EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3973-3993 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Innovative Perspective on Bipolar Fuzzy Fantastic Ideals in BCK/BCI-Algebras M. Balamurugan1, Khalil H. Hakami2,∗, Moin A. Ansari2,∗, K. Loganathan3 1 Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, Tamil Nadu, India 2 Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Kingdom of Saudi Arabia 3 Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur-303007, India Abstract. In this paper, we propose the concept of (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy ideals in BCK/BCI-algebras. We show that an (∈,∈ ∨q̌φ)-bipolar fuzzy ideal is an (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy ideal. For a BCK/BCI-algebra, it has been shown that an (∈,∈ ∨(φ⋇, q̌φ)-bipolar fuzzy ideal is an (∈,∈ ∨q̌)-bipolar fuzzy ideal of ℵ̌, but not conversely, and then an example is given. We introduce the concept of (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals in BCK/BCI-algebras. It has been shown that an (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideal is an (∈,∈)-bipolar fuzzy ideal in BCK/BCI-algebras. Furthermore, the connection between (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals and fantastic ideals are established. 2020 Mathematics Subject Classifications: 06F35, 03G25, 03B52 Key Words and Phrases: BCK/BCI-algebra, fuzzy logic, bipolar fuzzy ideal (BFI), bipolar fuzzy fantastic ideal (BFFI), (∈,∈ ∨(φ⋇, q̌φ))-BFI, (∈,∈ ∨(φ⋇, q̌φ))-BFFI 1. Introduction Zadeh [45] introduced the concept of fuzzy set theory in 1965 as a mathematical frame- work for handling uncertainty and vagueness. Rather than being merely inside or outside a set, it broadens the classical thought of set theory to allow for varying degrees of mem- bership among its elements, from 0 to 1. This method is especially suitable for areas with non-binary information or where there is imprecision, such as artificial intelligence, control systems, and decision-making. Axiomatic systems of propositional calculi are formal sys- tems that use axioms and inference rules to come up with theorems in propositional logic. ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5527 Email addresses: drbalamuruganm@veltech.edu.in (M. Balamurugan), khakami@jazanu.edu.sa (Khalil H. Hakami), maansari@jazanu.edu.sa (Moin A. Ansari), loganathankaruppusamy304@gmail.com (K. Loganathan) https://www.ejpam.com 3973 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3974 This has been first described by Imai et al. [25, 26]. These systems are foundational in the study of logic and are crucial for understanding the formal properties and limitations of logical systems. A bipolar fuzzy set (BFS) was introduced by Zhang [47] to extend the classical fuzzy set theory by incorporating both positive and negative membership degrees. This method can better reflect real-life situations where an element can show different levels of belong- ing and not belonging to a set at the same time. It also has a more complex semantic interpretation compared to traditional fuzzy sets. Bipolar logic and bipolar fuzzy logic are conceptual frameworks introduced by Zhang [46]. These frameworks extend classical binary logic and traditional fuzzy logic by incorporating bipolarity, which means that they handle positive and negative information separately. This allows for a more nuanced rep- resentation of reality, reflecting the inherent dualities found in many real-world situations. Bipolar fuzzy subalgebras and bipolar fuzzy ideals are concepts introduced by Lee [31–33] to extend the theory of BCK/BCI-algebras into the realm of fuzzy logic. Saied et al. [44] studied bipolar-valued fuzzy BCK/BCI-algebras, which is a niche but fascinating area within mathematical logic and algebra. Rosenfeld [43] extended the classical concept of a group in algebra to accommodate the notion of fuzziness, which is characterized by elements having degrees of membership rather than crisp membership. Aslam [17] presented the idea of bipolar fuzzy ideals in LA-semigroups, which gives the study of algebraic structures using fuzzy set theory a big new dimension. Akram et al. [2, 3] introduced the concept of m-polar fuzzy Lie ideals of Lie algebras. Al-Masarwah et al. (See [5, 10–14]) extended the concept of bipolar/m-polar fuzzy subalgebras and ideals to BCK/BCI-algebras. The concept of (∈,∈,∨q)-fuzzy subgroup was introduced by Bhakat [20] as a generalization in the context of fuzzy group theory. Chen et al. [21] worked on m-polar fuzzy sets as an extension of bipolar fuzzy sets, which is a significant contribution to the field of fuzzy set theory. Farooq et al. [22] presented a topic related to m-polar fuzzy groups. Ibrar [24] studied ordered semigroups and their structures using (α, β)-bipolar fuzzy generalized bi-ideals. Jana [28] studied (∈,∈ ∨q)- bipolar fuzzy BCK-algebras, which belong to a specialized branch of mathematics that deals with algebraic structures and fuzzy logic. Bipolar fuzzy UP-algebras are a concept in mathematics introduced by Kawila et al. [29]. Balamurugan et al. [18, 42] investigated (∈́, ∈́∨ q́ǩ)-uni-intuitionistic fuzzy soft h-Ideals in subtraction BG-algebras and (∈,∈ ∨q̌)-bipolar fuzzy-ideals of BCK/BCI-algebras. Al- Maswarwah et al. [6–9] developed multipolar fuzzy ideals of BCK/BCI-algebras. Balamu- rugan [19]presented the concept of complex fuzzy ideals in BCK/BCI-algebras. Iampan et al. [23] discussed anti-intuitionistic fuzzy soft b-ideals in BCK/BCI algebras. Moin [15] introduced roughness in JU-algebras. Mohseni et al. [35] studied the concept of multi- polar fuzzy p-ideals of BCI-algebras. Muhiuddin et al. (see [36–41]) applied the concept to multipolar fuzzy ideals in BCK/BCI algebras. Al-Kadi et al. [4] looked into a group of BCI-algebras called bipolar fuzzy BCI-implicative