POSITIVE IMPLICATIVE (∈, ∈ ∨q)-FUZZY IDEALS ¯, ∈ ¯ ∨ q¯)-FUZZY IDEALS, FUZZY IDEALS WITH THRESHOLDS) ((∈ OF BCK-ALGEBRAS MUHAMMAD ZULFIQAR and MUHAMMAD SHABIR Communicated by the former editorial board In this paper, we introduce the concepts of positive implicative (∈, ∈ ∨q)-fuzzy ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of BCK-algebra and investiideal and positive implicative (∈ gate some of their related properties. AMS 2010 Subject Classification: 03G10, 03B05, 03B52, 06F35. Key words: BCK-algebra, positive implicative (∈, ∈ ∨q)-fuzzy ideal, positive ¯, ∈ ¯ ∨ q¯)-fuzzy ideal, fuzzy ideals with threshold. implicative (∈ 1. INTRODUCTION The concept of BCK-algebras was initiated by Imai and Iseki in [9]. The notion of a fuzzy set, which was published by Zadeh in his classical paper [27] of 1965, was applied by many researchers to generalize some of the basic concepts of algebra. The fuzzy algebraic structures play a vital role in Mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, real analysis, measure theory etc. In 1991, Xi applied fuzzy subsets in BCK-algebras [26] and studied fuzzy BCK-algebras. He defined the concepts of fuzzy ideal and fuzzy positive implicative ideal. For the general development of BCK-algebras, the fuzzy ideal theory plays an important role [21–22]. In the ideal theory of BCK-algebras, generation of an ideal by a subset in BCK-algebras is an important problem [11, 18, 20]. In 1971, Rosenfeld laid the foundations of fuzzy groups in [25]. Murali, defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset in [23]. The idea of quasi-coincidence of a fuzzy point with a fuzzy set given in [24], plays a vital role to generate some different types of fuzzy subgroups, called (α, β)-fuzzy subgroups, introduced by Bhakat and Das [5]. In particular, (∈, ∈ ∨q)-fuzzy subgroup is an important and useful generalization of the Rosenfeld’s fuzzy subgroups [25]. In [6], Biswas defined Rosenfeld’s fuzzy subgroups with interval valued membership MATH. REPORTS 16(66), 2 (2014), 219–241 220 Muhammad Zulfiqar and Muhammad Shabir 2 functions. In [1, 2], Bhakat and Bhakat and Das [3, 4], studied the concept of (∈ ∨q)-level subsets, (∈, ∈ ∨q)-fuzzy normal, quasi-normal and maximal subgroups. Zhan et al. [28], discussed (∈, ∈ ∨q)-fuzzy ideals in BCI-algebras. In [7], Davvaz discussed (∈, ∈ ∨q)-fuzzy subnearrings and ideals. Jun [13, 14] introduced the concept of (α, β)-fuzzy subalgebras (ideals) of BCK/BCIalgebras. Davvaz and Corsini redefined fuzzy Hv-submodule and many-valued implications in [8]. Jun [15] defined (∈, ∈ ∨q)-fuzzy subalgebras in BCK/BCIalgebras. Zulfiqar initiated the notion of (α, β)-fuzzy positive implicative ideals ¯, ∈ ¯ ∨ q¯)-fuzzy filters of BLin BCK-algebras [29]. In [17], Ma et al. studied (∈ algebras. In this paper, we show that every positive implicative (∈, ∈ ∨q)-fuzzy ideal of a BCK-algebra X is an (∈, ∈ ∨q)-fuzzy ideal of X. We prove that a fuzzy set µ of a BCK-algebra X is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X if and only if [µ]t (6= φ) is a positive implicative ideal of X for all t ∈ (0, 1]. We show that a fuzzy set µ of a BCK-algebra X is a positive implicative ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X if and only if it satisfies the conditions (C) and (L), (∈ where (C) µ(0) ∨ 0.5 ≥ µ(x), (L) µ(x ∗ z) ∨ 0.5 ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z), for all x, y, z ∈ X. We show that a fuzzy set µ of a BCK-algebra X is a positive implicative fuzzy ideal with thresholds ε and δ of X, with ε < δ if and only if µt = {x ∈ X | µ(x) ≥ t} is a positive implicative ideal of X for all ε < t ≤ δ. In Section 2, we recall the notions of ideal and positive implicative ideal of BCK-algebra; in Section 3, we review some fuzzy logic concepts; in Section 4, we define the concept of positive implicative (∈, ∈ ∨q)-fuzzy ideal of a BCKalgebra and investigate some of their properties; in Section 5, we introduce the ¯, ∈ ¯ ∨ q¯)-fuzzy ideal and positive implicative (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal concept of (∈ of BCK-algebras and discuss some of their properties; in Section 6, we define the concepts of fuzzy ideal with thresholds and positive implicative fuzzy ideal with thresholds of BCK-algebras and investigate some of their properties. 