EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1831-1841 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bipolar Fuzzy Commutative Ideals in BCK-algebras Areej Almuhaimeed1,∗, Halimah Alshehri2 1 Department of Mathematics, College of Science, Taibah University, Madinah, Saudi Arabia 2 Department of Computer Science and Engineering, Faculty Applied Studies and Community Service, King Saud University, Riyadh, Saudi Arabia Abstract. This paper presents the notions of bipolar fuzzy commutative ideals of a BCK-algebra. We also study various properties regarding this concept. We prove various characterisations for bipolar fuzzy commutative ideals. Moreover, a relationship between bipolar fuzzy commutative ideals and bipolar fuzzy positive implicative ideals is presented. In addition, we present the notation of bipolar fuzzy characteristic commutative ideal. Then a relationship between a bipolar fuzzy characteristic commutative ideal and its level-cut commutative ideals is provided. 2020 Mathematics Subject Classifications: 4D05, 03E72, 08A72 Key Words and Phrases: Bipolar fuzzy ideal, commutative ideal, positive implicative ideal, characteristic ideal 1. Introduction Zadeh introduced the notation of fuzzy sets [16]. In [6], BCK-algebra was provided as a logical class of algebras. This encouraged many researchers to apply fuzzy sets to BCK-algebras, see [8], [12], [13], [14], [15] and [10] Then many concepts which related to fuzzy sets have been widely investigated. One of the most interesting concepts is a bipolar fuzzy set. The importance of it lies behind its properties and applications, for example see [3]. Many papers provide the study of bipolar fuzzy set theory in several algebraic structures see [5], [2], [11], [1], [9] and [4]. This paper provides bipolar fuzzy commutative ideals of a BCK-algebra. We also study various properties regarding this concept. We prove various characterisations for bipolar fuzzy commutative ideals. Moreover, we provide a relationship between bipolar fuzzy commutative ideals and bipolar fuzzy positive implicative ideals. In addition, we present the notation of bipolar fuzzy characteristic commutative ideal and give a relationship ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5255 Email addresses: aamuhaimeed@taibahu.edu.sa (A. Almuhaimeed), haalshehri@ksu.edu.sa (H. Alshehri) https://www.ejpam.com 1831 © 2024 EJPAM All rights reserved. A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1832 between a bipolar fuzzy characteristic commutative ideal and its level-cut commutative ideals. Our motivation is to investigate bipolar fuzzy commutative ideals in BCK-algebras and their properties. This can lead to study this concept in soft case and apply the results in decision making algorithms. 