ideals. These ideals probably share some properties with both fuzzy logic and BCI-algebras. Abuhijileh et al. [1] developed the concept of complex fuzzy group based on Rosenfeld’s approach. Mahmood [34] studied bipolar complex fuzzy soft sets and their applications in decision-making. Jaleel et al. [27] K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3975 demonstrated interval-valued bipolar complex fuzzy soft set as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set, and soft set. Ali et al. [30] extended KU-algebras and investigated properties based on them whereas Moin et al. [16] introduced intersectional soft ideals and their quotients on KU-algebras, The article is organized as follows: Section 2 proceeds with a recapitulation of all required definitions and properties. In Section 3, we present (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy ideals in BCK/BCI-algebras. In Section 4, (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals in BCK/BCI-algebras are proposed and their properties are discussed in detail. Finally, in Section 5, the conclusions and scope of future research are given. 2. Preliminaries BCK-algebras and BCI-algebras are types of algebraic structures used in the study of non-classical logics, particularly in the context of certain types of implication algebras. These algebras generalize certain aspects of set theory, logic and have applications in some areas, such as theoretical computer science and mathematical logic. Definition 1. [25, 26] A BCK-algebra is a structure (ℵ̌; ≬, 0) consisting of a non-empty set ℵ̌, a binary operation ≬ on ℵ̌, and a constant 0 ∈ ℵ̌, satisfying the following axioms: (C1) ((ϱ̌0 ≬ ϱ̌1) ≬ (ϱ̌0 ≬ ϱ̌2)) ≬ (ϱ̌2 ≬ ϱ̌1) = 0, (C2) (ϱ̌0 ≬ (ϱ̌0 ≬ ϱ̌1)) ≬ ϱ̌1 = 0, (C3) ϱ̌0 ≬ ϱ̌0 = 0, (C4) 0 ≬ ϱ̌0 = 0, (C5) ϱ̌0 ≬ ϱ̌1 = 0 and ϱ̌1 ≬ ϱ̌0 = 0 ⇒ ϱ̌0 = ϱ̌1, ∀ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌ A non-void subset Ǎ is an ideal of ℵ̌ if (I1) 0 ∈ Ǎ, (I2) ∀ϱ̌0, ϱ̌1 ∈ ℵ̌, ϱ̌0 ≬ ϱ̌1 ∈ Ã, ϱ̌1 ∈ Ã ⇒ ϱ̌0 ∈ Ǎ. A non-void subset Ǎ is a fantastic ideal of ℵ̌ if (I1) and (I3) ∀ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌, (ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2 ∈ Ǎ, ϱ̌2 ∈ Ǎ ⇒ ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ Ǎ. A bipolar fuzzy set (BFS) is denoted by ζ̄ = (ζ̄−, ζ̄+), where ζ̄− : ℵ̌ → [−1, 0] and ζ̄+ : ℵ̌ → [0, 1]. Definition 2. [42] A BFS ζ̄ is a BFI of ℵ̌ if it meets the ensuing assertions: (i) ζ̄−(0) ≤ ζ̄−(ϱ̌0), ζ̄+(0) ≥ ζ̄+(ϱ̌0). (ii) ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1), (iii) ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1),∀ϱ̌0, ϱ̌1 ∈ ℵ̌. Definition 3. [42] A BFS ζ̄ is a (∈,∈ ∨q̌)-BFI of ℵ̌ if it meets the ensuing assertions: (i) ζ̄−(0) ≤ ζ̄−(ϱ̌0), ζ̄+(0) ≥ ζ̄+(ϱ̌0). (ii) ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ −1 2 , (iii) ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ 1 2 , ∀ϱ̌0, ϱ̌1 ∈ ℵ̌. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3976 3. (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy ideals In this section, we investigate an (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy ideal of BCK/BCI- algebras. Definition 4. A BFS ζ̄ is an (∈,∈ ∨q̌φ)-BFI of ℵ̌ if it meets the ensuing assertions: (i) ζ̄−(0) ≤ ζ̄−(ϱ̌0), ζ̄+(0) ≥ ζ̄+(ϱ̌0). (ii) ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ (φ2 − 1 2), (iii) ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + 1 2),∀ϱ̌0, ϱ̌1 ∈ ℵ̌. Definition 5. A BFS ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌ if it meets the ensuing assertions: (i) ζ̄−(0) ≤ ζ̄−(ϱ̌0), ζ̄+(0) ≥ ζ̄+(ϱ̌0). (ii) ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ (φ2 − φ⋇ 2 ), (iii) ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ), ∀ϱ̌0, ϱ̌1 ∈ ℵ̌. Definition 6. A BFS ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌ if it fulfills the ensuing asser- tions: (i) (ϱ̌0 ≬ ϱ̌1, š) ∈ ζ̄−, (ϱ̌1, ť) ∈ ζ̄− ⇒ (ϱ̌0, š ∨ ť) ∈ ∨(φ⋇, q̌φ)ζ̄−, (ii) (ϱ̌0 ≬ ϱ̌1, ǔ) ∈ ζ̄+, (ϱ̌1, v̌) ∈ ζ̄+ ⇒ (ϱ̌0, ǔ ∧ v̌) ∈ ∨(φ⋇, q̌φ)ζ̄+, for all ϱ̌0, ϱ̌1 ∈ ℵ̌, š, ť ∈ [−1, 0) and ǔ, v̌ ∈ (0, 1]. Example 1. Take a BCK/BCI-algebra ℵ̌ = {0, ǔ, ϱ̌0, ϱ̌1, ϱ̌2} with the subsequent Cayley table: ≬ 0 ǔ ϱ̌0 ϱ̌1 ϱ̌2 0 0 0 0 0 0 ǔ ǔ 0 ǔ 0 ǔ ϱ̌0 ϱ̌0 ϱ̌0 0 0 0 ϱ̌1 ϱ̌1 ǔ ϱ̌1 0 ϱ̌1 ϱ̌2 ϱ̌2 ϱ̌2 ϱ̌2 ϱ̌2 0 Define a BFS ζ̄ of ℵ̌ as follows: ζ̄(ǔ) =  (−0.72, 0.52), ǔ = 0; (−0.42, 0.22), ǔ = ǔ; (−0.22, 0.06), ǔ = ϱ̌0; (−0.52, 0.12), ǔ = ϱ̌1; (−0.12, 0.22), ǔ = ϱ̌2. It is easy to show that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌. Theorem 1. A BFS ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌ if and only if satisfies ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ (φ2 − φ⋇ 2 ) K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3977 and ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ) for all ϱ̌0, ϱ̌1 ∈ ℵ̌. Proof. Let ζ̄ be an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌. If ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) > φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) < −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) and ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1). Assume that ζ̄−(ϱ̌0) > ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) and ζ̄+(ϱ̌0) < ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1). Let us take š ∈ ¬ζ̄ and ǔ ∈ ζ̄ such that ζ̄−(ϱ̌0) > š > ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) and ζ̄+(ϱ̌0) < ǔ < ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1). Then (ϱ̌0 ≬ ϱ̌1, š) ∈ ζ̄−, (ϱ̌1, š) ∈ ζ̄− and (ϱ̌0 ≬ ϱ̌1, ǔ) ∈ ζ̄+, (ϱ̌1, ǔ) ∈ ζ̄+ but (ϱ̌0, š ∨ š) = (ϱ̌0, š)∈ ∨(φ⋇, q̌φ)ζ̄− and (ϱ̌0, ǔ ∧ ǔ) = (ϱ̌0, ǔ)∈ ∨(φ⋇, q̌φ)ζ̄+, a contradiction. Hence, ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) whenever ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) > φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(y̌) whenever ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) < −φ 