2. SECTION 2 (CRISP SETS - LEVEL 0) Throughout this paper, X always denotes a BCK-algebra unless otherwise specified. Definition 2.1 ([12]). By a BCK-algebra, we mean an algebra (X, ∗, 0) of type (2, 0) satisfying the axioms: (BCK-I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0 (BCK-II) (x ∗ (x ∗ y)) ∗ y = 0 3 Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 221 (BCK-III) x ∗ x = 0 (BCK-IV) 0 ∗ x = 0 (BCK-V) x ∗ y = 0 and y ∗ x = 0 imply x = y for all x, y, z ∈ X. We can define a partial order ≤ on X by x ≤ y if and only if x ∗ y = 0. Proposition 2.2 ([12, 20, 21]). In any BCK-algebra X, the following are true: (1) (x ∗ y) ∗ z = (x ∗ z) ∗ y (2) (x ∗ z) ∗ (y ∗ z) ≤ x ∗ y (3) (x ∗ y) ∗ (x ∗ z) ≤ z ∗ y (4) x ∗ 0 = x (5) x ∗ (x ∗ (x ∗ y)) = x ∗ y for all x, y, z ∈ X. A BCK-algebra X is called positive implicative [20] if it satisfies (x ∗ z) ∗ (y ∗ z) = (x ∗ y) ∗ z, for all x, y, z ∈ X. Note that positive implicative BCK-algebras coincide with Hilbert algebras [10]. Definition 2.3 ([20]). A nonempty subset S of a BCK-algebra X is called a subalgebra of X if it satisfies x ∗ y ∈ S, for all x, y ∈ S. Definition 2.4 ([11]). A nonempty subset I of a BCK-algebra X is called an ideal of X if it satisfies the conditions (I1) and (I2), where (I1) 0 ∈ I, (I2) x ∗ y ∈ I and y ∈ I imply x ∈ I, for all x, y ∈ X. Definition 2.5 ([16]). A nonempty subset I of a BCK-algebra X is called a positive implicative ideal of X if it satisfies the conditions (I1) and (I3), where (I1) 0 ∈ I, (I3) (x ∗ y) ∗ z ∈ I and y ∗ z ∈ I imply x ∗ z ∈ I, for all x, y, z ∈ X. Theorem 2.6 ([19]). A BCK-algebra X is a positive implicative if and only if every ideal of X is a positive implicative ideal. 222 Muhammad Zulfiqar and Muhammad Shabir 4 Proposition 2.7 ([20]). Any positive implicative ideal of a BCK-algebra X is an ideal of X, but the converse is not true in general. Theorem 2.8 ([20]). Let X be a BCK-algebra. Then an ideal I of X is positive implicative ideal of X if and only if the condition (for all x, y ∈ X) ((x ∗ y) ∗ y ∈ I ⇒ x ∗ y ∈ I) is satisfied. 3. SECTION 3 (LEVEL 1 OF FUZZIFICATION) We now review some fuzzy logic concepts. A fuzzy set µ of a universe X is a function from X into the unit closed interval [0, 1], that is µ : X → [0, 1]. Definition 3.1 ([26]). For a fuzzy set µ of a BCK-algebra X and t ∈ (0, 1], the crisp set µt = {x ∈ X | µ(x) ≥ t} is called the level subset of µ. Recall that ([0, 1], ∧ = min, ∨ = max, 0, 1) is a complete lattice (chain). Definition 3.2 ([26]). Let X be a BCK-algebra. A fuzzy set µ in X is said to be a fuzzy subalgebra of X if it satisfies µ(x ∗ y) ≥ µ(x) ∧ µ(y), for all x, y ∈ X. Definition 3.3 ([21]). A fuzzy set µ of a BCK-algebra X is called a fuzzy ideal of X if it satisfies the conditions (F1) and (F2), where (F1) µ(0) ≥ µ(x), (F2) µ(x) ≥ µ(x ∗ y) ∧ µ(y), for all x, y ∈ X. Definition 3.4 ([16]). A fuzzy set µ of a BCK-algebra X is called a positive implicative fuzzy ideal of X if it satisfies the conditions (F1) and (F3), where (F1) µ(0) ≥ µ(x), (F3) µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z), for all x, y, z ∈ X. Theorem 3.5 ([16]). Every positive implicative fuzzy ideal of a BCKalgebra X is a fuzzy ideal of X. Remark 3.6. A fuzzy ideal may not be a positive implicative fuzzy ideal, as shown in the following example. 