International Journal of Analysis and Applications Volume 18, Number 5 (2020), 859-875 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-859 NEW TYPES OF BIPOLAR FUZZY IDEALS OF BCK-ALGEBRAS G. MUHIUDDIN1,∗, D. AL-KADI2, A. MAHBOOB3, K.P. SHUM4 1Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia 2Department of Mathematics and Statistic, Taif University, Taif 21974, Saudi Arabia 3Department of Mathematics, Madanapalle Institute of Technology & Science, Madanapalle-517325, India 4Institute of Mathematics, Yunnan University, Kunming 650091, People’s Republic of China ∗Corresponding author: chishtygm@gmail.com Abstract. The notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are introduced, and related properties are investigated. Characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are given, and several properties are discussed. Finally, we prove that if T is an implicative BCK-algebra, then a fuzzy subset µ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T. 1. Introduction BCK-algebras and BCI-algebras are two classes of non-classical logic algebras which were introduced by Y. Imai and K. Iseki in 1966 (see [9, 10]). They are algebraic formulation of BCK-system and BCI-system in combinatory logic. Fuzzy sets, which were introduced by Zadeh [29], deal with possibilistic uncertainty, connected with imprecision of states, perceptions and preferences. After the introduction of fuzzy sets by Zadeh, fuzzy set theory has become an active area of research in various fields. These are widely scattered over many Received June 1st, 2020; accepted July 1st, 2020; published July 29th, 2020. 2010 Mathematics Subject Classification. 06D72, 06F35, 16D25, 94D05. Key words and phrases. BCK-algebras; fuzzy ideal; bipolar fuzzy ideal; bipolar fuzzy closed ideal; bipolar fuzzy positive implicative ideal; bipolar fuzzy implicative ideal. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 859 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-859 Int. J. Anal. Appl. 18 (5) (2020) 860 disciplines such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others. The elements in fuzzy sets have degrees of belonging which range between 0 and 1. If the membership degree of an element is 0 then the element does not belong to the fuzzy set and it completely belongs to the correspondent fuzzy set if the membership is 1. If the membership lies between (0, 1) then it belongs partially to the fuzzy set. Among such elements some have irrelevant characteristics to the property corresponding to a fuzzy set and the others have contrary characteristics to the property and the traditional fuzzy set representation cannot be described it. Moreover, Muhiuddin et al. studied the fuzzy set theoretical approach to the BCK/BCI-algebras on various aspects (see for e.g., [22–25]). Also, some related concepts of fuzzy sets in different algebras have been studied in [1, 3, 4, 6, 26–28]. As a generalization of traditional fuzzy sets, Zhang [30] first introduced the bipolar fuzzy sets concept . Also, Lee [16, 17] studied the bipolar fuzzy sets in which a negative (resp. positive) membership degree is given for each element in the fuzzy set that ranges over the interval [−1, 0] (resp. [0, 1]). Later on, a number of research papers have been devoted to the study of bipolar fuzzy set theory in several algebraic structures (see for e.g., [7, 12, 14, 15, 20, 21]. In this paper, we introduce the notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras and investigate related properties. We give characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras and several properties are discussed. Finally, we prove that if T is an implicative BCK-algebra, then a fuzzy subset µ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T. 2. Preliminaries We review some definitions and properties that will be useful in our results. Definition 2.1. An algebra (T ;∗, 0) of kind (2, 0) is called a BCK-algebra if it satisfies the following conditions: (K1) ((t∗u) ∗ (t∗v)) ∗ (v ∗u) = 0, (K2) (t∗ (t∗u)) ∗u = 0, (K3) t∗ t = 0, (K4) 0 ∗ t = 0, (K5) t∗u = 0 and u∗ t = 0 ⇒ t = u, for all t,u,v ∈ T In a BCK-algebra, the following are true. (K6) t∗ 0 = t, Int. J. Anal. Appl. 18 (5) (2020) 861 (K7) (t∗u) ∗v = (t∗v) ∗u. A nonempty subset X of a BCK-algebra T is called an ideal of T if it satisfies (I1) 0 ∈ X, (I2) ∀t,u ∈ T,t∗u ∈ X,u ∈ X ⇒ t ∈ X. A nonempty subset X of a BCK-algebra T is called an implicative ideal of T if it satisfies (I1) and (I3) ∀t,u,v ∈ T, ((t∗ (u∗ t)) ∗v ∈ X,v ∈ X ⇒ t ∈ X. A fuzzy set µ in T is called a fuzzy ideal of T if it satisfies (F1) µ(0) ≥ µ(t), (F2) µ(t) ≥ µ(t∗u) ∧µ(u). A fuzzy positive implicative ideal of T is a fuzzy set µ in T which satisfies (F1) and (F3) µ(t∗v) ≥ µ((t∗u) ∗v) ∧µ(u∗v). A fuzzy implicative ideal of T is a fuzzy set µ in T which satisfies (F1) and (F4) µ(t) ≥ µ((t∗ (u∗ t)) ∗v) ∧µ(v). For more information regarding BCK-algebras, we refer the reader to [19] Lemma 2.1. [15] In a BCK-algebra T , every bipolar fuzzy ideal of T is a bipolar fuzzy subalgebra of T . 3. Bipolar fuzzy ideal In the following sections, T denotes a BCK-algebra unless otherwise specified. For any family {γi | i ∈ Γ} of real numbers, we define ∨{γi | i ∈ Γ} :=   max{γi | i ∈ Γ} if Γ is finite, sup{γi | i ∈ Γ} otherwise, ∧{γi | i ∈ Γ} :=   min{γi | i ∈ Γ} if Γ is finite, inf{γi | i ∈ Γ} otherwise. Moreover, if Γ = {1, 2, ...,n}, then ∨{γi | i ∈ Γ} and ∧{γi | i ∈ Γ} are denoted by γ1 ∨ γ2 ∨ ... ∨ γn and γ1 ∧γ2 ∧ ...∧γn, respectively. For a bipolar fuzzy set µ = (T ; µn,µp) and (α,β) ∈ [−1, 0) × (0, 1], we define (N) N(µ; α) := {t ∈ T : µn(t) ≤ α} is called the negative α− cut of µ. (P) P(µ; β) := {t ∈ T : µp(t) ≥ β} is called positive β − cut of µ. Int. J. Anal. Appl. 18 (5) (2020) 862 The set C(µ; (α,β)) := N(µ; α) ∩P(µ; β) is called the (α,β) − cut of µ = (T ; µn,µp). For every k ∈ (0, 1), if (α,β) = (−k,k), then the set C(µ; k) := N(µ;−k) ∩P(µ; k) is called the k − cut of µ = (T ; µn,µp). Definition 3.1. [15] A bipolar fuzzy set µ = (T ; µn,µp) in a BCK-algebra