2 + φ⋇ 2 . Suppose that ζ̄−(ϱ̌0 ≬ ϱ̌1)∨ ζ̄−(ϱ̌1) ≤ φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ ϱ̌1)∧ ζ̄+(ϱ̌1) ≥ −φ 2 + φ⋇ 2 . Then, (ϱ̌0 ≬ ϱ̌1, φ 2 − φ⋇ 2 ) ∈ ζ̄−, (ϱ̌1, φ 2 − φ⋇ 2 ) ∈ ζ̄− and (ϱ̌0 ≬ ϱ̌1,−φ 2 + φ⋇ 2 ) ∈ ζ̄+, (ϱ̌1,−φ 2 + φ⋇ 2 ) ∈ ζ̄+, which imply that (ϱ̌0, ( φ 2 − φ⋇ 2 ) ∧ (φ2 − φ⋇ 2 )) = (ϱ̌0, ( φ 2 − φ⋇ 2 )) ∈ ∨(φ⋇, q̌φ)ζ̄− (ϱ̌0, (−φ 2 + φ⋇ 2 ) ∧ (−φ 2 + φ⋇ 2 )) = (ϱ̌0, (−φ 2 + φ⋇ 2 )) ∈ ∨(φ⋇, q̌φ)ζ̄+. Thus, ζ̄−(ϱ̌0) ≤ φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0) ≥ −φ 2 + φ⋇ 2 . Otherwise, ζ̄−(ϱ̌0) + φ 2 − φ⋇ 2 > φ 2 − φ⋇ 2 + φ 2 − φ⋇ 2 = φ−φ⋇ and ζ̄+(ϱ̌0)− φ 2 + φ⋇ 2 < −φ 2 + φ⋇ 2 − φ 2 + φ⋇ 2 = −φ+φ⋇, a contradiction. And so, ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ (φ2 − φ⋇ 2 ) and ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ) for all ϱ̌0, ϱ̌1 ∈ ℵ̌. On the contrary, let us assume that an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌ holds. Let ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌ and ǔ, v̌ ∈ (0, 1] and š, ť ∈ ¬ζ̄ such that (ϱ̌0 ≬ ϱ̌1, š) ∈ ζ̄−, (ϱ̌1, ť) ∈ ζ̄− and (ϱ̌0 ≬ ϱ̌1, ǔ) ∈ ζ̄+, (ϱ̌1, v̌) ∈ ζ̄+. Then, ζ̄−(ϱ̌0 ≬ ϱ̌1) ≤ š, ζ̄−(ϱ̌0 ≬ ϱ̌1) ≤ ť and ζ̄+(ϱ̌0 ≬ ϱ̌1) ≥ ǔ, ζ̄+(ϱ̌1) ≤ v̌. If ζ̄−(ϱ̌0) > š ∨ ť and ζ̄+(ϱ̌0) < ǔ ∨ v̌, then ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ≤ φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ≥ −φ 2 + φ⋇ 2 . Otherwise, we get K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3978 ζ̄−(ϱ̌0) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ (φ2 − φ⋇ 2 ) ≤ ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ≤ š ∨ ť and ζ̄+(ϱ̌0) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ) ≥ ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ≥ ǔ ∧ v̌, a contradiction. In that case ζ̄−(ϱ̌0) + š ∨ ť < 2ζ̄−(ϱ̌0) ≤ 2(ζ̄−(ϱ̌0 ≬ ϱ̌1) ∨ ζ̄−(ϱ̌1) ∨ (φ2 − φ⋇ 2 )) = φ− φ⋇, and ζ̄−(ϱ̌0) + v̌ ∧ ǔ > 2ζ̄+(ϱ̌0) ≥ 2(ζ̄+(ϱ̌0 ≬ ϱ̌1) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 )) = −φ+ φ⋇. Hence, (ϱ̌0, š ∨ ť) ∈ ∨(φ⋇, q̌φ)ζ̄− and (ϱ̌0, ǔ ∧ v̌) ∈ ∨(φ⋇, q̌φ)ζ̄+. Therefore, ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌. Lemma 1. Every (∈,∈ ∨q̌φ)-BFI is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌, but the converse may not be true in general. Proof. Straightforward. Example 2. Take a BCI-algebra ℵ̌ = {0, ǔ, ϱ̌0, ϱ̌1} with Cayley table: ≬ 0 ǔ ϱ̌0 ϱ̌1 0 0 ǔ ϱ̌0 ϱ̌1 ǔ ǔ 0 ϱ̌1 ϱ̌0 ϱ̌0 ϱ̌0 ϱ̌1 0 ǔ ϱ̌1 ϱ̌1 ϱ̌0 ǔ 0 Define a BFS ζ̄ of ℵ̌ as follows: ζ̄(ǔ) =  (−0.67, 0.67), ǔ = 0; (−0.17, 0.57), ǔ = ǔ; (−0.57, 0.57), ǔ = ϱ̌0; (−0.17, 0.47), ǔ = ϱ̌1. Hence, (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌, but is not BFI of ℵ̌ because ζ̄+(ϱ̌1) = 0.47 ⩾̸ 0.57 = ζ̄+(ϱ̌1 ≬ ǔ) ∧ ζ̄+(ǔ). 4. (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals This section investigates (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals of BCK/BCI- algebras. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3979 Definition 7. A BFS ζ̄ is a BFFI of ℵ̌ if it fulfills the Definition 5(i) and the resulting assertions: (i) ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)) ∨ ζ̄−(ϱ̌2), (ii) ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)) ∧ ζ̄+(ϱ̌2),∀ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Definition 8. A BFS ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌ if it fulfills the ensuing asser- tions: (i) ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2), š) ∈ ζ̄−, (ϱ̌2, ť) ∈ ζ̄− ⇒ (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), š ∨ ť) ∈ ∨(φ⋇, q̌φ)ζ̄−, (ii) ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2), ǔ) ∈ ζ̄−, (ϱ̌2, v̌) ∈ ζ̄− ⇒ (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ǔ ∧ v̌) ∈ ∨(φ⋇, q̌φ)ζ̄+, ∀ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌, š, ť ∈ [−1, 0) and ǔ, v̌ ∈ (0, 1]. Example 3. Consider ℵ̌ = {0, ǔ, ϱ̌0, ϱ̌1, ϱ̌2} be a BCK-algebra in Example 1, and now we define a BFS ζ̄ of ℵ̌ as ζ̄(ǔ) =  (−0.76, 0.53), ǔ = 0; (−0.46, 0.23), ǔ = ǔ; (−0.26, 0.32), ǔ = ϱ̌0; (−0.56, 0.13), ǔ = ϱ̌1; (−0.16, 0.03), ǔ = ϱ̌2. It is easy to show that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌ and BFFI of ℵ̌. Theorem 2. A BFS ζ̄ is a BFFI of ℵ̌ ⇔ the succeeding assertions are holds: (i) (ϱ̌0, š) ∈ ζ̄− ⇒ (0, š) ∈ ζ̄− and (ϱ̌0, ǔ) ∈ ζ̄+ ⇒ (0, ǔ) ∈ ζ̄+, (ii) ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2), š) ∈ ζ̄−, (ϱ̌2, ť) ∈ ζ̄− ⇒ (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), š ∨ ť) ∈ ζ̄−, (iii) ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2), ǔ) ∈ ζ̄+, (ϱ̌2, v̌) ∈ ζ̄+ ⇒ (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ǔ ∧ v̌) ∈ ζ̄+, for all ϱ̌0, š, ť ∈ [−1, 0)ϱ̌1, ϱ̌2 ∈ ℵ̌ and ǔ, v̌ ∈ (0, 1]. Proof. Suppose that Definition 5 (i) is hold and ϱ̌0 ∈ ℵ̌, ǔ ∈ (0, 1], š ∈ [−1, 0) such that (ϱ̌0, š) ∈ ζ̄− and (ϱ̌0, ǔ) ∈ ζ̄+. Then ζ̄−(0) ≤ ζ̄−(ϱ̌0) ≤ š and ζ̄+(0) ≥ ζ̄+(ϱ̌0) ≥ ǔ, and so (0, š) ∈ ζ̄−and(0, ǔ) ∈ ζ̄+. Since (ϱ̌0, ζ̄(ϱ̌0)) ∈ ζ̄− and (ϱ̌0, ζ̄(ϱ̌0)) ∈ ζ̄+ for all ϱ̌0 ∈ ℵ̌, it follows from (i) that (0, ζ̄(ϱ̌0)) ∈ ζ̄− and (0, ζ̄(ϱ̌0)) ∈ ζ̄+ so that ζ̄−(0) ≤ ζ̄−(ϱ̌0) and ζ̄+(0) ≥ ζ̄+(ϱ̌0) for all ϱ̌0 ∈ ℵ̌. Assume that Definition 7 holds. Let ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌, and š, ť ∈ [−1, 0), ǔ, v̌ ∈ (0, 1] be such that ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, š) ∈ ζ̄−, (ϱ̌2, ť) ∈ ζ̄−, and ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ǔ) ∈ ζ̄+, (ϱ̌2, v̌) ∈ ζ̄+. Then ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š, ζ̄−(ϱ̌2) ≤ ť and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ, ζ̄+(ϱ̌2) ≥ v̌. It follows from Definition 7, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)) ∨ ζ̄−(ϱ̌2) ≤ š ∨ ť K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3980 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)) ∧ ζ̄+(ϱ̌2) ≥ ǔ ∧ v̌. So, that (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), š ∨ ť) ∈ ζ̄− and (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ǔ ∧ v̌) ∈ ζ̄+. Next, suppose that (ii) and (iii) are holds. For every ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌, ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)) ∈ ζ̄−, (ϱ̌2, ζ̄−(ϱ̌2)) ∈ ζ̄− and ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)) ∈ ζ̄+, (ϱ̌2, ζ̄+(ϱ̌2)). Hence, (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2)) ∈ ζ̄− and (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2)) ∈ ζ̄+ by (ii), and (iii), respectively and thus, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2), and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2). Theorem 3. A BFS ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌ ⇔ satisfies the succeeding assertions: (i) ζ̄−(0) ≤ ζ̄−(ϱ̌0) ∨ (φ2 − φ⋇ 2 ) and ζ̄+(0) ≥ ζ̄+(ϱ̌0) ∧ (−φ 2 + φ⋇ 2 ), (ii) ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ (φ2 − φ⋇ 2 ), (iii) ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) for all ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Proof. Suppose ζ̄ be an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌. Let ϱ̌0 ∈ ℵ̌ be such that ζ̄−(ϱ̌0) > (φ2 − φ⋇ 2 ) and ζ̄+(ϱ̌0) < (−φ 2 + φ⋇ 2 ). If ζ̄−(0) > ζ̄−(ϱ̌0) and ζ̄+(0) < ζ̄+(ϱ̌0), ζ̄−(0) > š > ζ̄−(ϱ̌0) and ζ̄+(0) < ǔ < ζ̄+(ϱ̌0) for every š ∈ (φ2 − φ⋇ 2 , 0) and ǔ ∈ (0,−φ 2 + φ⋇ 2 ), so we get (ϱ̌0, š) ∈ ζ̄−, (0, š)∈ζ̄− and (ϱ̌0, ǔ) ∈ ζ̄+, (0, v̌)∈ζ̄+. Since ζ̄−(0)+š > φ−φ⋇ and ζ̄+(0)+ǔ < −φ+φ⋇, so we have (0, š)q̌φζ̄− and (0, ǔ)q̌φζ̄+. It follows that (0, š)∈ ∨q̌φζ̄− and (0, ǔ)∈ ∨q̌φζ̄+, a contradiction. Hence, ζ̄−(0) ≤ ζ̄−(ϱ̌0) and ζ̄+(0) ≥ ζ̄+(ϱ̌0). Now if ζ̄−(ϱ̌0) ≤ (φ2 − φ⋇ 2 ) and ζ̄+(ϱ̌0) ≥ (−φ 2 + φ⋇ 2 ), then (ϱ̌0, ( φ 2 − φ⋇ 2 )) ∈ ζ̄− and (ϱ̌0, (−φ 2 + φ⋇ 2 )) ∈ ζ̄+. Thus, (0, φ2 − φ⋇ 2 )) ∈ ∨q̌φζ̄− and (0,−φ 2 + φ⋇ 2 ) ∈ ∨q̌φζ̄+. Thus, ζ̄−(0) ≤ φ 2 − φ⋇ 2 and ζ̄+(0) ≥ −φ 2 + φ⋇ 2 . Otherwise, ζ̄−(ϱ̌0) + φ 2 − φ⋇ 2 > φ 2 − φ⋇ 2 + φ 2 − φ⋇ 2 = φ− φ⋇ and ζ̄+(ϱ̌0) + −φ 2 + φ⋇ 2 < −φ 2 + φ⋇ 2 +−φ 2 + φ⋇ 2 = −φ+φ⋇, a contradiction. Consequently, ζ̄−(0) ≤ {ζ̄−(ϱ̌0), φ2 − φ⋇ 2 } and ζ̄+(0) ≥ {ζ̄+(ϱ̌0),−φ 2 + φ⋇ 2 }, for all ϱ̌0 ∈ ℵ̌. Let ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Suppose that ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) > φ 2 − φ⋇ 2 and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)∧ ζ̄+(ϱ̌2) < −φ 2 + φ⋇ 2 . Then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)∨ ζ̄−(ϱ̌2) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2). If not, then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > š > ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ǔ < ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2), for some š ∈ (φ2 − φ⋇ 2 , 0), ǔ ∈ (0,−φ 2 + φ⋇ 2 ). K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3981 It follows that ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, š) ∈ ζ̄− and (ϱ̌2, š) ∈ ζ̄− but (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), š∨š) = (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), š) ∈ ∨q̌φζ̄− and ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ǔ) ∈ ζ̄+ and (ϱ̌2, ǔ) ∈ ζ̄+ but (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ǔ ∨ ǔ) = (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ǔ) ∈ ∨q̌φζ̄+ which is a contradiction. Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) whenever ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)∨ ζ̄−(ϱ̌2) > φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)∧ ζ̄+(ϱ̌2) whenever ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) < −φ 2 + φ⋇ 2 . If ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ≤ φ 2 − φ⋇ 2 , then ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, φ 2 − φ⋇ 2 ) ∈ ζ̄− and (ϱ̌2, φ 2 − φ⋇ 2 ) ∈ ζ̄−, which imply that (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), φ 2 − φ⋇ 2 ) = (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), φ 2 − φ⋇ 2 ) ∨ φ 2 − φ⋇ 2 ) ∈ ∨q̌φζ̄− and if ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ≥ φ 2 − φ⋇ 2 , then ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, φ 2 − φ⋇ 2 ) ∈ ζ̄+ and (ϱ̌2,−φ 2 + φ⋇ 2 ) ∈ ζ̄+, which imply that (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)),−φ 2 + φ⋇ 2 ) = (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)),−φ 2 + φ⋇ 2 ) ∧ (−φ 2 + φ⋇ 2 ) ∈ ∨q̌φζ̄+. Therefore, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ −φ 2 + φ⋇ 2 , because if ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + φ 2 − φ⋇ 2 > φ 2 − φ⋇ 2 + φ 2 − φ⋇ 2 = φ − φ⋇ and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)))− φ 2 + φ⋇ 2 < −φ 2 + φ⋇ 2 − φ 2 + φ⋇ 2 = −φ+ φ⋇, which is a contradiction. Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ (φ2 − φ⋇ 2 ), and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ), for all ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Conversely, assume that ζ̄ satisfies the conditions of (i), (ii), and (iii). Let ϱ̌0 ∈ ℵ̌ and ǔ ∈ (0, 1] and š ∈ [−1, 0) be such that (ϱ̌0, š) ∈ ζ̄− and (ϱ̌0, ǔ) ∈ ζ̄+. Then, ζ̄−(ϱ̌0) ≤ š and ζ̄+(ϱ̌0) ≥ ǔ. Suppose that ζ̄−(0) ≥ š and ζ̄+(0) ≤ ǔ. If ζ̄−(ϱ̌0) > φ 2− φ⋇ 2 and ζ̄+(ϱ̌0) < −φ 2+ φ⋇ 2 , then ζ̄−(0) ≤ ζ̄−(ϱ̌0) ∨ (φ2 − φ⋇ 2 ) = ζ̄−(ϱ̌0) ≤ š and ζ̄+(0) ≥ ζ̄+(ϱ̌0) ∧ (−φ 2 + φ⋇ 2 ) = ζ̄+(ϱ̌0) ≥ ǔ, a contradiction. Hence, we know that ζ̄−(ϱ̌0) ≤ φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0) ≥ −φ 2 + φ⋇ 2 and so we get ζ̄−(0) + š < 2ζ̄−(0) ≤ ζ̄−(ϱ̌0) ∨ (φ2 − φ⋇ 2 ) = φ− φ⋇ and ζ̄+(0) + ϱ̌2 > 2ζ̄−(0) ≥ ζ̄+(ϱ̌0) ∧ (−φ 2 + φ⋇ 2 ) = −φ+ φ⋇. Thus, (0, š) ∈ ∨ζ̄− and (0, ǔ) ∈ ∨ζ̄+. Let ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌, ǔ, v̌ ∈ (0, 1] and š, ť ∈ [1, 0) be such that ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, š) ∈ ζ̄−, (ϱ̌2, ť) ∈ ζ̄− and ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ǔ) ∈ ζ̄+, (ϱ̌1, v̌) ∈ ζ̄+. Then ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š, ζ̄−(ϱ̌2) ≤ ť and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ, ζ̄+(ϱ̌1) ≥ v̌. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3982 Suppose that ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > š ∨ ť and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ǔ ∧ v̌. If ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ≥ φ 2 − φ⋇ 2 and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌1) ≤ −φ 2 + φ⋇ 2 . Then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ≤ š ∨ ť and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ≥ ǔ ∧ v̌, a contradiction. Thus, ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ≤ φ 2 − φ⋇ 2 and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ≥ −φ 2 + φ⋇ 2 . In that case ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + š ∨ ť < 2ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ 2((ζ̄−(ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ φ 2 − φ⋇ 2 ) = φ− φ⋇, and ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ ∧ v̌ > 2ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ 2(ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ −φ 2 + φ⋇ 2 ) = −φ+ φ⋇. Hence, (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), š ∨ ť) ∈ ∨(k⋇, q̌φ)ζ̄− and (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)), ǔ ∧ v̌) ∈ ∨(k⋇, q̌φ)ζ̄+. So, ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌. Definition 9. Let ζ̄ be a BFS of ℵ̌ and (š, ǔ) ∈ [−1, 0] × [0, 1], we define U(ζ̄; š, ǔ) = {ϱ̌0 ∈ ℵ̌ | ζ̄−(ϱ̌0) ≤ š and ζ̄+(ϱ̌0) ≥ ǔ} is called a š-level cut of ζ̄− and ǔ-level cut of ζ̄+ of the BFS ζ̄. Theorem 4. A BFS ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌ ⇔ the level subset U(ζ̄; š, ǔ) = {ϱ̌0, ϱ̌2 ∈ ℵ̌ | ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ǔ} is a BFFI of ℵ̌ for all š ∈ [φ2 − φ⋇ 2 , 0) and for all ǔ ∈ (0,−φ 2 + φ⋇ 2 ]. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3983 Proof. Assume that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFI of ℵ̌. Let (ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ϱ̌2 ∈ U(ζ̄; š, ǔ) with š ∈ [φ2 − φ⋇ 2 , 0) and ǔ ∈ (0,−φ 2 + φ⋇ 2 ). Then ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š, ζ̄−(ϱ̌2) ≤ š and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ, ζ̄+(ϱ̌2) ≥ ǔ. Therefore from Theorem 1 that ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) ≤ š ∨ ( φ 2 − φ⋇ 2 ) = š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) ≥ ǔ ∧ (−φ 2 + φ⋇ 2 ) = ǔ, so that ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ U(ζ̄; š, ǔ). Therefore, U(ζ̄; š, ǔ) is a fantastic ideal of ℵ̌. Conversely, let ζ̄ be a BFS of ℵ̌ be such that U(ζ̄; š, ǔ) = {ϱ̌0 ∈ ℵ̌ | ζ̄− ≤ š and ζ̄+ ≥ ǔ} is a fantastic ideal of ℵ̌ for all š ∈ [φ2 − φ⋇ 2 , 0) and ǔ ∈ (0,−φ 2 + φ⋇ 2 ]. If there exist (ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ϱ̌2 ∈ ℵ̌ such that ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)∨ ζ̄−(ϱ̌2)∨ (φ2 − φ⋇ 2 ) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)∧ζ̄+(ϱ̌2)∧(−φ 2 + φ⋇ 2 ), then we take š ∈ (−1, 0) and ǔ ∈ (0, 1) such that ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > š > ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ (φ2 − φ⋇ 2 ) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ǔ < ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ). Thus, (ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2), ϱ̌1 ∈ U(ζ̄; š, ǔ) with ǔ < −φ 2 + φ⋇ 2 and š > −φ 2 + φ⋇ 2 , and so ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ U(ζ̄; š, ǔ), i.e., ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ǔ which is a contradiction. Therefore, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ), for all ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Using the Theorem 1, we conclude that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))- BFFI of ℵ̌. Theorem 5. Let ζ̄ be an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌, where ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > φ 2 − 1 2 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < −φ 2 + φ⋇ 2 for all ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Then ζ̄ is an (∈,∈)-BFFI of ℵ̌. Proof. The proof is simple with theorem 1. K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3984 Theorem 6. Let ∧ be an index set and {(ζ̄i− , ζ̄i+) | i ∈ ∧ } be a family of (∈,∈ ∨(φ⋇, q̌φ))- BFFI of ℵ̌. Then ζ̄ = ⋂ i∈ ∧(ζ̄i− , ζ̄i+) is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌. Proof. Let us take (ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ϱ̌2 ∈ ℵ̌ and š1, š2 ∈ [−1, 0), and ǔ1, ǔ2 ∈ (0, 1] be such that ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š1 and ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š2, ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ1 and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ2. Assume that ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1∨š2∈ ∨(φ⋇, q)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ1∧ǔ2∈ ∨(φ⋇, q)ζ̄+ . Then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > š1 ∨ š2 and ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)))+ š1 ∨ š2 ≥ φ−φ⋇, and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ǔ1 ∧ ǔ2 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ1 ∧ ǔ2 ≤ −φ+φ⋇, which implies ζ̄−(ϱ̌0) > φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0) < φ 2 − φ⋇ 2 . (1) Now, we define ∆1 = {i ∈ ∧ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1∨š2 ∈ ζ̄iť and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ1∧ǔ2 ∈ ζ̄ip̌} and ∆2 = {[{i ∈ ∧ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1∨š2(φ ⋇, q̌φ)ζ̄i−} ∩ {j ∈ ∧ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1∨š2∈ ζ̄i−}] and [{i ∈ ∧ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)ǔ1∧ǔ2qζ̄i+} ∩ {j ∈ ∧ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)ǔ1∧ǔ2∈ ζ̄i+}]}. Then ∧ = ∆1 ∪∆2 and ∆1 ∩∆2 = ∅. If ∆2 = ∅, then ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1∨š2 ∈ ζ̄i− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ1∧ǔ2 ∈ ζ̄i+ for all i ∈ ∧ , i.e., ζ̄i−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ š1 ∨ š2 and ζ̄i+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ǔ1 ∧ ǔ2 for all for all i ∈ ∧ , which indicate ζ̄i−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ š1 ∨ š2 and ζ̄i+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ǔ1 ∧ ǔ2. This is a contrary. Hence, for every i ∈ ∆2, and so, ∆2 ̸= ∅, we have ζ̄i−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > š1 ∨ š2 and ζ̄i−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + š1 ∨ š2 < φ− φ⋇, and ζ̄i+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ǔ1 ∧ ǔ2 and ζ̄i+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ1 ∧ ǔ2 > −φ+ φ⋇. It follows that š1 ∨ š2 < φ 2 − φ⋇ 2 and ǔ1 ∧ ǔ2 > −φ 2 + φ⋇ 2 . Now, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1 ∈ ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ1 ∈ ζ̄+ ⇒ ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ š1 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ǔ1, and thus, ζ̄i−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š1 ≤ š1 ∨ š2 < φ 2 − φ⋇ 2 K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3985 and ζ̄ip̌((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ1 ≥ ǔ1 ∧ ǔ2 > −φ 2 + φ⋇ 2 for all i ∈ ∧ . Similarly, we get ζ̄i−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) < φ 2 − φ⋇ 2 and ζ̄i+(ϱ̌2) > −φ 2 + φ⋇ 2 for all i ∈ ∧ . We suppose that š = ζ̄i−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) > φ 2 − φ⋇ 2 and ǔ = ζ̄i+(ϱ̌2) < −φ 2 + φ⋇ 2 . Taking that š > ť > φ 2 − φ⋇ 2 and ǔ < v̌ < −φ 2 + φ⋇ 2 , we get ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)ť ∈ ζ̄i−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) and ϱ̌2 ť ∈ ζ̄i−(ϱ̌0 ≬ ϱ̌2), but ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ť∨ť = (ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)))ť∈ ∨(φ⋇, q)ζ̄i− and ((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)v̌ ∈ ζ̄ip̌(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) and ϱ̌1v̌ ∈ ζ̄ip̌(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))), but ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))v̌∧v̌ = ϱ̌0v̌∈ ∨(φ⋇, q)ζ̄ip̌ . This contradicts that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ)) − BFFI of ℵ̌. Hence, ζ̄i+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ φ 2 − φ⋇ 2 and ζ̄i+(ϱ̌2) ≥ −φ 2 + φ⋇ 2 for all i ∈ ∧ , so ζ̄−(ϱ̌2) ≤ φ 2 − φ⋇ 2 and ζ̄+(ǧ) ≥ (−φ 2 + φ⋇ 2 ) which contradicts 1. Therefore, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š1∨š2 ∈ ∨(φ⋇, q)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ1∧ǔ2 ∈ ∨(φ⋇, q)ζ̄+ and consequently, ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))- BFFI of ℵ̌. For any BFS ζ̄ in ℵ̌, where š ∈ [1, 0) and ǔ ∈ (0, 1], we denote ζ̄š− = {ϱ̌0, ϱ̌2 ∈ ℵ̌ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š ∈ (φ⋇, q̌φ)ζ̄−}, ζ̄ǔ+ = {ϱ̌0, ϱ̌2 ∈ ℵ̌ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ ∈ (φ⋇, q̌φ)ζ̄+}, and [ζ̄](š,ǔ) = {ϱ̌0, ϱ̌2 ∈ ℵ̌ | ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)š ∈ (φ⋇, q̌φζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)ǔ ∈ (φ⋇, q̌φζ̄+}. Then it is obvious that [ζ̄](š,ǔ) = U(ζ̄; š, ǔ)∪ ζ̄šť ∪ ζ̄v̌p̌ . Here, [ζ̄](š,ǔ) is an (∈,∈ ∨(φ⋇, q̌φ))- level fantastic ideal of ζ̄. Theorem 7. Let ζ̄ be a BFS in ℵ̌. Then ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌ if and only if [ζ̄](š,ǔ) is a fantastic ideal of ℵ̌, for all š ∈ [−1, 0) and ǔ ∈ (0, 1]. Proof. Suppose that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌ and let ϱ̌0, ϱ̌1, ϱ̌2 ∈ [ζ̄](š,ǔ) for š ∈ [−1, 0) and ǔ ∈ (0, 1]. Then (ϱ̌0 ⋇ ϱ̌1) ⋇ ϱ̌2š ∈ q̌φζ̄−, ϱ̌2š ∈ (φ⋇, q̌φ)ζ̄− and (ϱ̌0 ⋇ ϱ̌1)⋇ ϱ̌2ǔ ∈ (φ⋇, q̌φ)ζ̄+, ϱ̌2ǔ ∈ (φ⋇, q̌φ)ζ̄+. That is ζ̄−((ϱ̌0 ⋇ ϱ̌1) ⋇ ϱ̌2) ≤ š or K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3986 ζ̄−((ϱ̌0⋇ϱ̌1)⋇ϱ̌2)+š < φ−φ⋇, ζ̄−(ϱ̌2) ≤ š or ζ̄−(ϱ̌2)+š < φ−φ⋇ and ζ̄+((ϱ̌0⋇ϱ̌1)⋇ϱ̌2) ≥ ǔ or ζ̄+((ϱ̌0 ⋇ ϱ̌1) ⋇ ϱ̌2) + ǔ > −φ + φ⋇, ζ̄+(ϱ̌2) ≥ ǔ or ζ̄+(ϱ̌2) + ǔ > −φ + φ⋇. Using the Theorem 1, we get, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) Case 1. ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š, ζ̄−(ϱ̌2) ≤ š and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ, ζ̄+(ϱ̌2) ≥ ǔ. If š < φ 2 − φ⋇ 2 and ǔ > −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) = φ 2 − φ⋇ 2 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ) = −φ 2 + φ⋇ 2 . Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + š < φ 2 − φ⋇ 2 + φ 2 − φ⋇ 2 = φ− φ⋇, and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ > −φ 2 + φ⋇ 2 − φ 2 + φ⋇ 2 = −φ+ φ⋇, and so, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ (φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ (φ⋇, q̌φ)ζ̄+. If š ≥ φ 2 − φ⋇ 2 and ǔ ≤ −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) ≤ š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) ≥ ǔ. Thus, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š ∈ ∨(φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ ∈ ∨(φ⋇, q̌φ)ζ̄+. Therefore, [ζ̄](š,ǔ). K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3987 Case 2. ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š, ζ̄−(ϱ̌2) + š < φ − φ⋇ and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ, ζ̄+(ϱ̌2) + ǔ > −φ+ φ⋇. If š < φ 2 − φ⋇ 2 and ǔ > −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) = ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) = (φ− φ⋇ − š) ∨ ( φ 2 − φ⋇ 2 ) = φ− φ⋇ − š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) = ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ) = (−φ+ φ⋇ − ǔ) ∧ (−φ 2 + φ⋇ 2 ) = −φ+ φ⋇ − ǔ. Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + š < −φ 2 + φ⋇ 2 +−φ 2 + φ⋇ 2 = φ− φ⋇ and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ > −φ 2 + φ⋇ 2 − φ 2 + φ⋇ 2 = −φ+ φ⋇, and so, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ (φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ (φ⋇, q̌φ)ζ̄+. If š ≥ φ 2 − φ⋇ 2 and ǔ ≤ −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) ≤ š ∨ (φ− φ⋇ − š) ∨ ( φ 2 − φ⋇ 2 ) = š. and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) ≥ ǔ ∧ (−φ+ φ⋇ − ǔ) ∧ (−φ 2 + φ⋇ 2 ) = ǔ. Thus, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š ∈ ∨(φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ ∈ ∨(φ⋇, q̌φ)ζ̄+. Therefore, [ζ̄](š,ǔ). K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3988 Case 3. ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)+ š < φ−φ⋇, ζ̄−(ϱ̌2) ≤ š and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2)+ ǔ > −φ+φ⋇, ζ̄+(ϱ̌2) ≥ ǔ. If š < φ 2 − φ⋇ 2 and ǔ > −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) = (φ− φ⋇ − š) ∨ ( φ 2 − φ⋇ 2 ) = φ− φ⋇ − š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) = (−φ+ 1− ǔ) ∧ (−φ 2 + φ⋇ 2 ) = −φ+ φ⋇ − ǔ. Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + š < φ 2 − φ⋇ 2 + φ 2 − φ⋇ 2 = φ− φ⋇ and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ > −φ 2 + φ⋇ 2 − φ 2 + φ⋇ 2 = −φ+ φ⋇, and so, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ (φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ (φ⋇, q̌φ)ζ̄+. If š ≥ φ 2 − φ⋇ 2 and ǔ ≤ −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) ≤ (φ− φ⋇ − š) ∨ š ∨ ( φ 2 − φ⋇ 2 ) = š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) ≥ (−φ+ φ⋇ − ǔ) ∧ ǔ ∧ (−φ 2 + φ⋇ 2 ) = ǔ. Thus, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š ∈ ∨(φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ ∈ ∨(φ⋇, q̌φ)ζ̄+. Therefore, [ζ̄](š,ǔ). K. H. Hakami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3973-3993 3989 Case 4. ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) + š < φ − φ⋇, ζ̄−(ϱ̌2) + š < φ − φ⋇ and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) + ǔ > −φ+ φ⋇, ζ̄+(ϱ̌2) + ǔ > −φ+ φ⋇. If š < φ 2 − φ⋇ 2 and ǔ > −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) = (φ− φ⋇ − š) ∨ ( φ 2 − φ⋇ 2 ) = φ− φ⋇ − š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌1) ∧ (−φ 2 + φ⋇ 2 ) = (−φ+ φ⋇ǔ)− ∧(−φ 2 + φ⋇ 2 ) = −φ+ φ⋇ − ǔ. Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + š < φ 2 − φ⋇ 2 + φ 2 − φ⋇ 2 = φ− φ⋇ and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) + ǔ > −φ 2 + φ⋇ 2 − φ 2 + φ⋇ 2 = −φ+ φ⋇, and so, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ q̌φζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ q̌φζ̄+. If š ≥ φ 2 − φ⋇ 2 and ǔ ≤ −φ 2 + φ⋇ 2 , then ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) ≤ (φ− φ⋇ − š) ∨ ( φ 2 − φ⋇ 2 ) = ( φ 2 − φ⋇ 2 ) ≤ š and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ) ≥ (−φ+ φ⋇ − ǔ) ∧ (−φ 2 + φ⋇ 2 ) = (−φ 2 + φ⋇ 2 ) ≥ ǔ. Therefore, ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))š ∈ ∨(φ⋇, q̌φ)ζ̄− and ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))ǔ ∈ ∨(φ⋇, q̌φ)ζ̄+. Hence, [ζ̄](š,ǔ). Therefore, [ζ̄](š,ǔ) is a fantastic ideal of ℵ̌. Conversely, let ζ̄ be a BFS in ζ̄ and š ∈ [−1, 0), ǔ ∈ (0, 1]. Then ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌ be such that [ζ̄](š,ǔ) is a fantastic ideal of ℵ̌. If possible, let ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) > š ≥ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) REFERENCES 3990 and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) < ǔ ≤ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ), for some š ∈ (−1, 0), ǔ ∈ (0, v̌). Then (ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2, ϱ̌2 ∈ U(ζ̄; š, ǔ) ⊆ [ζ̄](š,ǔ), which indicate ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0)) ∈ [ζ̄](š,ǔ). Thus, ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≤ š or ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) + š < φ − φ⋇, ζ̄−(ϱ̌2) ≤ š or ζ̄−(ϱ̌2) + š < φ − φ⋇ and ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ≥ ǔ or ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) + ǔ > −φ+ φ⋇, ζ̄+(ϱ̌2) ≥ ǔ or ζ̄+(ϱ̌2) + ǔ > −φ+ φ⋇, and these are a contradiction. Hence, ζ̄−(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≤ ζ̄−((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∨ ζ̄−(ϱ̌2) ∨ ( φ 2 − φ⋇ 2 ) and ζ̄+(ϱ̌0 ≬ (ϱ̌1 ≬ (ϱ̌1 ≬ ϱ̌0))) ≥ ζ̄+((ϱ̌0 ≬ ϱ̌1) ≬ ϱ̌2) ∧ ζ̄+(ϱ̌2) ∧ (−φ 2 + φ⋇ 2 ), for all ϱ̌0, ϱ̌1, ϱ̌2 ∈ ℵ̌. Now, by using the Theorem 1, we conclude that ζ̄ is an (∈,∈ ∨(φ⋇, q̌φ))-BFFI of ℵ̌. 5. Conclusion The concept of an (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy set combines elements from fuzzy sets and bipolar fuzzy sets. In this paper, we investigated (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy ideals and (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals and discussed their essential properties. We examined the connection between (∈,∈ ∨(φ⋇, q̌φ))-bipolar fuzzy fantastic ideals and fuzzy fantastic ideals. In our future study of the bipolar fuzzy structure, we may consider the following topics: (i) bipolar complex fuzzy q-ideals in BCK/BCI-algebra; (ii) bipolar complex intuitionistic fuzzy commutative ideals in BG-algebras, BE-algebras; and (iii) complex picture fuzzy ideals in BCC-algebras, JU-algebras etc. Acknowledgements The authors gratefully acknowledge the funding of the Deanship of Graduate Stud- ies and Scientific Research, Jazan University, Saudi Arabia, through Project Number: GSSRD-24. References [1] E.A. Abuhijileh, M. Massa’deh, A. Sheimat, and A. Alkouri. Complex fuzzy groups based on rosenfeld’s approach. WSEAS Trans. Math., 20:368–377, 2021. [2] M. Akram and A. Farooq. m-polar fuzzy lie ideals of lie algebras. Quasigroups Related Systems, 24(2):141–150, 2016. REFERENCES 3991 [3] M. Akram, A. Farooq, and K.P. Shum. On m-polar fuzzy lie subalgebras. Ital. J. Pure Appl. Math., 36:445–454, 2016. [4] D. Al-Kadi and G. Muhiuddin. Bipolar fuzzy bci-implicative ideals of bci-algebras. Ann. Commun. Math, 1(3):88–96, 2020. [5] A. Al-Masarwah and A. G. Ahmad. Doubt bipolar fuzzy subalgebras and ideals in bck/bci-algebras. J. Math. Anal., 9(3):9–27, 2018. [6] A. Al-Masarwah and A. G. Ahmad. Novel concepts of doubt bipolar fuzzy h-ideals of bck/bci-algebras. International Journal