5 Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 223 Example 3.7. Let X = {0, 1, 2, 3} in which ∗ is given by the table: * 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 1 2 0 Then X is a BCK-algebra [20]. Define a fuzzy set µ in X by µ(0) = 0.9, µ(1) = µ(2) = 0.7 and µ(3) = 0.4. Simple calculations show that µ is a fuzzy ideal of X, but it is not a positive implicative fuzzy ideal of X. Because for x = 2, y = 1, z = 1, (F3) becomes µ(2 ∗ 1) ≥ µ((2 ∗ 1) ∗ 1) ∧ µ(1 ∗ 1) µ(1) ≥ µ(1 ∗ 1) ∧ µ(0) 0.7 ≥ µ(0) ∧ µ(0) ≥ 0.9 ∧ 0.9 ≥ 0.9 but 0.7 0.9. Theorem 3.8 ([16]). A fuzzy ideal µ of a BCK-algebra X is a positive implicative fuzzy ideal of X if and only if it satisfies the condition µ(x ∗ y) ≥ µ((x ∗ y) ∗ y) for all x, y ∈ X. Theorem 3.9 ([29]). A fuzzy set µ of a BCK-algebra X is a positive implicative fuzzy ideal of X if and only if, for every t ∈ [0, 1], µt (6= φ) is a positive implicative ideal of X. Theorem 3.10 ([29]). Let µ be a fuzzy set of a BCK-algebra X. If µ is a positive implicative fuzzy ideal of X, then the set I = {x ∈ X | µ(x) = µ(0)} is a positive implicative ideal of X. 4. SECTION 4 (LEVEL 2.1 OF FUZZIFICATION) In this section, we introduce the concept of positive implicative (∈, ∈ ∨q)fuzzy ideal of a BCK-algebra and investigate some of their properties. 224 Muhammad Zulfiqar and Muhammad Shabir 6 Definition 4.1 ([13]). A fuzzy set µ of a BCK-algebra X having the form t ∈ (0, 1], if y = x µ(y) = 0, if y 6= x. is said to be a fuzzy point with support x and value t and is denoted by xt . A fuzzy point xt is said to belong to (resp., quasi-coincident with) a fuzzy set µ, written as xt ∈ µ (resp., xt qµ) if µ(x) ≥ t (resp., µ(x) + t > 1). If xt ∈ µ or xt qµ, then we write xt ∈ ∨qµ. Definition 4.2 ([13]). A fuzzy set µ of a BCK-algebra X is called an (∈, ∈ ∨q)-fuzzy ideal of X if it satisfies the conditions (A) and (B), where (A) xt ∈ µ ⇒ 0t ∈ ∨qµ, (B) (x ∗ y)t ∈ µ, yr ∈ µ ⇒ xt∧r ∈ ∨qµ, for all t, r ∈ (0, 1] and x, y ∈ X. Theorem 4.3 ([13]). Every fuzzy ideal of a BCK-algebra X is an (∈, ∈ ∨q)-fuzzy ideal of X. Lemma 4.4 ([13]). Let µ be a fuzzy set of a BCK-algebra X. Then µt is an ideal of X for all 0.5 < t ≤ 1 if and only if it satisfies the conditions (C) and (D), where (C) µ(0) ∨ 0.5 ≥ µ(x), (D) µ(x) ∨ 0.5 ≥ µ(x ∗ y) ∧ µ(y), for all x, y, z ∈ X. Definition 4.5. A fuzzy set µ of a BCK-algebra X is called a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X if it satisfies the conditions (A) and (E), where (A) xt ∈ µ ⇒ 0t ∈ ∨qµ, (E) ((x ∗ y) ∗ z)t ∈ µ, (y ∗ z)r ∈ µ ⇒ (x ∗ z)t∧r ∈ ∨qµ, for all t, r ∈ (0, 1] and x, y ∈ X. Example 4.6. Let X = {0, 1, 2, 3} be a BCK-algebra with the Cayley table as follow [20]: * 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 1 2 0 Let µ be a fuzzy set in X defined by µ(0) = 0.7, µ(1) = µ(2) = µ(3) = 0.6. Simple calculations show that µ is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. 7 Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 225 Theorem 4.7. Every positive implicative (∈, ∈ ∨q)-fuzzy ideal of a BCKalgebra X is an (∈, ∈ ∨q)-fuzzy ideal of X. Proof. Let µ be a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. Then for all t, r ∈ (0, 1] and for all x, y, z ∈ X, we have ((x ∗ y) ∗ z)t ∈ µ, (y ∗ z)r ∈ µ ⇒ (x ∗ z)t∧r ∈ ∨qµ. Put z = 0 in above, we get ((x ∗ y) ∗ 0)t ∈ µ, (y ∗ 0)r ∈ µ ⇒ (x ∗ 0)t∧r ∈ ∨qµ. This implies (x ∗ y)t ∈ µ, yr ∈ µ ⇒ xt∧r ∈ ∨qµ (by PROPOSITION 2.2(4)). This means that µ satisfies the condition (B). Combining with (A) implies that µ is an (∈, ∈ ∨q)-fuzzy ideal of X. The converse of the above theorem does not hold (see Example 4.10). Theorem 4.8. A (∈, ∈ ∨q)-fuzzy ideal µ of a BCK-algebra X is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X if and only if it satisfies the condition µ(x ∗ y) ≥ µ((x ∗ y) ∗ y) ∧ 0.5 for all x, y ∈ X. Proof. The proof is similar to the proof of THEOREM 3.8. Theorem 4.9 ([29]). A fuzzy set µ of a BCK-algebra X is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X if and only if it satisfies the conditions (F) and (G), where (F) µ(0) ≥ µ(x) ∧ 0.5, (G) µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5, for all x, y, z ∈ X. Example 4.10. Let X = {0, a, b, c} be a BCK-algebra in which ∗ is defined as follows [16]: * 0 a b c 0 0 a b c a 0 0 a c b 0 0 0 c c 0 a b 0 226 Muhammad Zulfiqar and Muhammad Shabir 8 Define a map µ : X → [0, 1] by µ(0) = 0.7, µ(a) = µ(b) = 0.4 and µ(c) = 0.3. Simple calculations show that µ is an (∈, ∈ ∨q)-fuzzy ideal of X, but it is not a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. Because for x = b, y = a, z = a, (G) becomes µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 µ(b ∗ a) ≥ µ((b ∗ a) ∗ a) ∧ µ(a ∗ a) ∧ 0.5 µ(a) ≥ µ(a ∗ a) ∧ µ(0) ∧ 0.5 µ(a) ≥ µ(0) ∧ µ(0) ∧ 0.5 0.4 ≥ 0.7 ∧ 0.7 ∧ 0.5 ≥ 0.5 but 0.4 0.5 Theorem 4.11. Let µ be a positive implicative (∈, ∈ ∨q)-fuzzy ideal of a BCK-algebra X. Then (1) If there exists x ∈ X, such that µ(x) ≥ 0.5, then µ(0) ≥ 0.5. (2) If µ(0) < 0.5, then µ is a positive implicative (∈, ∈)-fuzzy ideal of X. Proof. (1) It follows from THEOREM 4.9(F). (2) The proof is similar to [29]. Theorem 4.12 ([29]). A fuzzy set µ of a BCK-algebra X is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X if and only if the set µt (6= φ) is a positive implicative ideal of X, for all 0 < t ≤ 0.5. In the next theorem we can show a similar result for the case when µt is a positive implicative ideal of a BCK-algebra X for 0.5 < t ≤ 1. Theorem 4.13. Let µ be a fuzzy set of a BCK-algebra X. Then µt (6= φ) is a positive implicative ideal of X for all 0.5 < t ≤ 1 if and only if it satisfies the conditions (C) and (H), where (C) µ(0) ∨ 0.5 ≥ µ(x), (H) µ(x ∗ z) ∨ 0.5 ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z), for all x, y, z ∈ X. Proof. Suppose µt (6= φ) is a positive implicative ideal of X. From LEMMA 4.4, it follows that (C) hold. Assume that there exist x, y, z ∈ X such that µ(x ∗ z) ∨ 0.5 < µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) = t. Then µ(x ∗ z) < t and (x ∗ y) ∗ z ∈ µt , y ∗ z ∈ µt . Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 9 227 Since µt is a positive implicative ideal of X, we have x ∗ z ∈ µt . Thus, µ(x ∗ z) = t, a contradiction. Hence, (H) holds. Conversely, assume that the conditions (C) and (H) hold. By LEMMA 4.4, we know that µt is a positive implicative ideal of X. Suppose that 0.5 < t ≤ 1, (x ∗ y) ∗ z ∈ µt , y ∗ z ∈ µt Then 0.5 < t ≤ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ≤ µ(x ∗ z) ∨ 0.5 (by using condition (H)) ≤ µ(x ∗ z). Hence, µt is a positive implicative ideal of X. Theorem 4.14. Every positive implicative fuzzy ideal of a BCK-algebra X is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. Proof. Suppose µ is a positive implicative fuzzy ideal of X. Then it is also a fuzzy ideal of X by THEOREM 3.5. By using THEOREM 4.3, it follow that µ is an (∈, ∈ ∨q)-fuzzy ideal of X. Then by Definition 3.4 (F3), for any x, y, z ∈ X, we have µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z). (i) If µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ≥ 0.5, then µ((x ∗ y) ∗ z) ≥ 0.5 and µ(y ∗ z) ≥ 0.5. This implies that µ(x ∗ z) ≥ 0.5. Thus, µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5. (ii) If µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) < 0.5, then µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) = µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5. Thus, µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5. This satisfies the condition (G). Hence, µ is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. 