T is called a bipolar fuzzy ideal of T if it satisfies the following assertions: (BF1) (∀t ∈ T) (µn(0) ≤ µn(t), µp(0) ≥ µp(t)), (BF2) (∀t,u ∈ T) (µn(t) ≤ µn(t∗u) ∨µn(u), µp(t) ≥ µp(t∗u) ∧µp(u)). Example 3.1. Consider the BCK-algebra (T ;∗, 0) given in Table 1. ∗ 0 c d e 0 0 0 0 0 c c 0 0 c d d c 0 d e e e e 0 Table 1: Cayley table Define a bipolar fuzzy set µ = (T ; µn,µp) in T by ∗ 0 c d e µn -0.2 -0.3 -0.3 -0.8 µp 0 0.5 0.5 1 Then by routine calculations, µ = (T ; µn,µp) is a bipolar fuzzy ideal of T . Int. J. Anal. Appl. 18 (5) (2020) 863 Example 3.2. Consider the BCK-algebra (T ;∗, 0) given in Table 2. ∗ 0 a c d e 0 0 0 0 0 0 a a 0 a 0 0 c c c 0 0 0 d d d d 0 0 e e d e a 0 Table 2: Cayley table Define µ by ∗ 0 a c d e µn -0.7 -0.6 -0.7 -0.6 -.06 µp 0.6 0.1 0.6 0.1 0.1 Then by routine calculations, µ = (T ; µn,µp) is a bipolar fuzzy ideal of T . Theorem 3.1. If µ = (T ; µn,µp) is a bipolar fuzzy ideal in T and I(0) = {t ∈ T : µn(t) = µn(0),µp(t) = µp(0)}. Then I(0) is an ideal of T . Proof. Let t,u ∈ T be such that t ∗ u ∈ I(0) and u ∈ I(0). Then we have µn(t ∗ u) = µn(0),µp(t ∗ u) = µp(0),µn(u) = µn(0) and µp(u) = µp(0). Thus using Definition 3.1 (BF2), we get µn(t) ≤ µn(0) ∨µn(0) = µn(0) and µp(t) ≥ µn(0) ∧ µn(0) = µn(0). On the other hand, we know from Definition 3.1 (BF1) that µn(0) ≤ µn(t), µp(0) ≥ µp(t) and so µn(t) = µn(0), µp(t) = µp(0). Hence, t ∈ I(0). It is obvious that 0 ∈ I(0). Therefore, I(0) is an ideal of T . � If µ = (T ; µn,µp) is a bipolar fuzzy set in T which satisfies Definition 3.1 (BF1) then the following example shows that Definition 3.1 (BF2) is sufficient for I(0) to be an ideal of T. Example 3.3. Consider the BCK-algebra (T ;∗, 0) given in Table 3. ∗ 0 a c d e 0 0 0 0 0 0 a a 0 a 0 0 c c c 0 c 0 d d d d 0 d e e e e e 0 Table 3: Cayley table Int. J. Anal. Appl. 18 (5) (2020) 864 Define µ by ∗ 0 a c d e µn -0.7 -0.7 -0.5 -0.3 -.07 µp 0.8 0.8 0.4 0.2 0.8 Note that µ = (T ; µn,µp) is a bipolar fuzzy set in T which satisfies Definition 3.1 (BF1) and not (BF2) as there exist c,e ∈ T where µn(c) = −0.5 6≤−0.7 = µn(c∗e) ∨µn(e). Then I(0) = {0,a,e} is not an ideal of T as c∗e = 0 ∈ I(0) and e ∈ I(0), but c 6∈ I(0). Theorem 3.2. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T and let ∅ 6= I ⊆ T such that 0 ∈ I. Define µn(t) = α1 or α2 whenever t ∈ I or t 6∈ I respectively, and µp(t) = β1 or β2 whenever t ∈ I or t 6∈ I respectively where α1,α2 ∈ [−1, 0] such that α1 ≤ α2 and β1,β2 ∈ [0, 1] such that β1 ≥ β2. Then Definition 3.1 (BF2) and the implication (∀t,u ∈ T) (t∗u ∈ I and u ∈ I ⇒ t ∈ I) are equivalent. Proof. Let t ∗ u ∈ I and u ∈ I. Then µn(t ∗ u) = µn(u) = α1, and µp(t ∗ u) = µp(u) = β1. Therefore, µn(t) ≤ µn(t ∗ u) ∨ µn(u) = α1 ∨ α1 = α1 and so µn(t) = α1 as α1 ≤ α2. Similarly, µp(t) ≥ β1 and so µp(t) = β1 as β1 ≥ β2. So whenever t∗u ∈ I and u ∈ I we have t ∈ I. Conversely, if the implication is valid then for t,u ∈ T we have µn(t∗u) = µn(u) = µn(t) = α1, µp(t∗u) = µp(u) = µp(t) = β1 . Then it is obvious to see that Definition 3.1 (BF2) is satisfied. � Having 0 ∈ I in Theorem 6.1 and as (∀t ∈ T) (µn(0) = µn(t ∗ t) ≤ µn(t), µp(0) = µp(t ∗ t) ≥ µp(t)). Thus (∀t ∈ T) (µn(0) ≤ µn(t), µp(0) ≥ µp(t)). That is Definition 3.1 (BF1) is satisfied. Hence, we have the following theorem. Theorem 3.3. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T and let ∅ 6= I ⊆ T such that 0 ∈ I. Define µn(t) = α1 or α2 whenever t ∈ I or t 6∈ I respectively. Define µp(t) = β1 or β2 whenever t ∈ I or t 6∈ I respectively where α1,α2 ∈ [−1, 0] such that α1 ≤ α2 and β1,β2 ∈ [0, 1] such that β1 ≥ β2. Then µ = (T; µn,µp) is a bipolar fuzzy ideal of T if and only if I is an ideal of T . Theorem 3.4. [15] A bipolar fuzzy set µ = (T ; µn,µp) in T is a bipolar fuzzy ideal of T if both the nonempty negative α-cut and the nonempty positive β-cut of µ = (T ; µn,µp) are ideals of T for all (α,β) ∈ [−1, 0) × (0, 1]. Corollary 3.1. If µ = (T ; µn,µp) in T is a bipolar fuzzy ideal of T , then the k-cut of µ = (T ; µn,µp) is an ideal of T for all k ∈ (0, 1). Int. J. Anal. Appl. 18 (5) (2020) 865 The following example shows the converse of corollary 3.1 may not be true. Example 3.4. Let T = {0, 1, 2, 3, 4} be a set in which the operation ∗ is defined by the following Cayley table which is given in Table 4. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 2 0 3 3 3 3 0 0 4 4 4 3 2 0 Table 4: Cayley table Then (T ;∗, 0) is a BCK-algebra. Let µ = (T; µn,µp) be a bipolar fuzzy set in T given by ∗ 0 1 2 3 4 µn -0.7 -0.4 -0.6 -0.4 -0.2 µp 0.8 0.2 0.6 0.3 0.4 By routine calculations, we know that C(µ; 0.4) := N(µ;−0.4) ∩P(µ; 0.4) = {0, 2}. is an ideal of T , but µ = (T; µn,µp) is not an bipolar fuzzy ideal of T because µn(4) = −0.2 6≤−0.4 = µn(4 ∗ 2) ∨µn(2). Lemma 3.1. [15] Every bipolar fuzzy ideal µ = (T ; µn,µp) of T satisfies the following implication. (∀t,u ∈ T)(t ≤ u ⇒ µn(t) ≤ µn(u),µp(t) ≥ µp(u)). Proposition 3.1. [15] A bipolar fuzzy set µ = (T ; µn,µp) in T is a bipolar fuzzy ideal of T if and only if for all t,u,v ∈ T , (t∗u) ∗v = 0 implies µn(t) ≤ µn(u) ∨µn(v) and µp(t) ≥ µp(u) ∨µp(v). 4. Bipolar fuzzy closed ideal Definition 4.1. A bipolar fuzzy ideal µ = (T ; µn,µp) in T is said to a bipolar fuzzy closed ideal if µn(0∗t) ≤ µn(t) and µp(0 ∗ t) ≥ µp(t) for all t ∈ T . Example 4.1. Consider the BCI-algebra (T ;∗, 0) given in Table 5. Int. J. Anal. Appl. 18 (5) (2020) 866 ∗ 0 a c d e 0 0 0 c d e a a 0 c d e c c c 0 e d d d d e 0 c e e e d c 0 Table 5: Cayley table Define µ by ∗ 0 a c d e µn -0.8 -0.6 -0.8 -0.4 -.03 µp 0.6 0.6 0.4 0.3 0.1 Then µ = (T ; µn,µp) is a bipolar fuzzy closed ideal of T . As an extension to Theorem 3.3 we give the following theorem. Theorem 4.1. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T and let ∅ 6= I ⊆ T such that 0 ∈ I. Define µn(t),µp(t) as defined in Theorem 3.3. Then µ = (T ; µn,µp) is a bipolar fuzzy closed ideal of T if and only if I is a closed ideal of T . Proof. Assume µ = (T; µn,µp) is a bipolar fuzzy closed ideal of T and we show that 0 ∗ t ∈ I, ∀t ∈ I. For t ∈ I, we have µn(t) = α1,µp(t) = β1. As 0 ∗ t ≤ 0 then µn(0 ∗ t) ≤ µn(0), µp(0 ∗ t) ≥ µp(0). Therefore, µn(0 ∗ t) ≤ µn(0) = α1 and so µn(0 ∗ t) = α1 as α1 ≤ α2. Also µn(0 ∗ t) = β1 as β1 ≥ β2. That is, I is a closed ideal. For the converse, let I be a closed ideal of T and show that µn(0∗t) ≤ µn(t), µp(0∗t) ≥ µp(t). For t ∈ T we know that 0 ∗ t ≤ 0 and so µn(0 ∗ t) ≤ µn(0) ≤ µn(t). Also µp(0 ∗ t) ≥ µp(0) ≥ µp(t). This proves that µ = (T; µn,µp) is a bipolar fuzzy closed ideal of T. � Theorem 4.2. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T . If one of the following assertion is satisfied (1) (∀t,u ∈ T) (µn(t) ≤ µn(t∗u) ∨µn(u),µp(t) ≥ µp(t∗u) ∧µp(u)) (2) (∀t,u ∈ T) (µn(t) ≤ µn(u∗ t) ∨µn(u),µp(t) ≥ µp(u∗ t) ∧µp(u)) then µ = (T ; µn,µp) is a bipolar fuzzy closed ideal of T . Proof. We will show that using any of the assertions above we get µn(0 ∗ t) ≤ µn(t), µp(0 ∗ t) ≥ µp(t). Using (1), let u = 0 then we have µn(0 ∗ t) ≤ µn((0 ∗ t) ∗ 0) ∨ µn(0) = µn(0). As 0 ∗ t = 0 we know that 0 ≤ t and so µn(0) ≤ µn(t). Therefore, µn(0 ∗ t) ≤ µn(t). Also knowing that µp(0) ≥ µp(t) and µp(0 ∗ t) ≥ µp((0 ∗ t) ∗ 0) ∧µp(0) = µp(0) we get µp(0 ∗ t) ≥ µp(t). Use the same approach with assertion (2) and let u = 0 to have µn(0∗ t) ≤ µn(0∗(0∗ t))∨µn(0) = µn(0) ≤ µn(t). Therefore, µn(0 ∗ t) ≤ µn(t), µp(0 ∗ t) ≥ µp(t). � Int. J. Anal. Appl. 18 (5) (2020) 867 5. Bipolar fuzzy positive implicative ideal Definition 5.1. A bipolar fuzzy set µ = (T ; µn,µp) in T is called bipolar fuzzy positive implicative ideal of T if it satisfies (BF1) and the following assertions: (BF3) (∀t,u,v ∈ T) µn(t∗v) ≤ µn((t∗u) ∗v) ∨µn(u∗v) (BF4) (∀t,u,v ∈ T) µp(t∗v) ≥ µp((t∗u) ∗v) ∧µp(u∗v). Example 5.1. Let T = {0, 1, 2, 3} be a set in which the operation ∗ is given by the following Cayley table in Table 6. ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 1 0 2 3 3 3 3 0 Table 6: Cayley table Let µ = (T ; µn,µp) be a bipolar fuzzy set in T defined as follows ∗ 0 1 2 3 µn -0.6 -0.6 -0.6 -0.4 µp 0.8 0.8 0.8 0.6 Then by routine calculations µ = (T ; µn,µp) is a bipolar fuzzy positive implicative ideal of T . Note that every bipolar fuzzy positive implicative ideal is a bipolar fuzzy ideal, but converse is not true. Example 5.2. Consider the BCK-algebra (T,∗, 0) given in Example 5.2. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T given by: ∗ 0 1 2 3 µn -0.6 -0.5 -0.5 -0.4 µp 0.8 0.7 0.7 0.6 By direct calculations we know that µ = (T ; µn,µp) is a bipolar fuzzy ideal of T but is not a bipolar fuzzy positive implicative ideal as µn(2 ∗ 1) = µ(1) = −0.5 6≤−0.6 = µn((2 ∗ 1) ∗ 1) ∨µn(1 ∗ 1). We provide a condition for a bipolar fuzzy ideal to be a bipolar fuzzy positive implicative ideal. Int. J. Anal. Appl. 18 (5) (2020) 868 Theorem 5.1. If µ = (T ; µn,µp) is a bipolar fuzzy ideal of T satisfying for all t,u,v ∈ T (BF5) µn(t∗v) ≤ µn(((t∗u) ∗u) ∗v) ∨µn(v), (BF6) µp(t∗v) ≥ µp(((t∗u) ∗u) ∗v) ∧µp(v), then µ is a bipolar fuzzy positive implicative ideal of T . Proof. Using (K1) and (K7), we have ((t∗v) ∗v) ∗ (u∗v) ≤ (t∗v) ∗u = (t∗u) ∗v, ∀t,u,v ∈ T. Since µ is order reversing, it follows from (BF3) and (BF4) that µn(t∗v) ≤ µn(((t∗v) ∗v) ∗ (u∗v)) ∨µn(u∗v) ≤ µn((t∗u) ∗v) ∨µn(u∗v) and µp(t∗v) ≥ µp(((t∗v) ∗v) ∗ (u∗v)) ∧µp(u∗v) ≥ µp((t∗u) ∗v) ∧µp(u∗v) for all t,u,v ∈ T . Hence, µ is a bipolar fuzzy positive implicative ideal of T . � Theorem 5.2. Let w ∈ T . If µ = (T; µn,µp) is a bipolar fuzzy positive implicative ideal of T , then I(w) is a positive implicative ideal of T . Proof. Recall that 0 ∈ I(w). Let t,u ∈ T be such that (t ∗ u) ∗ v ∈ I(w) and u ∗ v ∈ I(w). Then µn(w) ≥ µn((t∗u)∗v), µp(w) ≤ µp((t∗u)∗v), µn(w) ≥ µn(u∗v) and µp(w) ≤ µp(u∗v). Since µ = (T; µn,µp) is a bipolar fuzzy positive implicative ideal of T , it follows from (BF3) and (BF4) that µn(t∗v) ≤ µn((t∗u) ∗v) ∨µn(u∗v) ≤ µn(w) and µp(t∗v) ≥ µp((t∗u) ∗v) ∧µp(u∗v) ≥ µp(w) so that t∗v ∈ I(w). Therefore I(w) is a positive implicative ideal of T . � Lemma 5.1. Let µ = (T ; µn,µp) be a bipolar fuzzy ideal of T . Then • µ is a bipolar fuzzy positive implicative ideal of T if and only if it satisfies (BF7) ∀t,u ∈ T,µn(t∗u) ≤ µn((t∗u) ∗u) and µp(t∗u) ≥ µp((t∗u) ∗u). • µ is a bipolar fuzzy positive implicative ideal of T if and only if it satisfies (BF8) ∀t,u,v ∈ T,µn((t∗v) ∗ (u∗v)) ≤ µn((t∗u) ∗v) and µp((t∗v) ∗ (u∗v)) ≥ µp((t∗u) ∗v). Theorem 5.3. If µ is a bipolar fuzzy positive implicative ideal of T , then (BF9) ∀t,u,a,b ∈ T, ((t∗u) ∗u) ∗a ≤ b ⇒ µn(t∗u) ≤ µn(a) ∨µn(b) and µp(t∗u) ≥ µp(a) ∧µp(b). (BF10) ∀t,u,v,a,b ∈ T, ((t∗u) ∗v) ∗a ≤ b ⇒ µn((t∗v) ∗ (u∗v)) ≤ µn(a) ∨µn(b) Int. J. Anal. Appl. 18 (5) (2020) 869 and µp((t∗v) ∗ (u∗v)) ≥ µp(a) ∧µp(b). Proof. Let t,u,a,b ∈ T be such that ((t∗u) ∗u) ∗a ≤ b. Using Proposition 3.1, we have µn((t∗u) ∗u) ≤ µn(a) ∨µn(b) and µp((t∗u) ∗u) ≥ µp(a) ∧µp(b). It follows that from (BF3), (BF4), (K3) and (BF1), µn(t∗u) ≤ µn((t∗u) ∗u) ∨µn(u∗u) = µn((t∗u) ∗u) ∨µn(0) = µn((t∗u) ∗u) µn(t∗u) ≤ µn(a) ∨µn(b) and µp(t∗u) ≥ µp((t∗u) ∗u) ∧µp(u∗u) = µp((t∗u) ∗u) ∧µp(0) = µp((t∗u) ∗u) µp(t∗u) ≥ µp(a) ∧µp(b). Now let t,u,v,a,b ∈ T be such that ((t∗u) ∗v) ∗a ≤ b, that is, (((t∗u) ∗v) ∗a) ∗ b = 0. Since µ is a bipolar fuzzy positive implicative ideal of T , it follows from Proposition 3.1 and Lemma 5.1 that µn((t∗v) ∗ (u∗v)) ≤ µn((t∗u) ∗v) ≤ µn(a) ∨µn(b) and µp((t∗v) ∗ (u∗v)) ≥ µp((t∗u) ∗v) ≥ µp(a) ∧µp(b). This completes the proof. � We now give conditions for a bipolar fuzzy set to be a bipolar fuzzy positive implicative ideal. Theorem 5.4. Let µ be a bipolar fuzzy set in T satisfying the condition (BF9). Then µ is a bipolar fuzzy positive implicative ideal of T . Proof. We first prove that µ is a bipolar fuzzy ideal of T. Let t,u,v ∈ T be such that t∗u ≤ v. Then (((t∗ 0) ∗ 0) ∗u) ∗v = (t∗u) ∗v = 0, that is, ((t∗ 0) ∗ 0) ∗u ≤ v, which implies from (K6) and (BF9) that µn(t) = µn(t∗ 0) ≤ µn(u) ∨µn(v) and µp(t) = µp(t∗ 0) ≥ µp(u) ∧µp(v). Therefore, by Proposition 3.1, we know that µ is a bipolar fuzzy ideal of T . Note that (((t∗u) ∗u) ∗ ((t∗ u) ∗u)) ∗ 0 = 0 for all t,u ∈ T . Using (BF9) and (BF1), we have µn(t∗u) ≤ µn((t∗u) ∗u) ∨µn(0) = µn((t∗u) ∗u) Int. J. Anal. Appl. 18 (5) (2020) 870 and µp(t∗u) ≥ µp((t∗u) ∗u) ∧µp(0) = µp((t∗u) ∗u). So µ is a bipolar fuzzy positive implicative ideal of T by Lemma 5.1. � Theorem 5.5. Let µ be a fuzzy set in T satisfying the condition (BF10). Then µ is a bipolar fuzzy positive implicative ideal of T . Proof. Let t,u,a,b ∈ T be such that ((t∗u) ∗u) ∗a = b, that is (((t∗u) ∗u) ∗a) ∗ b = 0, which implies from (K6), (K3) and (BF10) that µn(t∗u) = µn((t∗u) ∗ 0) = µn((t∗u) ∗ (u∗u)) ≤ µn(a) ∨µn(b) and µp(t∗u) = µp((t∗u) ∗ 0) = µp((t∗u) ∗ (u∗u)) ≥ µn(a) ∧µp(b). So µ is a bipolar fuzzy positive implicative ideal of T by Theorem 5.4. � Theorem 5.6. Let µ and λ be bipolar fuzzy ideals of T such that µn(0) = λn(0),µp(0) = λp(0) and µ ⊆ λ, that is µn(t) ≤ λn(t) and µp(t) ≥ λp(t) for all t ∈ T . If µ is bipolar fuzzy positive implicative ideal of T , then so is λ. Proof. Assume that µ is a bipolar fuzzy positive implicative ideal of T. For any t,u,v ∈ T, we have λn(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) = λn(((t∗v) ∗ ((t∗u) ∗v)) ∗ (u∗v)) [by (K7)] = λn(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [by (K7)] ≥ µn(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [since µ ⊆ λ] ≥ µn(((t∗ ((t∗u) ∗v)) ∗u) ∗v) = µn(((t∗u) ∗ ((t∗u) ∗v)) ∗v) [by (K7)] = µn(((t∗u) ∗v) ∗ ((t∗u) ∗v)) [by (K7)] = µn(0) = λn(0). [by (K3) and assumption] It follows from (BF1) and (BF2) that λn((t∗v) ∗ (u∗v)) ≤ λn(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) ∨λn((t∗u) ∗v) ≤ λn(0) ∨λn((t∗u) ∗v) = λn((t∗u) ∗v) and λp(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) = λp(((t∗v) ∗ ((t∗u) ∗v)) ∗ (u∗v)) [by (K7)] = λp(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [by (K7)] Int. J. Anal. Appl. 18 (5) (2020) 871 ≤ µp(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [since µ ⊆ λ] ≤ µp(((t∗ ((t∗u) ∗v)) ∗u) ∗v) = µp(((t∗u) ∗ ((t∗u) ∗v)) ∗v) [by (K7)] = µp(((t∗u) ∗v) ∗ ((t∗u) ∗v)) [by (K7)] = µp(0) = λp(0). [by (K3) and assumption] It follows from (BF1) and (BF2) that λp((t∗v) ∗ (u∗v)) ≥ λp(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) ∧λp((t∗u) ∗v) ≥ λp(0) ∧λp((t∗u) ∗v) = λp((t∗u) ∗v) for all t,u,v ∈ T . Hence, by Lemma 5.1, λ is a bipolar fuzzy positive implicative ideal of T . � 6. Bipolar fuzzy implicative ideal Definition 6.1. A bipolar fuzzy set µ = (T ; µn,µp) in T is called bipolar fuzzy implicative ideal of T if it satisfies (BF1) and the following assertions: (BF11) (∀t,u,v ∈ T) µn(t) ≤ µn((t∗ (u∗ t)) ∗v) ∨µn(v) (BF12) (∀t,u,v ∈ T) µp(t) ≥ µp((t∗ (u∗ t)) ∗v) ∧µp(v). Example 6.1. Let T = {0, 1, 2, 3} be a set in which the operation ∗ is defined by Table 7: ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 1 0 2 3 3 3 3 0 Table 7: Cayley table Hence, (T;∗, 0) is a BCK-algebra. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T defined by 0 1 2 3 µn -0.5 -0.5 -0.5 -0.4 µp 0.7 0.7 0.7 0.6 By routine calculations, µ = (T; µn,µp) is a bipolar fuzzy implicative ideal of T . Now, we give a relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal. Theorem 6.1. Every bipolar fuzzy implicative ideal of T is a bipolar fuzzy ideal of T . Int. J. Anal. Appl. 18 (5) (2020) 872 Proof. Let µ be a bipolar fuzzy implicative ideal of T .Then for all t,u,v ∈ T , µn(t) ≤ µn((t∗ (u∗ t)) ∗v) ∨µn(v) and µp(t) ≥ µp((t∗ (u∗ t)) ∗v) ∧µp(v). Putting u = t and v = u, µn(t) ≤ µn((t∗ (t∗ t)) ∗u) ∨µn(u) µn(t) ≤ µn((t∗ 0) ∗u) ∨µn(u) µn(t) ≤ µn(t∗u) ∨µn(u) and µp(t) ≥ µp((t∗ (t∗ t)) ∗u) ∧µp(u) µp(t) ≥ µp((t∗ 0) ∗u) ∧µp(u) µp(t) ≥ µp(t∗u) ∧µp(u). Thus µ satisfies (BF2). Consequently, µ is a bipolar fuzzy ideal of T by (BF1). � In view of Lemma 2.1 and Theorem 6.1, we conclude the following Corollary. Corollary 6.1. Every bipolar fuzzy implicative ideal of T is a bipolar fuzzy subalgebra of T . The following example shows that the converse of Theorem 6.1 is not true in general. Example 6.2. Let T = {0,a,c,d,e} be a set in which the operation ∗ is defined by Table 8: ∗ 0 a c d e 0 0 0 0 0 0 a a 0 a 0 0 c c c 0 0 0 d d d d 0 0 e e d e a 0 Table 8: Cayley table Hence, (T;∗, 0) is a BCK-algebra. Let µ = (T ; µn,µp) be a bipolar fuzzy set in T defined by ∗ 0 a c d e µn -0.8 -0.7 -0.8 -0.7 -.07 µp 0.7 0.2 0.7 0.2 0.2 By routine calculations, we know that µ = (T ; µn,µp) is a bipolar fuzzy ideal of T , but not a bipolar fuzzy implicative ideal of T since P(µ; 0.5) = {0, c} is not an implicative ideal of T . We provide a condition for bipolar fuzzy ideal to be a bipolar fuzzy implicative ideal. Int. J. Anal. Appl. 18 (5) (2020) 873 Theorem 6.2. If T is an implicative BCK-algebra, then every bipolar fuzzy ideal of T is a bipolar fuzzy implicative ideal of T . Proof. Since T is an implicative BCK-algebra, it follows that t = t ∗ (u ∗ t) for all t,u ∈ T . Let µ be a bipolar fuzzy ideal of T . Then by (BF2) µn(t) ≤ µn(t∗v) ∨µn(v) µn(t) ≤ µn((t∗ (u∗ t) ∗v)) ∨µn(v) and µp(t) ≥ µp(t∗v) ∧µp(v) µp(t) ≥ µp((t∗ (u∗ t) ∗v)) ∧µp(v). Hence, µ is a bipolar fuzzy implicative ideal of T . The proof is complete. � In view of Theorem 6.1 and Theorem 6.2, we have the following theorem. Theorem 6.3. If T is an implicative BCK-algebra, then a fuzzy subset µ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T . Theorem 6.4. Let µ be a bipolar fuzzy implicative ideal of T . Then the set Tµ = {t ∈ T : µn(t) = µn(0) and µp(t) = µp(0)} is an implicative ideal of T . Proof. Clearly 0 ∈ T . Let t,u,v ∈ Tµ be such that (t∗ (u∗ t)) ∗v ∈ Tµ and v ∈ Tµ. Then, we have µn((t∗ (u∗ t)) ∗v) = µn(v) = µn(0) and µp((t∗ (u∗ t)) ∗v) = µp(v) = µp(0). It follows that µn(t) ≤ µn((t∗ (u∗ t)) ∗v) ∨µn(v) ≤ µn(0) ∨µn(0) µn(t) ≤ µn(0) and µp(t) ≥ µp((t∗ (u∗ t)) ∗v) ∧µp(v) ≥ µp(0) ∨µp(0) µp(t) ≥ µp(0). By using (BF1), we get µn(t) = µn(0) and µp(t) = µp(0) and hence t ∈ Tµ. Consequently, Tµ is an implicative ideal of T . � Acknowledgement: The authors are grateful to the anonymous referees for a careful checking of the details and for helpful comments that improved the overall presentation of this paper. Int. J. Anal. 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