EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5699 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bipolar-Valued Intuitionistic Fuzzy Positive Implicative Ideals in BCK-Algebras D. Ramesh1, Shake Baji2, Aiyared Iampan3,∗, R. Durga Prasad4, B. Satyanarayana5 1 Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Andhra Pradesh-522302, India 2 Department of Mathematics, Sir C.R. Reddy College of Engineering, Eluru-534007, Andhra Pradesh, India 3 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 4 Department of Mathematics, KG Reddy College of Engineering and Technology, Hyderabad, Telangana-501504, India 5 Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur-522510, Andhra Pradesh, India Abstract. This study develops a novel framework for bipolar-valued intuitionistic fuzzy positive implicative ideals (BPVIFPIIs) in BCK-algebras by integrating bipolar-valued intuitionistic fuzzy set theory with algebraic structures. The primary objective is to define and explore the properties of BPVIFPIIs in BCK-algebras, providing rigorous theoretical foundations supported by illustra- tive examples. Key conditions under which a bipolar-valued intuitionistic fuzzy set qualifies as a BPVIFPII are established. The findings reveal significant connections between BPVIFPIIs and other fuzzy ideals, highlighting their role in advancing the understanding of uncertainty and alge- braic reasoning. This research opens avenues for further exploration of bipolar fuzzy structures in algebra and their practical implications in decision-making processes involving uncertain data. 2020 Mathematics Subject Classifications: 06F35; 03E72 Key Words and Phrases: Bipolar-valued fuzzy set (BPVFS), bipolar-valued intuitionistic fuzzy set (BPVIFS), bipolar-valued intuitionistic fuzzy ideal (BPVIFI), bipolar-valued intuitionistic fuzzy positive implicative ideal (BPVIFPII) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5699 Email addresses: ram.fuzzy@gmail.com (D. Ramesh), shakebaji6@gmail.com (S. Baji), aiyared.ia@up.ac.th (A. Iampan), durgaprasad.fuzzy@gmail.com (R. D. Prasad), drbsn63@yahoo.co.in (B. Satyanarayana) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 2 of 20 1. Introduction In this article, we will utilize the following list of abbreviations: • BCK-A: BCK-algebra • FS: Fuzzy set • BPVFS: Bipolar-valued fuzzy set • IFS: Intuitionistic fuzzy set • BPVIFS: Bipolar-valued intuitionistic fuzzy set • BPVIFSA: Bipolar-valued intuitionistic fuzzy subalgebra • BPVIFI: Bipolar-valued intuitionistic fuzzy ideal • BPVIFPII: Bipolar-valued intuitionistic fuzzy positive implicative ideal In many practical scenarios, the handling of information and the process of decision- making often encounter situations where data or results lack a clear definition. This intrinsic absence of accuracy and precision is commonly known as uncertainty. Uncertainty can come up for a few reasons, like not having all the data, mistakes in object measurement, errors in collecting data, and the natural differences in complex systems. Dealing with uncertainty and handling it properly is crucial for making good decisions, particularly in areas like business, engineering, artificial intelligence, etc. A very effective way to deal with the challenge of uncertainty is through the application of fuzzy set theory. This idea was introduced in the mid-20th century by Zadeh [31]. This theory is like a mathematical tool that helps us handle uncertainty and imprecision in a neat and structured way. Research on BCK/BCI algebras was initiated by Imai and Iséki [6, 7] in 1966, as evidenced by their work on set-theoretic difference and propositional logics. Several re- searchers, including Jun ([11, 21]), Liu [18], and Lee [16], have extensively explored the fuzzy structures inherent in BCK/BCI algebras. Others ([5, 8, 17, 27, 28]) have also made significant contributions to this field from various perspectives on various branches of algebra. BPVFSs (BPVFS), an extension of fuzzy sets ([32, 33]), are designed to ad- dress scenarios in which both negative and positive membership degrees hold significance. This extension allows us to consider the negative and positive aspects of membership and identify their respective roles. Unlike a conventional fuzzy set, where an element is ei- ther entirely connected (membership value = 1) or partially connected (membership value = (0, 1)), BPVFSs offer a more detailed representation. Alternatively, a BPVFS permits the assignment of degrees to members within the range