2. Preliminaries In [7], a BCK-algebra is defined as follows: Consider the algebra (X, ∗, 0). Suppose that, for all a, g, w ∈ X, we have (BCK1) ((a ∗ g) ∗ (a ∗ w) ∗ (w ∗ g) = 0), (BCK2) (a ∗ (a ∗ g)) ∗ g = 0, (BCK3) a ∗ a = 0, (BCK4) 0 ∗ a = 0, (BCK5) If a ∗ g = 0 and g ∗ a = 0, then we have f = a. Then X is a BCK-algebra. The subsequent properties are applicable to the operation ∗, for all a, g, w ∈ X: (a ∗ g) ∗ w = (a ∗ w) ∗ g (1) (a ∗ w) ∗ (a ∗ (a ∗ w)) = (a ∗ w) ∗ w (2) We will use these properties in later sections. A relation ≤ is defined on X as follows: a ≤ g ⇐⇒ a ∗ g = 0 Definition 1. [7] An ideal I of X is a non-empty subset satisfying two conditions: (1) 0 ∈ I. (2) If a ∗ g ∈ I and g ∈ I, then a ∈ I. Definition 2. [10] A bipolar fuzzy set ϱ := (X, ϱ+, ϱ−) that satisfies the following condi- tions: (1) ϱ−(0) ≤ ϱ−(a), ϱ+(0) ≥ ϱ+(a) for all a ∈ X. (2) ϱ−(a) ≤ max {ϱ−(a∗g), ϱ−(g)} and ϱ+(a) ≥ min {ϱ+(a∗g), ϱ+(g)} for all a, g ∈ X. is called a bipolar fuzzy ideal. Definition 3. [10] A bipolar fuzzy subset ϱ := (X, ϱ+, ϱ−) is called a bipolar fuzzy subal- gebra if it meets two criteria for all a, g ∈ X: (1) ϱ+(a ∗ g) ≥ min{ϱ+(a), ϱ+(g)}. (2) ϱ−(a ∗ g) ≤ max{ϱ−(a), ϱ−(g)}. This paper interests in some types of bipolar fuzzy ideals of a BCK-algebra. A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1833 3. Bipolar fuzzy commutative ideal Let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy set. A bipolar fuzzy set in X is said to be a bipolar fuzzy commutative ideal of X if (1) ϱ+(0) ≥ ϱ+(a) and ϱ−(0) ≤ ϱ−(a). (2) ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ min{ϱ+((a ∗ g) ∗ w), ϱ+(w)}. (3) ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ max{ϱ−((a ∗ g) ∗ w), ϱ−(w)}. for all a, g, w ∈ X: Example 1. Let (X; ∗, 0) be a BCK-algebra given in Table 1 as follows: * 0 r s t 0 0 0 0 0 r r 0 0 r s s r 0 s t t t t 0 Table 1: : Tabular representation of a BCK-algebra X in example 1 Consider the bipolar fuzzy set ϱ := (X, ϱ+, ϱ−) represented by: * 0 r s t ϱ− -0.5 -0.5 -0.5 -0.4 ϱ+ 0.8 0.8 0.8 0.6 Table 2: : Tabular representation of a bipolar fuzzy set in example 1 Direct calculations implies that ϱ := (X, ϱ+, ϱ−) is a bipolar fuzzy commutative ideal of X. The following theorem provides equivalent statements regarding a bipolar fuzzy com- mutative ideal. Theorem 1. Let X be a BCK-algebra and ϱ := (X, ϱ+, ϱ−) a bipolar fuzzy ideal of X. Then ϱ is a bipolar fuzzy commutative ideal if and only if ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ ϱ+(a ∗ g) ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ ϱ−(a ∗ g) (3) Proof. Suppose that ϱ is a bipolar fuzzy commutative ideal. Thus, by definition, we have ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(w)}, A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1834 ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(w)}. Set w = 0, use the fact that a ∗ 0 = 0 for all a ∈ X and condition 1 from definition, the condition in (1) holds. Conversely, assume that ϱ satisfies the conditions in (1). That ϱ is a bipolar fuzzy ideal implies that ϱ+(a ∗ g) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(w)}, ϱ−(a ∗ g) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(w)}. Applying (1), ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(w)}, ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(w)}, and so ϱ is a bipolar fuzzy commutative ideal. Theorem 2. Every bipolar fuzzy commutative ideal