228 Muhammad Zulfiqar and Muhammad Shabir 10 For any fuzzy set µ of a BCK-algebra X and t ∈ (0, 1], we denote Q(µ; t) = {x ∈ X | xt qµ} and [µ]t = {x ∈ X | xt ∈ ∨qµ}. It is clear that [µ]t = µt ∪ Q(µ; t). Theorem 4.15. A fuzzy set µ of a BCK-algebra X is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X if and only if [µ]t (6= φ) is a positive implicative ideal of X for all t ∈ (0, 1]. Proof. Suppose µ is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. Then µ(0) ≥ µ(x) ∧ 0.5, for all x ∈ [µ]t . Since x ∈ [µ]t . Then xt ∈ ∨qµ, i.e., µ(x) ≥ t or µ(x) + t > 1. Case 1: µ(x) ≥ t. (1) If t > 0.5, then µ(0) ≥ µ(x) ∧ 0.5 ≥ t ∧ 0.5 = 0.5. So that µ(0) + t > 1, i.e., 0t qµ. (2) If t ≤ 0.5, then µ(0) ≥ t, i.e., 0t ∈ µ. Case 2: µ(x) + t > 1. (1) If t > 0.5, then µ(0) ≥ µ(x) ∧ 0.5 > (1 − t) ∧ 0.5 = 1 − t. Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 11 Hence, µ(0) + t > 1. Thus, we have 0t qµ. (2) If t ≤ 0.5, then µ(0) > (1 − t) ∧ 0.5 = 0.5. So, 0t ∈ µ. Thus, in any case, we have 0 ∈ [µ]t . Suppose (x ∗ y) ∗ z, y ∗ z ∈ [µ]t for t ∈ (0, 1]. Then ((x ∗ y) ∗ z)t ∈ ∨qµ or (y ∗ z)t ∈ ∨qµ. Thus, µ((x ∗ y) ∗ z) ≥ t or µ((x ∗ y) ∗ z) + t > 1 and µ(y ∗ z) ≥ t or µ(y ∗ z) + t > 1. Since µ is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X, we have µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5. Case 1: µ((x ∗ y) ∗ z) ≥ t and µ(y ∗ z) ≥ t. (1) If t > 0.5, then µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 ≥ t ∧ t ∧ 0.5 ≥ t ∧ 0.5 = 0.5. So, we have (x ∗ z)t qµ. (2) If t ≤ 0.5, then µ(x ∗ z) ≥ t. Thus, we have (x ∗ z)t ∈ µ. Case 2: µ((x ∗ y) ∗ z) ≥ t and µ(y ∗ z) + t > 1. (1) If t > 0.5, then µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 ≥ t ∧ (1 − t) ∧ 0.5 = 1 − t. 229 230 Muhammad Zulfiqar and Muhammad Shabir i.e., µ(x ∗ z) + t > 1. Thus, we have (x ∗ z)t qµ. (2) If t < 0.5, then µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 ≥ t ∧ (1 − t) ∧ 0.5 = t. So, (x ∗ z)t ∈ µ. Case 3: µ((x ∗ y) ∗ z) + t > 1 and µ(y ∗ z) ≥ t. The proof is similar to Case 2. Case 4: µ((x ∗ y) ∗ z) + t > 1 and µ(y ∗ z) + t > 1. (1) If t > 0.5, then µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 > (1 − t) ∧ (1 − t) ∧ 0.5 > (1 − t) ∧ 0.5 = 1 − t. i.e., µ(x ∗ z) + t > 1 and thus, we have (x ∗ z)t qµ. (2) If t ≤ 0.5, then µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 ≥ (1 − t) ∧ (1 − t) ∧ 0.5 ≥ (1 − t) ∧ 0.5 = 0.5 = t. Thus, (x ∗ z)t ∈ µ. Therefore, in any case, we have (x ∗ z)t ∈ ∨qµ and so that x ∗ z ∈ [µ]t . 12 Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 13 231 Hence, [µ]t is a positive implicative ideal of X. Conversely, assume that µ is a fuzzy set of X and t ∈ (0, 1] be such that [µ]t is a positive implicative ideal of X. If µ(0) < t ≤ µ(x) ∧ 0.5 for some t ∈ (0, 0.5), then µ(0) < t < 0.5. Since 0 ∈ [µ]t , then µ(0) ≥ t or µ(0) + t > 1, a contradiction. If µ(x ∗ z) < t ≤ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ 0.5 for some t ∈ (0, 0.5). Then µ((x ∗ y) ∗ z) ≥ t and µ(y ∗ z) ≥ t. i.e., (x ∗ y) ∗ z, y ∗ z ∈ µt ⊆ [µ]t . Thus, x ∗ z ∈ [µ]t . Hence, we have µ(x ∗ z) ≥ t or µ(x ∗ z) + t > 1, a contradiction. Hence, µ is a positive implicative (∈, ∈ ∨q)-fuzzy ideal of X. 5. SECTION 5 (LEVEL 2.2 OF FUZZIFICATION) ¯, ∈ ¯ ∨ q¯)-fuzzy ideal and In this section, we introduce the concepts of (∈ ¯ ¯ positive implicative (∈, ∈ ∨ q¯)-fuzzy ideal of BCK-algebras and investigate some of their properties. Definition 5.1. Let µ be a fuzzy set of a BCK-algebra X. Then µ is called ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X if it satisfies the conditions (I) and (J), where a (∈ ¯ µ ⇒ xt ∈ ¯ ∨ q¯µ, (I) 0t ∈ ¯ µ ⇒ (x ∗ y)t ∈ ¯ ∨ q¯µ or yr ∈ ¯ ∨ q¯µ, (J) xt∧r ∈ for all x, y ∈ X and t, r ∈ [0, 1]. Definition 5.2. Let µ be a fuzzy set of a BCK-algebra X. Then µ is called ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X if it satisfies the conditions (I) a positive implicative (∈ and (K), where ¯ µ ⇒ xt ∈ ¯ ∨ q¯µ, (I) 0t ∈ ¯ µ ⇒ ((x ∗ y) ∗ z)t ∈ ¯ ∨ q¯µ or (y ∗ z)r ∈ ¯ ∨ q¯µ, (K) (x ∗ z)t∧r ∈ for all x, y, z ∈ X and t, r ∈ [0, 1]. 