of Innovative Computing, Information and Control, 14(06):2025–2041, 2018. [7] A. Al-Masarwah and A. G. Ahmad. Subalgebras of type (α, β) based on m-polar fuzzy points in bck/bci-algebras. AIMS Mathematics, 5(2):1035–1050, 2020. [8] A. Al-Masarwah and A. G. Ahmad. A new interpretation of multi-polarity fuzziness subalgebras of bck/bci-algebras. Fuzzy Information and Engineering, 14(3):243–254, 2022. [9] A. Al-Masarwah, A. G. Ahmad, G. Muhiuddin, and D. Al-Kadi. Generalized m- polar fuzzy positive implicative ideals of bck-algebras. Journal of Mathematics, 2021(1):6610009, 2021. [10] A. Al-Masarwah and A.G. Ahmad. On some properties of doubt biplar fuzzy h-ideals in bck/bci-algebras. Eur. J. Pure Appl. Math., 11(3):652–670, 2018. [11] A. Al-Masarwah and A.G. Ahmad. m-polar (α, β)-fuzzy ideals in bck/bci-algebras. Symmetry, 11(1):44, 2019. [12] A. Al-Masarwah and A.G. Ahmad. m-polar fuzzy ideals of bck/bci-algebras. J. King Saud Univ.-Sci., 31(4):1220–1226, 2019. [13] A. Al-Masarwah and A.G. Ahmad. A new form of generalized m-pf ideals in bck/bci- algebras. Ann. Commun. Math., 2(1):11–16, 2019. [14] A. Al-Masarwah and A.G. Ahmad. On (complete) normality of m-pf subalgebras in bck/bci-algebras. AIMS Math., 4(3):740–750, 2019. [15] Moin A. Ansari. Rough set theory applied to ju-algebras. Int. J. Math. Comput. Sc., 16:1371–1384, 2021. [16] Moin A. Ansari, Ali N. A. Koam, and A. Haider. Intersection soft ideals and their quotients on ku-algebras. AIMS Mathematics, 6(11):12077–12084, 2021. [17] M. Aslam, S. Abdullah, and M. Masood. Bipolar fuzzy ideals in la-semigroups. World Appl. Sci. J., 17(12):1769–1782, 2012. REFERENCES 3992 [18] M. Balamurugan, N. Alessa, K. Loganathan, and N. Amar Nath. (∈́, ∈́ ∨ q́ǩ)-uni- intuitionistic fuzzy soft h-ideals in subtraction bg-algebras. Mathematics, 11(10):2296, 2023. [19] M. Balamurugan, T. Ramesh, A. Al-Masarwah, and K. Alsager. A new approach of complex fuzzy ideals in bck/bci-algebras. Mathematics, 12(10):1583, 2024. [20] S.K. Bhakat and P. Dasi. (∈,∈ ∨ q)-fuzzy subgroup. Fuzzy Sets and Systems, 30(3):359–368, 1996. [21] J. Chen, S. Li, S. Ma, and X. Wang. m-polar fuzzy sets: An extension of bipolar fuzzy sets. Sci. World J., 2014. [22] A. Farooq, G. Ali, and M.Akram. On m-polar fuzzy groups. International Journal of Algebra and Statistics, 5(2):115–127, 2016. [23] A. Iamapan, M. Balamurugan, and V. Govindan. (∈,∈ ∨qk̃)-anti-intuitionistic fuzzy soft b-ideals in bck/bci-algebras. Mathematics and Statistics, 10(3):515–522, 2022. [24] M. Ibrar, A. Khan, and B. Davvazi. Characterizations of regular ordered semigroups in terms of (α, β)-bipolar fuzzy generalized bi-ideals. Journal of Intelligent & Fuzzy Systems, 33(1):365–376, 2017. [25] Y. Imai and K. Isék. On axiom systems of propositional calculi. xiv. Proc. Japan Acad., 42(1):19–22, 1966. [26] K. Isék. An algebra related with a propositional calculus. Linear Algebra and its Applications, 42(1):26–29, 1996. [27] A. Jaleel, T. Mahmood, W. Emam, and S. Yin. Interval valued bipolar complex fuzzy soft sets and their applications in decision making. Scientific Reports, 14:1–9, 2024. [28] C. Jana, M. Pal, and A.B. Saiedi. (∈,∈ ∨q)-bipolar fuzzy bck-algebras. Missouri Journal of Mathematical Sciences, 29(2):139–160, 2017. [29] K. Kawila, C. Udomsetchai, and A. Iampan. Bipolar fuzzy up-algebras. Math. Com- put. Appl., 23(4):69, 2018. [30] Ali N. A. Koam, Azeem Haider, and Moin A. Ansari. On an extension of ku-algebras. AIMS Mathematics, 6(2):1249–1257, 2021. [31] K. J. Lee. Bipolar fuzzy subalgebras and bipolar fuzzy ideals of bck/bci-algebras. Bull. Malays. Math. Sci. Soc., 32(3):361–373, 2009. [32] K. J. Lee and Y. B. Jun. Bipolar fuzzy a-ideals of bci-algebras. Commun. Korean Math. Soc., 26(4):531–542, 2011. [33] K. M. Lee. Bipolar-valued fuzzy sets and their basic operations. Proc. Int. Conf. on Intelligent Technologies, Bangkok, Thailand, pages 307–312, 2000. REFERENCES 3993 [34] T. Mahmood, U. Rehman, A. Jaleel, J. Ahmmad, and R. Chinram. Bipolar complex fuzzy soft sets and their applications in decision-making. Mathematics, 10:1048, 2022. [35] M.T. Mohseni, S.S. Ahn, R.A. Borzooei, and Y.B. Jun. Multipolar fuzzy p-ideals of bci-algebras. Mathematics, 7(11):1094, 2019. [36] G. Muhiuddin. Bipolar fuzzy ku-subalgebras/ideals of ku-algebras. Annals of Fuzzy Mathematics and Informatics, 8(3):409–418, 2014. [37] G. Muhiuddin, N. Abughazalah, A. Aljuhani, and M. Balamurugan. Tripolar picture fuzzy ideals of bck-algebras. Symmetry, 14(8):1562, 2022. [38] G. Muhiuddin, D. Al-Kadi, and M. Balamurugan. Anti-intuitionistic fuzzy soft a- ideals applied to bci-algebras. Axioms, 9(3):79, 2020. [39] G. Muhiuddin, D. Al-Kadi, A. Mehboob, and K.P. Shum. New types of bipolar fuzzy ideals of bck-algebras. International Journal of Analysis and Applications, 18(5):859– 875, 2020. [40] G. Muhiuddin, H. Harizavi, and Y.B. Jun. Bipolar-valued fuzzy soft hyper bck-ideals in bck-algebras. Discrete Mathematics Algorithms and Applications, 12(2):295998, 2019. [41] G. Muhiuddin, M.M. Takallo, R.A. Borzooei, and Y.B. Jun. m-polar fuzzy q-ideals in bci-algebras. Journal of King Saud Univeristy Science, 32(6):2803–2809, 2020. [42] M. Mursaleen, M. Balamurugan, K. Loganathan, and K.S. Nisar. (∈,∈ ∨q̌)-bipolar fuzzy-ideals of bck/bci-algebras. Journal of Function Spaces, (10):6615288, 2021. [43] A. Rosenfeld. Fuzzy groups. J. Math. Anal. Appl., 35:512–517, 1971. [44] A.B. Saeid. Bipolar-valued fuzzy bck/bci-algebras. World Applied Sci. J., 7(11):1404– 1411, 2009. [45] L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965. [46] W. R. Zhang, L. Zhang, and Y. Yang. Bipolar logic and bipolar fuzzy logic. Inform. Sci., 165(3):265–287, 2004. [47] W.R. Zhang. Bipolar fuzzy sets and relations: A computational framework for cogni- tive and modeling and multiagent decision analysis. Proc. of IEEE Conf., 2:305–309, 1994.