of a BCK-algebra X is a bipolar fuzzy ideal of X. Proof. Let ϱ be a bipolar fuzzy commutative ideal of X and a,w ∈ X, then we have min{ϱ+(a ∗ w), ϱ+(w)} = min{ϱ+((a ∗ 0) ∗ w), ϱ+(w)} ≤ ϱ+(a ∗ (0 ∗ (0 ∗ a))) = ϱ+(a) and max{ϱ−(a ∗ w), ϱ−(w)} = max{ϱ−((a ∗ 0) ∗ w), ϱ−(w)} ≥ ϱ−(a ∗ (0 ∗ (0 ∗ w))) = ϱ−(a) Therefore, ϱ is a bipolar fuzzy ideal. The following corollary results from the aforementioned theorem: Corollary 1. Every bipolar fuzzy commutative ideal of a BCK-algebra X is a bipolar fuzzy subalgebra of X. Proof. Clear. Remark 1. A bipolar fuzzy ideal of a BCK-algebra need not be a bipolar fuzzy commutative ideal. A counter example is given as follows: Consider the BCK-algebra X given in Table 3: A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1835 * 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 0 3 3 3 3 0 0 4 4 4 4 3 0 Table 3: : Tabular representation of a BCK-algebra X described above * 0 1 2 3 4 ϱ− -0.6 -0.5 -0.5 -0.4 -0.4 ϱ+ 0.8 0.7 0.7 0.7 0.6 Table 4: : Tabular representation of a bipolar fuzzy set ϱ described above Let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy set in X given by Table 4: Thus ϱ is a bipolar fuzzy ideal. However, ϱ is not a bipolar fuzzy commutative ideal as ϱ+(2 ∗ (3 ∗ (3 ∗ 2))) = 0.7 < 0.8 = min{ϱ+((2 ∗ 3) ∗ 0), ϱ+(0)}. Proposition 1. Every bipolar fuzzy commutative ideal of a BCK-algebra is order preserv- ing. Proof. Assume that ϱ := (X, ϱ+, ϱ−) is a bipolar fuzzy commutative ideal and let a, g, w ∈ X in which a ≤ w. Then a ∗ w = 0. That ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(w)}, ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ max{ϱ−((a ∗ g) ∗ w), ϱ−(w)}, and setting g = 0 implies that ϱ+(a) = ϱ+(a ∗ (0 ∗ (0 ∗ a))) ≥ min{ϱ+((a ∗ 0) ∗ w), ϱ+(w)} = min{ϱ+(a ∗ w), ϱ+(w)} = min{ϱ+(0), ϱ+(w)} = ϱ+(w), and ϱ−(a) = ϱ−(a ∗ (0 ∗ (0 ∗ a))) ≤ max {ϱ−((a ∗ 0) ∗ w), ϱ−(w)} = max {ϱ−(a ∗ w), ϱ−(w)} = max {ϱ−(0), ϱ−(w)} A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1836 = ϱ−(w) Therefore, ϱ := (X, ϱ+, ϱ−) is order preserving as required. Lemma 1. Suppose that ϱ := (X, ϱ+, ϱ−) is a bipolar fuzzy ideal. If a ∗ g ≤ w holds for some a, g, w ∈ X, then ϱ+(a) ≥ {ϱ+(g), ϱ+(w)}, ϱ−(a) ≤ max {ϱ−(g), ϱ−(w)}. Recall that a BCK-algebra X is commutative if it satisfies the condition: f ∗ (f ∗ g) = g ∗ (g ∗ f). Theorem 3. Let X be a commutative BCK-algebra. Then every bipolar fuzzy ideal of X is a bipolar fuzzy commutative ideal. Proof. Let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy ideal and a, g, w ∈ X. Then ((a ∗ (g ∗ (g ∗ a))) ∗ ((a ∗ g) ∗ w)) ∗ w = ((a ∗ (g ∗ (g ∗ a))) ∗ w) ∗ ((a ∗ g) ∗ w) ≤ (a ∗ (g ∗ (g ∗ a))) ∗ (a ∗ g) = (a ∗ (a ∗ g)) ∗ (g ∗ (g ∗ a)) = 0, and so (a ∗ (g ∗ (g ∗ a))) ∗ ((a ∗ g) ∗ w) ≤ w. Applying lemma 1, ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ min{ϱ+((a ∗ g) ∗ w), ϱ+(w)}, ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(w)}. Therefore, ϱ is a bipolar fuzzy commutative ideal. Suppose that ϱ := (X, ϱ+, ϱ−) is a bipolar fuzzy set. Let (α, β) ∈ [−1, 0)×(0, 1]. Recall that N(ϱ;α) = {a ∈ X : ϱ−(a) ≤ α} is said to be the negative α-cut of ϱ, and P (ϱ;β) = {r ∈ X : ϱ+(r) ≥ β} is called the positive β-cut of ϱ. Theorem 4. Let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy set. Then ϱ is a bipolar fuzzy commutative ideal of X if and only if both the non-empty negative α-cut and the non- empty positive β-cut of ϱ are commutative ideals of X for all (α, β) ∈ [−1, 0)× (0, 1]. A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1837 Proof. Let ϱ be a bipolar fuzzy commutative ideal of X. For any fixed α ∈ [−1, 0) and β ∈ (0, 1], if ϱ−(0) ≤ α and ϱ+(0) ≥ β, it implies that 0 ∈ ϱ−(α) and 0 ∈ ϱ+(β). Thus the first condition holds. Let ((a ∗ g) ∗ w) ∈ ϱ−(α) ∩ ϱ+(β) and t ∈ ϱ−(α) ∩ ϱ+(β). It follows that ϱ−((a ∗ g) ∗ w) ≤ α, ϱ+((a ∗ g) ∗ w) ≥ β and ϱ−(w) ≤ α, ϱ+(w) ≥ β. By definition, we obtain ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(w)} ≥ β, ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(w)} ≤ α. Therefore, ϱ is a commutative ideal of X. Now, assume that ϱ := (X, ϱ+, ϱ−) is a commutative ideal of X for all (α, β) ∈ [−1, 0)(0, 1], ϱ−(0) ≤ α and ϱ+(0) ≥ β. Suppose that ϱ−((a∗g)∗w) = α1 and ϱ−(w) = α2 for some a, g, w ∈ X. This implies that ((a ∗ g) ∗ w) ∈ ϱ−(α1) and w ∈ ϱ−(α2). Without loss of generality, let α1 ≤ α2. Let ϱ+((a ∗ g) ∗ w) = β1, ϱ+(w) = β2 and β1 ≥ β2. Then ϱ−(α2) ≤ ϱ−(α1) which means that w ∈ ϱ−(α1). Now, ϱ+(β1) ≥ ϱ+(β2) and hence w ∈ ϱ+(β1). That ϱ −(α1) and ϱ+(β1) are commutative ideals of X, implies that a ∗ (g ∗ (g ∗ a)) ∈ (ϱ−(α1) ∩ ϱ+(β1)). Hence ϱ−(a ∗ (g ∗ (g ∗ a))) ≤ α1 = min {ϱ−((a ∗ g) ∗ w), ϱ−(h)}, ϱ+(a ∗ (g ∗ (g ∗ a))) ≥ β1 = max {ϱ+((a ∗ g) ∗ w), ϱ+(w)}. Thus ϱ−(0) ≤ ϱ−(a) and ϱ+(0) ≥ ϱ+(a) . Hence, ϱ is a bipolar fuzzy commutative ideal as required. 4. Characteristic Commutative Ideals Recall that a commutative ideal A of a BCK-algebra X that satisfies: if for every φ ∈ Aut(X), the set of all automorphisms of X, we have φ(A) = A, is said to be a characteristic commutative ideal of X A bipolar fuzzy commutative ideal A of a BCK-algebra X is said to be a bipolar fuzzy characteristic commutative ideal of X if for every φ ∈ Aut(X) we have ϱ+(φ(a)) = ϱ+(a) and ϱ−(φ(a)) = ϱ−(a). Theorem 5. Let X be a BCK-algebra and let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy charac- teristic commutative ideal of X. Then any negative α-cut and positive β-cut commutative ideals of ϱ are characteristic commutative ideals of X. A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1838 Proof. Consider a negative α-cut, N(ϱ;α) = ϱα and a positive β-cut, P (ϱ;β) = ϱβ of ϱ. Let α, β ∈ Im(f), the image of ϱ, φ ∈ Aut(X), r ∈ ϱβ and s ∈ ϱα. That ϱ is a bipolar fuzzy characteristic commutative ideal of X, implies that ϱ+(φ(r)) = ϱ+(r) ≥ β, ϱ−(φ(s)) = ϱ−(s) ≤ α. Then φ(r) ∈ ϱβ and φ(s) ∈ ϱα which means that φ(ϱβ) ⊆ ϱβ and φ(ϱα) ⊆ ϱα. On the other hand, let r ∈ ϱβ, s ∈ ϱα and a, b ∈ X in which φ(a) = r and φ(b) = s. It follows that ϱ+(a) = ϱ+(φ(a)) = ϱ+(r) ≥ β, ϱ−(b) = ϱ−(φ(b)) = ϱ−(s) ≤ α, and thus a ∈ ϱβ and b ∈ ϱα. Now, r = φ(a) ∈ φ(ϱβ), s = φ(b) ∈ φ(ϱα), and hence ϱβ ⊆ φ(ϱβ) and ϱα ⊆ φ(ϱα). Therefore, ϱβ and ϱα are bipolar characteristic commutative ideals of X as