232 Muhammad Zulfiqar and Muhammad Shabir * 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 14 3 0 1 2 0 Example 5.3. Let X = {0, 1, 2, 3} be a BCK-algebra with Cayley table as follows [20]: Let µ be a fuzzy set in X defined by µ(0) = 0.5, µ(3) = 0.2 and µ(1) = ¯, ∈ ¯ ∨ q¯)-fuzzy ideal as well µ(2) = 0.4. Simple calculations show that µ is an (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X. as a positive implicative (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of a BCKTheorem 5.4. Every positive implicative (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X. algebra X is an (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X. Then Proof. Let µ be a positive implicative (∈ for all t, r (0, 1] and for all x, y, z ∈ X, we have ¯ µ ⇒ ((x ∗ y) ∗ z)t ∈ ¯ ∨ q¯µ or (x ∗ z)t∧r ∈ ¯ ∨ q¯µ. (y ∗ z)r ∈ Put z = 0 in above, we get ¯ µ ⇒ ((x ∗ y) ∗ 0)t ∈ ¯ ∨ q¯µ or (x ∗ 0)t∧r ∈ ¯ ∨ q¯µ. (y ∗ 0)r ∈ This implies ¯ µ ⇒ (x ∗ y)t ∈ ¯ ∨ q¯µ or yr ∈ ¯ ∨ q¯µ (by Proposition 2.2(4)). xt∧r ∈ This means that µ satisfies the condition (J). Combining with (I) implies ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X. that µ is an (∈ The converse of the above theorem does not hold (see Example 5.7). Theorem 5.5. A fuzzy set µ of a BCK-algebra X is a positive implicative ¯ ¯ (∈, ∈ ∨ q¯)-fuzzy ideal of X if and only if it satisfies the conditions (C) and (L), where (C) µ(0) ∨ 0.5 ≥ µ(x), (L) µ(x ∗ z) ∨ 0.5 ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z), for all x, y, z ∈ X. Proof. (I) ⇒ (C) Let x ∈ X be such that µ(x) > µ(0) ∨ 0.5. Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 15 233 Select t such that µ(x) ≥ t > µ(0) ∨ 0.5. ¯ µ. But µ(x) ≥ t and µ(x) + t > 1, that is xt ∈ µ and xt qµ, Then 0t ∈ which is a contradiction. Hence, µ(0) ∨ 0.5 ≥ µ(x). (C) ⇒ (I) ¯ µ. Then µ(0) < t. Let 0t ∈ If µ(0) ≥ 0.5, then by condition (C) µ(0) ≥ µ(x) ¯ µ. If µ(0) < 0.5, then by condition (C) and so µ(x) < t, that is xt ∈ 0.5 ≥ µ(x). Suppose xt ∈ µ. Then µ(x) ≥ t. Thus, 0.5 ≥ t. Hence, µ(x) + t ≤ 0.5 + 0.5 = 1, that is xt q¯µ. This implies that ¯ ∨ q¯µ. xt ∈ (K) ⇒ (L) Suppose there exist x, y, z ∈ X such that µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) > µ(x ∗ z) ∨ 0.5. Select t such that µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ≥ t > µ(x ∗ z) ∨ 0.5. ¯ µ but ((x ∗ y) ∗ z)t ∈ µ and (y ∗ z)t ∈ µ, which is a Then (x ∗ z)t ∈ contradiction. Hence, µ(x ∗ z) ∨ 0.5 ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z). (L) ⇒ (K) Let x, z ∈ X and t, r ∈ (0, 1] be such that ¯ µ. (x ∗ z)t∧r ∈ Then µ(x ∗ z) < t ∧ r. (a) If µ(x ∗ z) ≥ 0.5, then by condition (L) µ(x ∗ z) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) 234 Muhammad Zulfiqar and Muhammad Shabir 16 ¯ µ. ¯ µ or (y ∗ z)r ∈ and so µ((x ∗ y) ∗ z) < t or µ(y ∗ z) < r, that is ((x ∗ y) ∗ z)t ∈ Hence, ¯ ∨ q¯µ. ¯ ∨ q¯µ or (y ∗ z)r ∈ ((x ∗ y) ∗ z)t ∈ (b) If µ(x ∗ z) < 0.5, then by condition (L) 0.5 ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z). Suppose ((x ∗ y) ∗ z)t ∈ µ, (y ∗ z)r ∈ µ. Then µ((x ∗ y) ∗ z) ≥ t and µ(y ∗ z) ≥ r. Thus, 0.5 ≥ t ∧ r. Hence, µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) + t ∧ r ≤ 0.5 + 0.5 = 1, that is ((x ∗ y) ∗ z)t q¯µ or (y ∗ z)r q¯µ. This implies that ¯ ∨ q¯µ or(y ∗ z)r ∈ ¯ ∨ q¯µ. ((x ∗ y) ∗ z)t ∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal µ of a BCK-algebra X is a positive Theorem 5.6. A (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X if and only if it satisfies the condition implicative (∈ µ(x ∗ y) ∨ 0.5 ≥ µ((x ∗ y) ∗ y) for all x, y ∈ X. ¯, ∈ ¯ ∨ q¯) - fuzzy ideal of X. If z is Proof. Let µ be a positive implicative (∈ replaced by y in (L), we have µ(x ∗ y) ∨ 0.5 ≥ µ((x ∗ y) ∗ y) ∧ µ(y ∗ y) ≥ µ((x ∗ y) ∗ y) ∧ µ(0) (BCK-III) ≥ µ((x ∗ y) ∗ y) (by condition (C)) ¯, ∈ ¯ ∨ q¯) - fuzzy ideal Conversely, assume that µ is a positive implicative (∈ of X and µ(x ∗ y) ∨ 0.5 ≥ µ((x ∗ y) ∗ y). ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X, we have Since µ is an (∈ µ(0) ∨ 0.5 ≥ µ(x), for all x ∈ X. Now, µ(x ∗ z) ∨ 0.5 ≥ µ((x ∗ z) ∗ z) µ(x ∗ z) ∨ 0.5 ∨ 0.5 ≥ µ((x ∗ z) ∗ z) ∨ 0.5. ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X, we have Since µ is an (∈ µ(x ∗ z) ∨ 0.5 ≥ µ(((x ∗ z) ∗ z) ∗ (y ∗ z)) ∧ µ(y ∗ z) ≥ µ((x ∗ z) ∗ y) ∧ µ(y ∗ z) (by PROPOSITION 2.2(2)) ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) (by PROPOSITION 2.2(1)). ¯, ∈ ¯ ∨ q¯) - fuzzy ideal of X. Hence, µ is a positive implicative (∈ 17 Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 235 Example 5.7. Let X = {0, a, b, c} be a BCK-algebra with Cayley table as follows [29]: * 0 a b c 0 0 a b c a 0 0 a c b 0 0 0 c c 0 a b 0 Let µ be a fuzzy set in X defined by µ(0) = 0.6, µ(a) = µ(b) = 0.4 and ¯, ∈ ¯ ∨ q¯) - fuzzy ideal of X, µ(c) = 0.3. Simple calculations show that µ is an (∈ ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X. but µ is not a positive implicative (∈ Because for x = b, y = a, z = a, (L) becomes µ(b ∗ a) ∨ 0.5 ≥ µ((b ∗ a) ∗ a) ∧ µ(a ∗ a) µ(a) ∨ 0.5 ≥ µ(a ∗ a) ∧ µ(0) 0.4 ∨ 0.5 ≥ µ(0) ∧ µ(0) 0.5 ≥ 0.6 ∧ 0.6 0.5 ≥ 0.6 but 0.5 0.6. Theorem 5.8. A fuzzy set µ of a BCK-algebra X is a positive implicative ¯, ∈ ¯ ∨ q¯)-fuzzy ideal of X if and only if for any t ∈ (0.5, 1], µt = {x ∈ X | (∈ µ(x) ≥ t} is a positive implicative ideal of X. Proof. The proof is similar to the proof of THEOREM 4.12. Remark 5.9. Let µ be a fuzzy set of a BCK - algebra X and It = {t | t ∈ (0, 1] and µt is a positive implicative ideal of X}. In particular, (1) If It = (0, 1], then µ is a positive implicative fuzzy ideal of a BCK - algebra X (THEOREM 3.9). (2) If It = (0, 0.5], then µ is a positive implicative (∈, ∈ ∨q) - fuzzy ideal of a BCK - algebra X (THEOREM 4.12). ¯, ∈ ¯ ∨ q¯) - fuzzy ideal of a (3) If It = (0.5, 1], then µ is a positive implicative (∈ BCK - algebra X (THEOREM 5.8). Corollary 5.10. Every positive implicative fuzzy ideal of a BCK - alge¯, ∈ ¯ ∨ q¯) - fuzzy ideal of X. bra X is a positive implicative (∈ 236 Muhammad Zulfiqar and Muhammad Shabir 18 6. SECTION 6 (LEVEL 3 OF FUZZIFICATION) In this section, we introduce the concepts of fuzzy ideal with thresholds and positive implicative fuzzy ideal with thresholds of BCK - algebras and investigate some of their properties. Definition 6.1. A fuzzy set µ of a BCK - algebra X is called a fuzzy ideal with thresholds ε and δ of X, ε, δ ∈ (0, 1] with ε < δ, if it satisfies the conditions (M) and (N), where (M) µ(0) ∨ ε ≥ µ(x) ∧ δ, (N) µ(x) ∨ ε ≥ µ(x ∗ y) ∧ µ(y) ∧ δ, for all x, y ∈ X. Example 6.2. Let X = {0, 1, 2, 3, 4} in which the operation ∗ is defined as follows: * 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 3 2 0 1 0 3 4 3 0 0 0 0 3 4 0 0 0 0 0 Then (X, ∗ , 0) is a BCK - algebra [20]. Let s0 , s1 , s2 ∈ [0, 1] be such that s0 > s1 > s2 . We define a map µ : X → [0, 1] by µ(0) = s0 , µ(1) = s1 and µ(2) = µ(3) = µ(4) = s2 . Simple calculations show that µ is a fuzzy ideal of X with thresholds ε = s2 and δ = s0 . Definition 6.3. A fuzzy set µ of a BCK-algebra X is called a positive implicative fuzzy ideal with thresholds ε and δ of X, ε, δ ∈ (0, 1] with ε < δ, if it satisfies the conditions (M) and (O), where (M) µ(0) ∨ ε ≥ µ(x) ∧ δ, (O) µ(x ∗ z) ∨ ε ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ δ, for all x, y, z ∈ X. Example 6.4. Let X = {0, 1, 2, 3, 4} be a BCK - algebra in which ∗ is defined as follows [20]: * 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 3 2 0 1 0 3 4 3 0 0 0 0 0 4 0 0 0 0 0 19 Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 237 Let µ be a fuzzy set of a BCK - algebra X defined by µ(0) = µ(1) = 0.6, µ(2) = 1, µ(3) = 0 and µ(4) = 0.2. Simple calculations show that µ is a positive implicative fuzzy ideal of X with thresholds ε = 0.2 and δ = 0.6. Theorem 6.5. Every positive implicative fuzzy ideal with thresholds of a BCK-algebra X is a fuzzy ideal with thresholds of X. Proof. Let µ be a positive implicative fuzzy ideal with thresholds of X. Then for all ε, δ ∈ (0, 1] and for all x, y, z ∈ X, we have µ(x ∗ z) ∨ ε ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ δ. Put z = 0 in above, we get µ(x ∗ 0) ∨ ε ≥ µ((x ∗ y) ∗ 0) ∧ µ(y ∗ 0) ∧ δ. This implies µ(x) ∨ ε ≥ µ(x ∗ y) ∧ µ(y) ∧ δ (by PROPOSITION 2.2(4)). This means that µ satisfies the condition (O). Combining with (M) implies that µ is fuzzy ideal with thresholds of X. Example 6.6. Let X = {0, 1, 2, 3, 4} in which the operation ∗ is defined as follows: * 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 3 2 0 1 0 3 4 3 0 0 0 0 3 4 0 0 0 0 0 Then (X, ∗ , 0) is a BCK-algebra [20]. Let s0 , s1 , s2 ∈ [0, 1] be such that s0 > s1 > s2 . We define a map µ : X → [0, 1] by µ(0) = s0 , µ(1) = s1 and µ(2) = µ(3) = µ(4) = s2 . Routine calculations give that µ is a fuzzy ideal of X with thresholds ε = s2 and δ = s0 . But µ is not a positive implicative fuzzy ideal of X with thresholds ε = s2 and δ = s0 , because Put x = 2, y = 3, z = 4 in (O) we get µ(2 ∗ 4) ∨ ε ≥ µ((2 ∗ 3) ∗ 4) ∧ µ(3 ∗ 4) ∧ δ µ(0) ∨ ε ≥ µ(0 ∗ 4) ∧ (0) ∧ δ µ(0) ∨ ε ≥ µ(0) ∧ µ(0) ∧ δ s0 ∨ s2 ≥ s0 ∧ s0 ∧ s0 s2 ∨ s2 ≥ s0 s2 ≥ s0 238 Muhammad Zulfiqar and Muhammad Shabir 20 but s2 s0 . Theorem 6.7. A fuzzy ideal µ with thresholds ε and δ of a BCK - algebra X is a positive implicative fuzzy ideal with thresholds of X if and only if it satisfies the condition µ(x ∗ y) ∨ ε ≥ µ((x ∗ y) ∗ y) ∧ δ for all x, y ∈ X. Proof. The proof is similar to the proof of THEOREM 3.8 and THEOREM 4.8. Theorem 6.8. A fuzzy set µ of a BCK - algebra X is a positive implicative fuzzy ideal with thresholds ε and δ of X, with ε < δ if and only if µt = {x ∈ X | µ(x) ≥ t} is a positive implicative ideal of X for all ε < t ≤ δ. Proof. Suppose µ is a positive implicative fuzzy ideal of X and ε < t ≤ δ. Let x ∈ µt . Then µ(x) ≥ t. So µ(0) ∨ ε ≥ µ(x) ∧ δ ≥t∧δ ≥t > ε. Thus, µ(0) ≥ t. We get 0 ∈ µt . Let (x ∗ y) ∗ z ∈ µt and y ∗ z ∈ µt . Then µ((x ∗ y) ∗ z) ≥ t and µ(y ∗ z) ≥ t. By using condition (O), we have µ(x ∗ z) ∨ ε ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ δ ≥t∧t∧δ ≥t∧δ ≥t > ε. It follows that µ(x ∗ z) ≥ t, and so that x ∗ z ∈ µt . Positive implicative (∈, ∈ ∨q)-fuzzy ideals of BCK-algebras 21 239 Hence, µt is a positive implicative ideal of X. Conversely, assume that µt is a positive implicative ideal of X for all ε < t ≤ δ. If there exist x, y, z ∈ X such that µ(x ∗ z) ∨ ε < t0 = µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ δ. Then ε < t0 ≤ δ and (x ∗ y) ∗ z ∈ µt0 , y ∗ z ∈ µt0 , µ(x ∗ z) < t0 . Since µt0 is a positive implicative ideal of X, so x ∗ z ∈ µt0 and µ(x ∗ z) ≥ t0 . This is a contradiction with µ(x ∗ z) < t0 . Therefore, µ(x ∗ z) ∨ ε ≥ µ((x ∗ y) ∗ z) ∧ µ(y ∗ z) ∧ δ. Similarly, we can prove that µ(0) ∨ ε ≥ µ(x) ∧ δ. From Definition 6.3, the following result holds: Theorem 6.9. Let µ be a positive implicative fuzzy ideal with thresholds of ε and δ of a BCK - algebra X, with ε < δ. Then (i) µ is a positive implicative fuzzy ideal when ε = 0, δ = 1. (ii) µ is a positive implicative (∈, ∈ ∨q) - fuzzy ideal when ε = 0, δ = 0.5. ¯, ∈ ¯ ∨ q¯) - fuzzy ideal when ε = 0.5, δ = 1. (iii) µ is a positive implicative (∈ Proof. Straightforward. 7. CONCLUSION In the study of fuzzy algebraic system, we see that the positive implicative fuzzy ideals with special properties always play a central role. In this paper, we define the concepts of positive implicative (∈, ∈ ∨q) ¯, ∈ ¯ ∨ q¯) - fuzzy ideal of BCK - algebra fuzzy ideal and positive implicative (∈ and give several characterizations of a positive implicative fuzzy ideal in BCK - algebras in terms of these notions. We believe that the research along this direction can be continued, and in fact, some results in this paper have already constituted a foundation for further investigation concerning the further development of fuzzy BCK - algebras and their applications in other branches of algebra. 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