly, H-ACC (resp. H-DCC)) if for every ascending (resp. descending) sequence I1 ⊆ I2 ⊆ I3 ⊆ ... (resp. I1 ⊇ I2 ⊇ I3 ⊇ ...) of H-ideals of X there exists a natural number n such that In = Ik for all n ≥ k. If X satisfies H-DCC, we say that X is an H-Artin BCK/BCI-algebras. In the next two theorems, we investigate characterizations of H-Artin BCK/BCI- algebras in terms of doubt bipolar fuzzy H-ideals. Theorem 9. Let X be a BCK/BCI-algebra satisfying H-DCC and A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. If a sequence of elements of Im(µPA) is strictly decreasing and a sequence of elements of Im(µNA ) is strictly increasing, then A = (µPA, µ N A ) has finite number of values. Proof. Let {tn} be a strictly decreasing sequence of Im(µPA), then 0 ≤ ... < t2 < t1 ≤ 1. Define APtr = {x ∈ X | µPA(x) ≤ tr}, r = 1, 2, 3, ... . Then APtr is an H-ideal by Theorem 7. Let x ∈ APtr , then µPA(x) ≤ tr < tr−1, which implies that x ∈ APtr−1 . Hence, APtr ⊆ APtr−1 . Since tr−1 ∈ Im(µPA), there exists xr−1 ∈ X such that µPA(xr−1) = tr−1. It follows that xr−1 ∈ APtr−1 , but xr−1 6∈ APtr . Thus, APtr ⊂ APtr−1 , and so we obtain a strictly decreasing sequence APt1 ⊃ A P t2 ⊃ A P t3 ⊃ ... of H-ideals of X which is not terminating. a contradiction. Similar for Im(µNA ). This completes the proof. Now we consider the converse of Theorem 9. Theorem 10. Let X be a BCK/BCI-algebra. If every doubt bipolar fuzzy H-ideal of X has finite number of values, then X satisfies H-DCC. Proof. Suppose that X does not satisfy H-DCC, then there exists a strictly descending chain I◦ ⊃ I1 ⊃ I2 ⊃ ... of H-ideals of X. Define a bipolar fuzzy set A = (µPA, µ N A ) in X by µPA(x) = { 1 n+1 , if x ∈ In − In+1, n = 0, 1, 2, ... 0, if x ∈ ⋂∞ n=0 In, µNA (x) = −µPA(x). Where I◦ stands for X. A. Al-Masarwah, A. G. Ahmad / Eur. J. Pure Appl. Math, 11 (3) (2018), 652-670 668 We prove that A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal of X. Clearly, µPA(0) = 0 ≤ µPA(x) and µNA (0) = 0 ≥ µNA (x) for all x ∈ X. Let x, y, z ∈ X. Assume that x∗ (y ∗z) ∈ In − In+1 and y ∈ Ik − Ik+1 for n = 0, 1, 2, ...; k = 0, 1, 2, ... . Without loss of generality, we may assume that n ≤ k. Then clearly y ∈ In. Since In is an H-ideal, we have x∗z ∈ In. Hence, µPA(x ∗ z) ≤ 1 n+1 = max{µPA(x ∗ (y ∗ z)), µPA(y)} and µNA (x ∗ z) ≥ −1 n+1 = min{µNA (x ∗ (y ∗ z)), µNA (y)}. If x ∗ (y ∗ z), y ∈ ⋂∞ n=0 In, then x ∗ z ∈ ⋂∞ n=0 In. Thus, µPA(x ∗ z) = 0 = max{µPA(x ∗ (y ∗ z)), µPA(y)} and µNA (x ∗ z) = 0 = min{µNA (x ∗ (y ∗ z)), µNA (y)}. If x ∗ (y ∗ z) 6∈ ⋂∞ n=0 In and y ∈ ⋂∞ n=0 In, then there exists k ∈ N such that x ∗ (y ∗ z) 6∈ Ik − Ik+1. It follows that x ∗ z ∈ Ik, so that µPA(x ∗ z) ≤ 1 k+1 = max{µPA(x ∗ (y ∗ z)), µPA(y)} and µNA (x ∗ z) ≥ −1 k+1 = min{µNA (x ∗ (y ∗ z)), µNA (y)}. Finally, assume that x ∗ (y ∗ z) ∈ ⋂∞ n=0 In and y 6∈ ⋂∞ n=0 In, then y ∈ Ir − Ir+1 for some r ∈ N. It follows that x ∗ z ∈ Ir and hence µPA(x ∗ z) ≤ 1 r+1 = max{µPA(x ∗ (y ∗ z)), µPA(y)} and µNA (x ∗ z) ≥ −1 r+1 = min{µNA (x ∗ (y ∗ z)), µNA (y)}. Consequently, we find that A = (µPA, µ N A ) is a doubt bipolar fuzzy H-ideal and A = (µPA, µ N A ) has infinite number of different values. This is a contradiction and the proof is complete. 5. Conclusions In the study of a BCK/BCI-algebra, we know that doubt bipolar fuzzy H-ideals with special properties always play a vital role in the structure theory of a BCK/BCI- algebra. 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