required. Theorem 6. Let X be a BCK-algebra and let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy com- mutative ideal of X. If each negative α-cut and each positive β-cut commutative ideal of ϱ is a characteristic commutative ideal of X, then ϱ is a bipolar fuzzy characteristic commutative ideal of X. Proof. Let r ∈ X, φ ∈ Aut(X) and f(r) = β. It follows that r ∈ ϱβ and r ∈ ϱγ for all γ > β. That φ(ϱβ) = ϱβ, implies that φ(r) ∈ ϱβ. Thus ϱ+(φ(r)) ≥ β. Let γ = ϱ+(φ(r)). Then φ(r) ∈ ϱγ = φ(ϱγ) by hypothesis. That φ ∈ Aut(X), implies that φ is injective and so r ∈ ϱγ . This means that γ cannot be greater than β and hence ϱ+(φ(r)) = β = ϱ+(r). Similarly, we can show that ϱ−(φ(r)) = α = ϱ−(r) for every r ∈ X, φ ∈ Aut(X) and f(r) = α. Therefore, ϱ is a bipolar fuzzy characteristic commutative ideal of X. 5. Bipolar fuzzy commutative positive ideal Recall that a bipolar fuzzy ideal of a BCK-algebra X is said to be a bipolar fuzzy positive implicative ideal of X if ϱ+(a ∗ w) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(g ∗ w)}, ϱ−(a ∗ w) ≤ max {ϱ−((a ∗ g) ∗ w)), ϱ−(g ∗ w)}. If X is positive implicative, then we have (a ∗ w) ∗ (g ∗ w) = ((a ∗ g) ∗ w) (4) A. Almuhaimeed, H. Alshehri / Eur. J. Pure Appl. Math, 17 (3) (2024), 1831-1841 1839 Theorem 7. Let X be positive implicative. A bipolar fuzzy commutative ideal of X is a bipolar fuzzy positive implicative ideal of X if and only if: ϱ+(a ∗ g) ≥ ϱ+((a ∗ g) ∗ w) ϱ−(a ∗ g) ≤ ϱ−((a ∗ g) ∗ w) (5) Proof. Let ϱ := (X, ϱ+, ϱ−) be a bipolar fuzzy commutative ideal of X. Assume that ϱ is a bipolar fuzzy positive implicative ideal. Then ϱ+(m ∗ p) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(n ∗ p)}, ϱ−(m ∗ p) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(n ∗ p)}. Put p = n, we obtain ϱ+(m ∗ n) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(0)}, ϱ−(m ∗ n) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(0)}. Thus we have ϱ+(m ∗ n) ≥ ϱ+((a ∗ g) ∗ w) ϱ−(m ∗ n) ≤ ϱ−((a ∗ g) ∗ w) Conversely, suppose that (5) holds. We have ϱ+(a ∗ w) ≥ ϱ+((a ∗ g) ∗ w) by(5) = ϱ+((a ∗ w) ∗ (a ∗ (a ∗ w))) by(2) ≥ ϱ+((a ∗ w) ∗ (a ∗ (a ∗ (a ∗ w)))) X is positive implicative ≥ min {ϱ+(((a ∗ w) ∗ a) ∗ g), ϱ+(g)} ϱ is a bipolar fuzzy commutative ideal = min {ϱ+(((a ∗ w) ∗ g) ∗ a), ϱ+(g)} by(1) = min {ϱ+((a ∗ g) ∗ w) ∗ a), ϱ+(g)} by(1) ≥ min {ϱ+((a ∗ g) ∗ w), ϱ+(g)} by Proposition 1 = min {ϱ+((a ∗ (g ∗ w)) ∗ w), ϱ+(g ∗ w)} (Set g = g ∗ w) = min {ϱ+((a ∗ w) ∗ (g ∗ w)), ϱ+(g ∗ w)} by(1) = min {ϱ+((a ∗ g) ∗ w), ϱ+(g ∗ w)} by(4) On the other hand, ϱ−(a ∗ w) ≤ ϱ−((a ∗ w) ∗ w) by(5) = ϱ−((a ∗ w) ∗ (a ∗ (a ∗ w))) by(2) ≤ ϱ−((a ∗ w) ∗ (a ∗ (a ∗ (a ∗ w)))) X is positive implicative ≤ max {ϱ−((a ∗ w) ∗ a) ∗ g), ϱ−(g)} ϱ is a bipolar fuzzy commutative ideal REFERENCES 1840 = max {ϱ−(((a ∗ w) ∗ g) ∗ a), ϱ−(g)} by(1) = max {ϱ−(((a ∗ g) ∗ w) ∗ a), ϱ−(g)} by(1) ≤ max {ϱ−((a ∗ g) ∗ w), ϱ−(g)} by Proposition 1 = max {ϱ−(((a ∗ (g ∗ w)) ∗ w), ϱ−(g ∗ w)} (Set g = g ∗ w) = max {ϱ−((a ∗ w) ∗ (g ∗ w)), ϱ−(g ∗ w)} by(1) = max {ϱ−((a ∗ g) ∗ w), ϱ−(g ∗ w)} by(4) This completes the proof. 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