of [−1, 1]. These degrees signify the extent to which an element is connected positively or negatively to the set. This exten- sion offers a complete view of uncertain and changing information, particularly valuable in practical scenarios where both negative and positive aspects matter. The concept of a BPVFS was applied to study different ideas in BCK/BCI-algebras, like a-ideals of BCI-algebras [16], subalgebras and ideals of BCK/BCI-algebras [15], and D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 3 of 20 many others, as explained in [9, 10]. Recent research in [1] explores bipolar-valued fuzzy BCI-implicative ideals of BCI-algebras. Muhiuddin et al. [22] look at positive implicative and closed bipolar-valued fuzzy ideals in BCK-As. Many other scholars have also added to this field, exploring different aspects of algebra in various ways ([12, 13, 23, 25, 29]). After presenting the idea of the fuzzy set concept, many studies have been carried out to investigate the extension of this concept. In 1986, Atanasov introduced the idea of IFSs, representing an advancement in fuzzy set theory. Ezhilmaran and Shankar [29] present the idea of BPVIFSs. This classification of fuzzy sets encompasses not only negative and positive levels of belongingness but also negative and positive levels of non-belongingness for elements within a given set. Recently, in [26], Satyanarayana et al. introduced the concept of BPVIFII in BCK-A. The proposed theory of bipolar-valued intuitionistic fuzzy positive implicative ideals (BPVIFPIIs) in BCK-algebras is motivated by the need to address limitations in existing frameworks for handling uncertainty in algebraic structures. Traditional fuzzy and intu- itionistic fuzzy set theories provide tools for modeling uncertainty, yet they often fail to account for the dual nature of positive and negative membership degrees simultaneously. This duality becomes crucial in applications where both positive and negative aspects of membership and non-membership must be analyzed, such as in decision-making scenarios involving conflicting or imprecise data. By extending these concepts into the domain of BCK-algebras, the study aims to enrich the theoretical foundations of algebraic reasoning under uncertainty, offering a more comprehensive mathematical model. The existence of previous studies on related bipolar fuzzy structures, such as bipolar complex fuzzy subgroups [30], bipolar complex fuzzy semigroups [24], and T -bipolar soft groups and their fundamental laws [19], provides a solid foundation for advancing this field. These works have successfully demonstrated the versatility of bipolar fuzzy sets in addressing complex algebraic problems, including Γ-semigroups [20] and bipolar com- plex fuzzy submodules [2]. However, to the best of our knowledge, no existing literature has explored the bipolar-valued intuitionistic fuzzification of positive implicative ideals in BCK/BCI algebras. This absence motivated us to initiate theoretical research on this specific topic, aiming to bridge the gap in the current body of knowledge. In addition to its theoretical contributions, the proposed framework has significant practical potential. BPVIFPIIs can be applied in decision-making systems where con- flicting or dual aspects of data must be considered, such as in medical diagnostics, where symptoms may simultaneously support and contradict potential diagnoses. Similarly, this framework is relevant in machine learning and artificial intelligence, particularly in bipolar sentiment analysis or systems that require evaluating both positive and negative influences on decisions. By enabling the simultaneous analysis of positive and negative membership values, BPVIFPIIs provide a robust tool for applications where traditional fuzzy models are insufficient. To enhance clarity, we have illustrated the research process in a flowchart (Figure 1), which provides a structured overview of the development of BPVIFPIIs. We believe this graphical representation will facilitate a deeper understanding of the proposed concepts and their significance within the broader context of algebraic and fuzzy set theory. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 4 of 20 This article presents a comprehensive exploration of BPVIFPIIs within the framework of BCK-algebras, accompanied by illustrative examples that illuminate the core concepts. The study meticulously establishes the conditions under which a BPVIFS qualifies as a BPVIFPII and when a BPVIFI attains the structure of a BPVIFPII. By combining rigor- ous theoretical analysis with practical examples, this work not only expands the algebraic understanding of fuzzy structures but also highlights the critical connections between these fuzzy ideals and their broader implications in algebraic reasoning and uncertainty modeling. Figure 1: The research process 2. Preliminaries Definition 1. [7] A BCK-A G = (G, ⋄, 0) is an algebra of type (2, 0), where G is a nonempty set, ⋄ is a binary operation on G, and 0 is a fixed element of G if it satisfies the following axioms: for all g1, è1, 11 ∈ G, (BCK-1) ((g1 ⋄ è1) ⋄ (g1 ⋄ 11)) ⋄ (11 ⋄ è1) = 0, (BCK-2) (g1 ⋄ (g1 ⋄ è1)) ⋄ è1 = 0, (BCK-3) g1 ⋄ g1 = 0, (BCK-4) 0 ⋄ g1 = 0, (BCK-5) g1 ⋄ è1 = 0 and è1 ⋄ g1 = 0 ⇒ g1 = è1. For convenience, we will let G represent the BCK-A G = (G, ⋄, 0) until otherwise spec- ified. We are able to define a binary operation ≤ on G by assuming g1 ≤ è1 if and only if g1 ⋄ è1 = 0. In a BCK-A G, the following properties hold. g1 ⋄ 0 = g1, (1) D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 5 of 20 g1 ⋄ è1 ≤ g1, (2) (g1 ⋄ è1) ⋄ 11 = (g1 ⋄ 11) ⋄ è1, (3) (g1 ⋄ 11) ⋄ (è1 ⋄ 11) ≤ g1 ⋄ è1, (4) g1 ⋄ (g1 ⋄ (g1 ⋄ è1)) = g1 ⋄ è1, (5) g1 ≤ è1 ⇒ g1 ⋄ 11 ≤ è1 ⋄ 11 and 11 ⋄ è1 ≤ 11 ⋄ g1, (6) g1 ⋄ è1 ≤ 11 ⇒ g1 ⋄ 11 ≤ è1, (7) for all g1,è1, 11 ∈ G. Theorem 1. [6] In a BCK-A G, the following holds for all g1,è1, 11 ∈ G, (i) ((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11) ≤ (g1 ⋄ è1) ⋄ 11, (ii) (g1 ⋄ 11) ⋄ (g1 ⋄ (g1 ⋄ 11)) = (g1 ⋄ 11) ⋄ 11, (iii) (g1 ⋄ (è1 ⋄ (è1 ⋄ g1))) ⋄ (è1 ⋄ (g1 ⋄ (è1 ⋄ (è1 ⋄ g1)))) ≤ g1 ⋄ è1. Definition 2. [6] A BCK-A G is considered to be a positive implicative if the following condition holds (g1 ⋄ 11) ⋄ (è1 ⋄ 11) = (g1 ⋄ è1) ⋄ 11, (8) for all g1,è1, 11 ∈ G. Definition 3. [6] A BCK-A G is considered to be a commutative if the following condition holds g1 ⋄ (g1 ⋄ è1) = è1 ⋄ (è1 ⋄ g1), (9) for all g1,è1 ∈ G. Definition 4. [6] A BCK-A G is considered to be an implicative if the following condition holds g1 ⋄ (è1 ⋄ g1) = g1, (10) for all g1,è1 ∈ G. Definition 5. [6] A subalgebra of a BCK-A G is defined as a non-empty subset I of G satisfying the following condition: g1 ⋄ è1 ∈ I, (11) for all g1,è1 ∈ I. Definition 6. [6] An ideal of a BCK-A G is defined as a non-empty subset I of G satisfying the following condition: (I1) 0 ∈ I, (I2) g1 ⋄ è1 ∈ I,è1 ∈ I ⇒ g1 ∈ I, D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 6 of 20 for all g1,è1 ∈ G. Definition 7. [6] A commutative ideal of a BCK-A G is defined as a non-empty subset I of G satisfying the condition (I1) and the following (CI1) (g1 ⋄ è1) ⋄ 11 ∈ I, 11 ∈ I ⇒ g1 ⋄ (è1 ⋄ (è1 ⋄ g1)) ∈ I, for all g1,è1, 11 ∈ G. Definition 8. [6] A positive implicative ideal of a BCK-A G is defined as a non-empty subset I of G satisfying the condition (I1) and the following (PII1) (g1 ⋄ è1) ⋄ 11 ∈ I,è1 ⋄ 11 ∈ I ⇒ g1 ⋄ 11 ∈ I, for all g1,è1, 11 ∈ G. Definition 9. [6] An implicative ideal of a BCK-A G is defined as a non-empty subset I of G satisfying the condition (I1) and the following (II1) (g1 ⋄ (è1 ⋄ g1)) ⋄ 11 ∈ I, 11 ∈ I ⇒ g1 ∈ I, for all g1,è1, 11 ∈ G. Definition 10. [31] Let G be a non-empty set. An FS in G is a mapping M : G → [0, 1]. Definition 11. [31] The complement of an FS M denoted by M is also an FS defined as M(g1) = 1−M(g1) for all g1 ∈ G. Also (M) = M. Definition 12. [14] A BPVFS B of G is defined as B = {(g1,M+ B(g1),M− B(g1))|g1 ∈ G}, where M+ B : G → [0, 1] and M− B : G → [−1, 0] are mappings. The degree of positive membership M+ B indicates how the member of G satisfies the property related to the BPVFS B, while the negative degree of membership M− B indicates the grade of satisfaction that the member of G has toward some implicit counter property of B. We shall use the symbol B = (g1,M+ B,M− B) for a BPVFS B = {(g1,M+ B(g1),M− B(g1))|g1 ∈ G}. Definition 13. [15] A BPVFS B = (g1,M+ B,M− B) in G is said to be a bipolar fuzzy subalgebra if the following conditions are satisfied: M+ B(g1 ⋄ è1) ≥ min{M+ B(g1),M+ B(è1)}, M− B(g1 ⋄ è1) ≤ max{M− B(g1),M− B(è1)}, for all g1,è1 ∈ G. Definition 14. [15] A BPVFS B = (g1,M+ B,M− B) in G is said to be a bipolar fuzzy ideal if the following conditions are satisfied: M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1) and M+ B(g1) ≥ min{M+ B(g1 ⋄ è1),M+ B(è1)}, M− B(g1) ≤ max{M− B(g1 ⋄ è1),M− B(è1)}, for all g1,è1 ∈ G. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 7 of 20 Definition 15. [22] A BPVFS B = (g1,M+ B,M− B) in G is said to be a bipolar fuzzy implicative ideal of G if it satisfies M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1) and M+ B(g1) ≥ min{M+ B((g1 ⋄ (è1 ⋄ g1)) ⋄ 11),M+ B(11)}, M− B(g1) ≤ max{M− B((g1 ⋄ (è1 ⋄ g1)) ⋄ 11),M− B(11)}, for all g1,è1, 11 ∈ G. Definition 16. A BPVFS B = (g1,M+ B,M− B) in G is said to be a bipolar fuzzy commu- tative ideal of G if it satisfies M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1) and M+ B(g1 ⋄ (è1 ⋄ (è1 ⋄ g1))) ≥ min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(11)}, M− B(g1 ⋄ (è1 ⋄ (è1 ⋄ g1))) ≤ max{M− B((g1 ⋄ è1) ⋄ 11),M− B(11)}, for all g1,è1, 11 ∈ G. Definition 17. [22] A BPVFS B = (g1,M+ B,M− B) in G is said to be a bipolar fuzzy positive implicative ideal of G if it satisfies M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1) and M+ B(g1 ⋄ 11) ≥ min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(è1 ⋄ 11)}, M− B(g1 ⋄ 11) ≤ max{M− B((g1 ⋄ è1) ⋄ 11),M− B(è1 ⋄ 11)}, for all g1,è1, 11 ∈ G. Definition 18. [3] An IFS B in a non-empty set G is an object having the form B = {(g1,M(g1),N (g1))|g1 ∈ G} where M(g1),N (g1) are level of belongingness and level of non-belongingness of g1 ∈ G respectively and 0 ≤ M(g1) + N (g1) ≤ 1 for all g1 ∈ G. We shall use the symbol B = (g1,M,N ) for an IFS B = {(g1,M(g1),N (g1))|g1 ∈ G}. Definition 19. [4] A BPVIFS B in a non-empty set G is an object having the form B = {(g1,M+ B(g1),M− B(g1),N+ B (g1),N− B (g1))|g1 ∈ G}, where M+ B(g1) : G → [0, 1], M− B(g1) : G → [−1, 0], N+ B (g1) : G → [0, 1], and N− B (g1) : G → [−1, 0] are such that 0 ≤ M+ B(g1) +N+ B (g1) ≤ 1 and −1 ≤ M− B(g1) +N− B (g1) ≤ 0. In this, M+B is used to indicate the level of positive membership level, showing the extent to which a member of G satisfies a property within a BPVIFS B. On the other side, M−B indicates the level of negative membership, showing the extent to which a member of G satisfies the implicit counter property associated with the BPVIFS. The terms N+ B (g1) and N− B (g1) refer to the level of positive non-membership and level negative non- membership respectively. We calculate N+ B (g1) and N− B (g1) as N+ B (g1) = 1 − M+ B(g1) and N− B (g1) = −1−M− B(g1). D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 8 of 20 Definition 20. A BPVIFS B = (M+ B,M− B,N+ B , N− B ) in G is a BPVIFI of G if it satisfies M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1), N+ B (0) ≤ N+ B (g1), N− B (0) ≥ N− B (g1), and M+ B(g1) ≥ min{M+ B(g1 ⋄ è1),M+ B(è1)}, M− B(g1) ≤ max{M− B(g1 ⋄ è1),M− B(è1)}, N+ B (g1) ≤ max{N+ B (g1 ⋄ è1),N+ B (è1)}, N− B (g1) ≥ min{N− B (g1 ⋄ è1),N− B (è1)}, for all g1,è1 ∈ G. Theorem 2. A BPVIFS B = (M+ B,M− B,N+ B , N− B ) in G is a BPVIFI of G if and only if it satisfies the condition if g1 ⋄ è1 ≤ 11 for all g1,è1, 11 ∈ G, then M+ B(g1) ≥ min{M+ B(è1),M+ B(11)}, M− B(g1) ≤ max{M− B(è1),M− B(11)}, N+ B (g1) ≤ max{N+ B (è1),N+ B (11)}, N− B (g1) ≥ min{N− B (è1),N− B (11)},  (12) Proof. Assume that B = (M+ B,M− B,N+ B ,N− B ) is a BPIFI of G. Let g1,è1, 11 ∈ G be such that g1 ⋄ è1 ≤ 11. Then (g1 ⋄ è1) ⋄ 11 = 0. Thus, M+ B(g1) ≥ min{M+ B(g1 ⋄ è1),M+ B(è1)} ≥ min{min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(11)},M+ B(è1)} ≥ min{min{M+ B(0),M+ B(11)},M+ B(è1)} = min{M+ B(è1),M+ B(11)}, M− B(g1) ≤ max{M− B(g1 ⋄ è1),M− B(è1)} ≤ max{max{M− B((g1 ⋄ è1) ⋄ 11),M− B(11)},M− B(è1)} ≤ max{max{M− B(0),M− B(11)},M− B(è1)} = max{M− B(è1),M− B(11)}, N+ B (g1) ≤ max{N+ B (g1 ⋄ è1),N+ B (è1)} ≤ max{max{N+ B ((g1 ⋄ è1) ⋄ 11),N+ B (11)},N+ B (è1)} ≤ max{max{N+ B (0),N+ B (11)},N+ B (è1)} = max{N+ B (è1),N+ B (11)}, N− B (g1) ≥ min{N− B (g1 ⋄ è1),N− B (è1)} ≥ min{min{N− B ((g1 ⋄ è1) ⋄ 11),N− B (11)},N− B (è1)} ≥ min{min{N− B (0),N− B (11)},N− B (è1)} = min{N− B (è1),N− B (11)}. Hence, (12) is valid. Conversely, let B = (M+ B,M− B,N+ B ,N− B ) be a BPIFS in G that satisfies (12). Since 0 ⋄ g1 ≤ g1 for all g1 ∈ G, we have M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1), N+ B (0) ≤ D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 9 of 20 N+ B (g1), and N− B (0) ≥ N− B (g1). Also, since g1 ⋄ (g1 ⋄ è1) ≤ è1 for all g1,è1 ∈ G, we have M+ B(g1) ≥ min{M+ B(g1 ⋄ è1),M+ B(è1)}, M− B(g1) ≤ max{M− B(g1 ⋄ è1),M− B(è1)}, N+ B (g1) ≤ max{N+ B (g1 ⋄ è1),N+ B (è1)}, N− B (g1) ≥ min{N− B (g1 ⋄ è1),N− B (è1)}. Therefore, B = (M+ B,M− B,N+ B ,N− B ) is a BPIFI of G. 3. Bipolar Valued Intuitionistic Fuzzy Positive Implicative Ideals Definition 21. A BPVIFS B = (M+ B,M− B,N+ B , N− B ) in G is a BPVIFPII of G if it satisfies for all g1,è1, 11 ∈ G, M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1), N+ B (0) ≤ N+ B (g1), N− B (0) ≥ N− B (g1), and M+ B(g1 ⋄ 11) ≥ min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(è1 ⋄ 11)}, M− B(g1 ⋄ 11) ≤ max{M− B((g1 ⋄ è1) ⋄ 11),M− B(è1 ⋄ 11)}, N+ B (g1 ⋄ 11) ≤ max{N+ B ((g1 ⋄ è1) ⋄ 11),N+ B (è1 ⋄ 11)}, N− B (g1 ⋄ 11) ≥ min{N− B ((g1 ⋄ è1) ⋄ 11),N− B (è1 ⋄ 11)}. Example 1. Consider G = {0, 1, 2, 3} be a set in which the binary operation ⋄ is defined as follows: 0 ⋄ g1 = 0 ∀g1 ∈ G 1 ⋄ g1 = { 0, if g1 ∈ {1, 3} 1, if g1 ∈ {0, 2} 2 ⋄ g1 = { 0, if g1 ∈ {2, 3} 2, if g1 ∈ {0, 1} 3 ⋄ g1 =  0, if g1 = 3 1, if g1 = 2 2, if g1 = 1 3, if g1 = 0 Then G is a BCK-A. Let us define a BPVIFS B = (M+ B,M− B,N+ B ,N− B ) in G as shown in Table 1. By using standard computation, it is clear that B = (M+ B,M− B,N+ B ,N− B ) is a BPV- IFPII of G. Example 2. Consider G = {0, 1, 2, 3, 4} be a set in which the binary operation ⋄ is defined as follows: 0 ⋄ g1 = 0 ∀g1 ∈ G D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 10 of 20 Table 1: BPVIFPII g1 M+ B(g1) M− B(g1) N+ B (g1) N− B (g1) 0 0.65 −0.85 0.35 −0.15 1 0.53 −0.53 0.47 −0.47 2 0.31 −0.23 0.69 −0.77 3 0.31 −0.23 0.69 −0.77 1 ⋄ g1 = { 0, if g1 ∈ {1, 2, 3, 4} 1, if g1 = 0 2 ⋄ g1 = { 0, if g1 ∈ {2, 3} 2, if g1 ∈ {0, 1, 4} 3 ⋄ g1 = { 0, if g1 = 3 3, if g1 ∈ {0, 1, 2, 4} 4 ⋄ g1 = { 0, if g1 = 4 4, if g1 ∈ {0, 1, 2, 3} Then G is a BCK-A. Let us define a BPVIFS B = (M+ B,M− B,N+ B ,N− B ) in G as shown in Table 2. Table 2: BPVIFPII g1 M+ B(g1) M− B(g1) N+ B (g1) N− B (g1) 0 0.91 −0.79 0.09 −0.21 1 0.75 −0.59 0.25 −0.41 2 0.53 −0.38 0.47 −0.62 3 0.32 −0.29 0.68 −0.71 4 0.11 −0.11 0.89 −0.89 By using standard computation, it is clear that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Theorem 3. Every BPVIFPII of G is also a BPVIFI of G. Proof. Let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFPII of G, and put 11 = 0 in Definition 21. Then, using (1), we obtain M+ B(g1 ⋄ 0) ≥ min{M+ B((g1 ⋄ è1) ⋄ 0),M+ B(è1 ⋄ 0)} ⇒ M+ B(g1) ≥ min{M+ B(g1 ⋄ è1),M+ B(è1)}, M− B(g1 ⋄ 0) ≤ max{M− B((g1 ⋄ è1) ⋄ 0),M− B(è1 ⋄ 0)} ⇒ M− B(g1) ≤ max{M− B(g1 ⋄ è1),M− B(è1)}, N+ B (g1 ⋄ 0) ≤ max{N+ B ((g1 ⋄ è1) ⋄ 0),N+ B (è1 ⋄ 0)} ⇒ N+ B (g1) ≤ max{N+ B (g1 ⋄ è1),N+ B (è1)}, D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 11 of 20 N− B (g1 ⋄ 0) ≥ min{N− B ((g1 ⋄ è1) ⋄ 0),N− B (è1 ⋄ 0)} ⇒ N−(g1) ≥ min{N− B (g1 ⋄ è1),N− B (è1)}, for all g1,è1, 11 ∈ G. This shows that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. The following example shows that the converse of Theorem 3 may not be true. Example 3. Let g1 = {0, 1, 2, 3} be a set in which the binary operation ⋄ is defined as given below 0 ⋄ g1 = 0 ∀g1 ∈ G 1 ⋄ g1 = { 0, if g1 ∈ {1, 2} 1, if g1 ∈ {0, 3} 2 ⋄ g1 =  0, if g1 = 2 1, if g1 = 1 2, if g1 ∈ {0, 3} 3 ⋄ g1 = { 0, if g1 = 3 3, if g1 ∈ {0, 1, 2} Then G is a BCK-A. Let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFS in G defined as shown in Table 3. Table 3: BPVIFI g1 M+ B(g1) M− B(g1) N+ B (g1) N− B (g1) 0 0.95 −0.87 0.05 −0.13 1 0.73 −0.43 0.27 −0.57 2 0.73 −0.43 0.27 −0.57 3 0.33 −0.15 0.67 −0.85 It is easy to check that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFI of G. However, it is not a BPVIFPII of G because M+ B(2 ⋄ 1) = M+ B(1) = 0.73 < 0.95 = M+ B(0) = min{M+ B((2 ⋄ 1) ⋄ 1),M+ B(1 ⋄ 1)}, M− B(2⋄1) = M− B(1) = −0.43 > −0.87 = M− B(0) = max{M− B((2⋄1)⋄1),M− B(1⋄1)}, N+ B (2 ⋄ 1) = N+ B (1) = 0.27 > 0.05 = N+ B (0) = max{N+ B ((2 ⋄ 1) ⋄ 1),N+ B (1 ⋄ 1)}, N− B (2 ⋄ 1) = N− B (1) = −0.57 < −0.13 = N− B (0) = min{N− B ((2 ⋄ 1) ⋄ 1),N− B (1 ⋄ 1)}. Corollary 1. Let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFPII of G. If g1 ≤ è1 in G, then M+ B(g1) ≥ M+ B(è1), M− B(g1) ≤ M− B(è1), N+ B (g1) ≤ N+ B (è1), and N− B (g1) ≥ N− B (è1). i.e., M+ B,N− B are order-reversing and M− B,N+ B are order-preserving. Corollary 2. In G, every BPVIFPII of G is a BPVIFSA. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 12 of 20 Theorem 4. If B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII in G, and J (0) = {g1 ∈ G|M+ B(g1) = M+ B(0),M− B(g1) = M− B(0),N+ B (g1) = N+ B (0),N− B (g1)= N−(0)}, then J (0) is a positive implicative ideal of G. Proof. Let g1,è1 ∈ G be such that (g1⋄è1)⋄11,è1⋄11 ∈ J (0). Since B = (g1,M+ B,M− B, N+ B ,N− B ) is a BPVIFPII in G, we have M+ B(g1⋄11) ≥ min{M+ B((g1⋄è1)⋄11),M+ B(è1⋄11)} = min{M+ B(0),M+ B(0)} = M+ B(0), M− B(g1 ⋄ 11) ≤ max{M− B((g1 ⋄ è1) ⋄ 11),M− B(è1 ⋄ 11)} = max{M− B(0),M− B(0)} = M− B(0), N+ B (g1 ⋄ 11) ≤ max{N+ B ((g1 ⋄ è1) ⋄ 11),N+ B (è1 ⋄ 11)} = max{N+ B (0),N+ B (0)} = N+ B (0), N− B (g1 ⋄ 11) ≥ min{N− B ((g1 ⋄ è1) ⋄ 11),N− B (è1 ⋄ 11)} = min{N− B (0),N− B (0)} = N− B (0). On the other hand, we know from (BPVIFPII-1) that M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1), N+ B (0) ≤ N+ B (g1), and N− B (0) ≥ N− B (g1) for all g1 ∈ G. Thus, M+ B(g1 ⋄ 11) = M+ B(0), M− B(g1 ⋄ 11) = M− B(0),N+ B (g1 ⋄ 11) = N+ B (0), and N− B (g1 ⋄ 11) = N− B (0). This implies g1 ⋄ 11 ∈ J (0). Obviously, 0 ∈ J (0). Therefore, J (0) is a positive implicative ideal of G. Theorem 5. In a positive implicative BCK-A G, every BPVIFI of G is a BPVIFPII of G. Proof. Let G be a positive implicative BCK-A, and B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFI of G. If we replace g1 with g1 ⋄ 11 and è1 with è1 ⋄ 11 in BPVIFI-2, 3, 4, 5, then M+ B(g1 ⋄ 11) ≥ min{M+ B((g1 ⋄ 11) ⋄ (è1 ⋄ 11)),M+ B(è1 ⋄ 11)} = min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(è1 ⋄ 11)}, M− B(g1 ⋄ 11) ≤ max{M− B((g1 ⋄ 11) ⋄ (è1 ⋄ 11)),M− B(è1 ⋄ 11)} = max{M− B((g1 ⋄ è1) ⋄ 11),M− B(è1 ⋄ 11)}, N+ B (g1 ⋄ 11) ≤ max{N+ B ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)),N+ B (è1 ⋄ 11)} = max{N+ B ((g1 ⋄ è1) ⋄ 11),N+ B (è1 ⋄ 11)}, N− B (g1 ⋄ 11) ≥ min{N− B ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)),N− B (è1 ⋄ 11)} = min{N− B ((g1 ⋄ è1) ⋄ 11),N− B (è1 ⋄ 11)}, for all g1,è1, 11 ∈ G. Obviously, M+ B(0) ≥ M+ B(g1), M− B(0) ≤ M− B(g1), N+ B (0) ≤ N+ B (g1), and N− B (0) ≥ N− B (g1) for all g1 ∈ G. Therefore, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Theorem 6. A BPVIFS B = (M+ B,M− B,N+ B ,N− B ) in G is a BPVIFPII of G if and only if it is a BPVIFI satisfying the following condition: M+ B(g1 ⋄ è1) ≥ M+ B((g1 ⋄ è1) ⋄ è1), M− B(g1 ⋄ è1) ≤ M− B((g1 ⋄ è1) ⋄ è1), N+ B (g1 ⋄ è1) ≤ N+ B ((g1 ⋄ è1) ⋄ è1), N− B (g1 ⋄ è1) ≥ N− B ((g1 ⋄ è1) ⋄ è1),  (13) for all g1,è1 ∈ G. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 13 of 20 Proof. Assume that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Write 11 = è1 in Definition 21, we obtain M+ B(g1 ⋄ è1) ≥ min{M+ B((g1 ⋄ è1) ⋄ è1),M+ B(è1 ⋄ è1)} = min{M+ B((g1 ⋄ è1) ⋄ è1),M+ B(0)} = M+ B((g1 ⋄ è1) ⋄ è1), M− B(g1 ⋄ è1) ≤ max{M− B((g1 ⋄ è1) ⋄ è1),M− B(è1 ⋄ è1)} = max{M− B((g1 ⋄ è1) ⋄ è1),M− B(0)} = M− B((g1 ⋄ è1) ⋄ è1), N+ B (g1 ⋄ è1) ≤ max{N+ B ((g1 ⋄ è1) ⋄ è1),N+ B (è1 ⋄ è1)} = max{N+ B ((g1 ⋄ è1) ⋄ è1),N+ B (0)} = N+ B ((g1 ⋄ è1) ⋄ è1), N− B (g1 ⋄ è1) ≥ min{N− B ((g1 ⋄ è1) ⋄ è1),N− B (è1 ⋄ è1)} = min{N− B ((g1 ⋄ è1) ⋄ è1),N− B (0)} = N− B ((g1 ⋄ è1) ⋄ è1), for all g1,è1 ∈ G. Thus, condition (13) holds. Conversely, let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFI of G satisfying the condition (13). By using BCK-1, (3), and (6), we obtain for all g1,è1, 11 ∈ G, ((g1 ⋄ 11) ⋄ (g1 ⋄ è1)) ≤ (è1 ⋄ 11) ⇒ ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ≤ (g1 ⋄ è1) ⇒ ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ⋄ 11 ≤ (g1 ⋄ è1) ⋄ 11 ⇒ ((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11) ≤ (g1 ⋄ è1) ⋄ 11. It follows from Corollary 1 that M+ B(((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)) ≥ M+ B((g1 ⋄ è1) ⋄ 11), M− B(((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)) ≤ M− B((g1 ⋄ è1) ⋄ 11), N+ B (((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)) ≤ N+ B ((g1 ⋄ è1) ⋄ 11), N− B (((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)) ≥ N− B ((g1 ⋄ è1) ⋄ 11),  (14) for all g1,è1, 11 ∈ G. Now, by using (13), Definition 20, and (14), we obtain M+ B(g1 ⋄ 11) ≥ M+ B((g1 ⋄ 11) ⋄ 11) ≥ min{M+ B(((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)),M+ B(è1 ⋄ 11)} ≥ min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(è1 ⋄ 11)}, M− B(g1 ⋄ 11) ≤ M− B((g1 ⋄ 11) ⋄ 11) ≤ max{M− B(((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)),M− B(è1 ⋄ 11)} ≤ max{M− B((g1 ⋄ è1) ⋄ 11),M− B(è1 ⋄ 11)}, N+ B (g1 ⋄ 11) ≤ N+ B ((g1 ⋄ 11) ⋄ 11) ≤ max{N+ B (((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)),N+ B (è1 ⋄ 11)} D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 14 of 20 ≤ max{N+ B ((g1 ⋄ è1) ⋄ 11),N+ B (è1 ⋄ 11)}, N− B (g1 ⋄ 11) ≥ N− B ((g1 ⋄ 11) ⋄ 11) ≥ min{N− B (((g1 ⋄ 11) ⋄ 11) ⋄ (è1 ⋄ 11)),N− B (è1 ⋄ 11)} ≥ min{N− B ((g1 ⋄ è1) ⋄ 11),N− B (è1 ⋄ 11)}. Therefore, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Theorem 7. A BPVIFS B = (M+ B,M− B,N+ B ,N− B ) in G is a BPVIFPII of G if and only if it satisfies (BPVIFI-1) and the following condition: M+ B(g1 ⋄ è1) ≥ min{M+ B(((g1 ⋄ è1) ⋄ è1) ⋄ 11),M+ B(11)}, M− B(g1 ⋄ è1) ≤ max{M− B(((g1 ⋄ è1) ⋄ è1) ⋄ 11),M− B(11)}, N+ B (g1 ⋄ è1) ≤ max{N+ B (((g1 ⋄ è1) ⋄ è1) ⋄ 11),N+ B (11)}, N− B (g1 ⋄ è1) ≥ min{N− B (((g1 ⋄ è1) ⋄ è1) ⋄ 11),N− B (11)},  (15) for all g1,è1, 11 ∈ G. Proof. Assume that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Then, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFI of G by Theorem 3, and thus, it satisfies (BPVIFI-1). Now, M+ B(g1 ⋄ è1) ≥ min{M+ B((g1 ⋄ è1) ⋄ 11),M+ B(11)} = min{M+ B(((g1 ⋄ è1) ⋄ 11) ⋄ (è1 ⋄ è1)),M+ B(11)} = min{M+ B(((g1 ⋄ 11) ⋄ è1) ⋄ (è1 ⋄ è1)),M+ B(11)} ≥ min{M+ B((g1 ⋄ 11) ⋄ è1),M+ B(11)} ≥ min{M+ B(((g1 ⋄ 11) ⋄ è1) ⋄ è1),M+ B(11)} = min{M+ B(((g1 ⋄ è1) ⋄ è1) ⋄ 11),M+ B(11)}, M− B(g1 ⋄ è1) ≤ max{M− B((g1 ⋄ è1) ⋄ 11),M− B(11)} = max{M− B(((g1 ⋄ è1) ⋄ 11) ⋄ (è1 ⋄ è1)),M− B(11)} = max{M− B(((g1 ⋄ 11) ⋄ è1) ⋄ (è1 ⋄ è1)),M− B(11)} ≤ max{M− B((g1 ⋄ 11) ⋄ è1),M− B(11)} ≤ max{M− B(((g1 ⋄ 11) ⋄ è1) ⋄ è1),M− B(11)} = max{M− B(((g1 ⋄ è1) ⋄ è1) ⋄ 11),M− B(11)}, N+ B (g1 ⋄ è1) ≤ max{N+ B ((g1 ⋄ è1) ⋄ 11),N+ B (11)} = max{N+ B (((g1 ⋄ è1) ⋄ 11) ⋄ (è1 ⋄ è1)),N+ B (11)} = max{N+ B (((g1 ⋄ 11) ⋄ è1) ⋄ (è1 ⋄ è1)),N+ B (11)} ≤ max{N+ B ((g1 ⋄ 11) ⋄ è1),N+ B (11)} ≤ max{N+ B (((g1 ⋄ 11) ⋄ è1) ⋄ è1),N+ B (11)} D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 15 of 20 = max{N+ B (((g1 ⋄ è1) ⋄ è1) ⋄ 11),N+ B (11)}, N− B (g1 ⋄ è1) ≥ min{N− B ((g1 ⋄ è1) ⋄ 11),N− B (11)} = min{N− B (((g1 ⋄ è1) ⋄ 11) ⋄ (è1 ⋄ è1)),N− B (11)} = min{N− B (((g1 ⋄ 11) ⋄ è1) ⋄ (è1 ⋄ è1)),N− B (11)} ≥ min{N− B ((g1 ⋄ 11) ⋄ è1),N− B (11)} ≥ min{N− B (((g1 ⋄ 11) ⋄ è1) ⋄ è1),N− B (11)} = min{N− B (((g1 ⋄ è1) ⋄ è1) ⋄ 11),N− B (11)}, for all g1,è1, 11 ∈ G. Hence, (15) holds. Conversely, let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFS in G which satisfies (BPVIFI- 1) and (15). Then, M+ B(g1) = M+ B(g1 ⋄ 0) ≥ min{M+ B(((g1 ⋄ 0) ⋄ 0) ⋄ 11),M+ B(11)} = min{M+ B(g1 ⋄ 11),M+ B(11)}, M− B(g1) = M− B(g1 ⋄ 0) ≤ max{M− B(((g1 ⋄ 0) ⋄ 0) ⋄ 11),M− B(11)} = max{M− B(g1 ⋄ 11),M− B(11)}, N+ B (g1) = N+ B (g1 ⋄ 0) ≤ max{N+ B (((g1 ⋄ 0) ⋄ 0) ⋄ 11),N+ B (11)} = max{N+ B (g1 ⋄ 11),N+ B (11)}, N− B (g1) = N− B (g1 ⋄ 0) ≥ min{N− B (((g1 ⋄ 0) ⋄ 0) ⋄ 11),N− B (11)} = min{N− B (g1 ⋄ 11),N− B (11)}, for all g1, 11 ∈ G. Thus, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFI of G. Taking 11 = 0 in (15), we obtain M+ B(g1 ⋄ è1) ≥ min{M+ B(((g1 ⋄ è1) ⋄ è1) ⋄ 0),M+ B(0)} = M+ B((g1 ⋄ è1) ⋄ è1), M− B(g1 ⋄ è1) ≤ max{M− B(((g1 ⋄ è1) ⋄ è1) ⋄ 0),M− B(0)} = M− B((g1 ⋄ è1) ⋄ è1), N+ B (g1 ⋄ è1) ≤ max{N+ B (((g1 ⋄ è1) ⋄ è1) ⋄ 0),N+ B (0)} = N+ B ((g1 ⋄ è1) ⋄ è1), N− B (g1 ⋄ è1) ≥ min{N− B (((g1 ⋄ è1) ⋄ è1) ⋄ 0),N− B (0)} = N− B ((g1 ⋄ è1) ⋄ è1), for all g1,è1 ∈ G. It follows from Theorem 6 that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 16 of 20 Theorem 8. Let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFI of G. Then B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G if and only if it satisfies the condition M+ B((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ≥ M+ B((g1 ⋄ è1) ⋄ 11), M− B((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ≤ M− B((g1 ⋄ è1) ⋄ 11), N+ B ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ≤ N+ B ((g1 ⋄ è1) ⋄ 11), N− B ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ≥ N− B ((g1 ⋄ è1) ⋄ 11),  (16) for all g1,è1, 11 ∈ G. Proof. Assume that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Then, by Theorem 3, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFI of G and satisfies (13). Since ((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11 = ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) ⋄ 11 ≤ (g1 ⋄ è1) ⋄ 11 for all g1,è1, 11 ∈ G, it follows from Corollary 1 that M+ B(((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≥ M+ B((g1 ⋄ è1) ⋄ 11), M− B(((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≤ M− B((g1 ⋄ è1) ⋄ 11), N+ B (((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≤ N+ B ((g1 ⋄ è1) ⋄ 11), N− B (((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≥ N− B ((g1 ⋄ è1) ⋄ 11),  (17) for all g1,è1, 11 ∈ G. Now, by using (3), (13), and (17), we obtain M+ B((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) = M+ B((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ≥ M+ B(((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≥ M+ B((g1 ⋄ è1) ⋄ 11), M− B((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) = M− B((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ≤ M− B(((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≤ M− B((g1 ⋄ è1) ⋄ 11), N+ B ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) = N+ B ((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ≤ N+ B (((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≤ N+ B ((g1 ⋄ è1) ⋄ 11), N− B ((g1 ⋄ 11) ⋄ (è1 ⋄ 11)) = N− B ((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ≥ N− B (((g1 ⋄ (è1 ⋄ 11)) ⋄ 11) ⋄ 11) ≥ N− B ((g1 ⋄ è1) ⋄ 11), for all g1,è1, 11 ∈ G. Hence, (16) is valid. Conversely, let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFI of G which satisfies (16). Write 11 = è1 in (16), we obtain M+ B((g1 ⋄ è1) ⋄ (è1 ⋄ è1)) = M+ B((g1 ⋄ è1) ⋄ 0) = M+ B(g1 ⋄ è1) ≥ M+ B((g1 ⋄ è1) ⋄ è1), D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 17 of 20 M− B((g1 ⋄ è1) ⋄ (è1 ⋄ è1)) = M− B((g1 ⋄ è1) ⋄ 0) = M− B(g1 ⋄ è1) ≤ M− B((g1 ⋄ è1) ⋄ è1), N+ B ((g1 ⋄ è1) ⋄ (è1 ⋄ è1)) = N+ B ((g1 ⋄ è1) ⋄ 0) = N+ B (g1 ⋄ è1) ≤ N+ B ((g1 ⋄ è1) ⋄ è1), N− B ((g1 ⋄ è1) ⋄ (è1 ⋄ è1)) = N− B ((g1 ⋄ è1) ⋄ 0) = N− B (g1 ⋄ è1) ≥ N− B ((g1 ⋄ è1) ⋄ è1). Therefore, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G by Theorem 6. Theorem 9. Let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFS in G. Then B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G if and only if it satisfies the condition (((g1 ⋄ è1) ⋄ è1) ⋄ u) ≤ v ⇒  M+ B(g1 ⋄ è1) ≥ min{M+ B(u),M+ B(v)}, M− B(g1 ⋄ è1) ≤ max{M− B(u),M− B(v)}, N+ B (g1 ⋄ è1) ≤ max{N+ B (u),N+ B (v)}, N− B (g1 ⋄ è1) ≥ min{N− B (u),N− B (v)},  (18) for all g1,è1, u, v ∈ G. Proof. Assume that B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. Then, by Theorem 3, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFI of G. Let g1,è1, u, v ∈ G be such that (((g1 ⋄ è1) ⋄ è1) ⋄ u) ≤ v. Now, by applying (13) and Theorem 2, we obtain M+ B(g1 ⋄ è1) ≥ M+ B((g1 ⋄ è1) ⋄ è1) ≥ min{M+ B(u),M+ B(v)}, M− B(g1 ⋄ è1) ≤ M− B((g1 ⋄ è1) ⋄ è1) ≤ max{M− B(u),M− B(v)}, N+ B (g1 ⋄ è1) ≤ N+ B ((g1 ⋄ è1) ⋄ è1) ≤ max{N+ B (u),N+ B (v)}, N− B (g1 ⋄ è1) ≥ N− B ((g1 ⋄ è1) ⋄ è1) ≥ min{N− B (u),N− B (v)}. Therefore, (18) is valid. Conversely, let B = (M+ B,M− B,N+ B ,N− B ) be a BPVIFS in G that satisfies (18). Let g1, u, v ∈ G be such that g1 ⋄ u ≤ v. Then, (((g1 ⋄ 0) ⋄ 0) ⋄ u) ⋄ v = 0, and so M+ B(g1) = M+ B(g1 ⋄ 0) ≥ min{M+ B(u),M+ B(v)}, M− B(g1) = M− B(g1 ⋄ 0) ≤ max{M− B(u),M− B(v)}, N+ B (g1) = N+ B (g1 ⋄ 0) ≤ max{N+ B (u),N+ B (v)}, N− B (g1) = N− B (g1 ⋄ 0) ≥ min{N− B (u),N− B (v)}. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 18 of 20 Thus, by Theorem 2, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFI of G. Since (((g1 ⋄è1)⋄è1)⋄ ((g1 ⋄ è1) ⋄ è1)) ⋄ 0 = 0 for all g1,è1 ∈ G, it follows from (18) that M+ B(g1 ⋄ è1) ≥ min{M+ B((g1 ⋄ è1) ⋄ è1),M+ B(0)} = M+ B((g1 ⋄ è1) ⋄ è1), M− B(g1 ⋄ è1) ≤ max{M− B((g1 ⋄ è1) ⋄ è1),M− B(0)} = M− B((g1 ⋄ è1) ⋄ è1), N+ B (g1 ⋄ è1) ≤ max{N+ B ((g1 ⋄ è1) ⋄ è1),N+ B (0)} = N+ B ((g1 ⋄ è1) ⋄ è1), N− B (g1 ⋄ è1) ≥ min{N− B ((g1 ⋄ è1) ⋄ è1),N− B (0)} = N− B ((g1 ⋄ è1) ⋄ è1). Therefore, by Theorem 6, B = (M+ B,M− B,N+ B ,N− B ) is a BPVIFPII of G. 4. Conclusion This study introduces the concept of bipolar-valued intuitionistic fuzzy positive im- plicative ideals (BPVIFPIIs) within BCK-algebras, offering a robust theoretical framework that expands the algebraic treatment of fuzzy structures. The research thoroughly exam- ines the conditions under which a bipolar-valued intuitionistic fuzzy set qualifies as a BPVIFPII and explores its connections with BPVIFIs, supported by illustrative examples and rigorous proofs. To provide a clearer understanding of the research process, Figure 1 presents a detailed flowchart outlining the logical progression of this study. This diagram highlights key stages, starting from the foundational definitions of fuzzy and intuitionistic fuzzy sets, extending through the development of bipolar-valued intuitionistic fuzzy structures, and culminating in the formalization of BPVIFPIIs in BCK-algebras. The flowchart serves as a visual guide, summarizing the theoretical steps and linking them to their practical implications, thereby facilitating a comprehensive grasp of the study’s contributions. This work not only establishes new theoretical foundations but also underscores the practical potential of BPVIFPIIs in decision-making systems, sentiment analysis, and arti- ficial intelligence. Future research will focus on exploring these applications and extending the proposed framework to related structures, such as bipolar-valued intuitionistic fuzzy soft ideals and (∈,∈ ∨q)-BPVIFIs, further enriching the field of algebraic reasoning under uncertainty. Acknowledgements This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567). D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5699 19 of 20 References [1] D. Al-Kadi and G. Muhiuddin. Bipolar fuzzy BCI-implicative ideals of BCI-algebras. Annals of Communications of Mathematics, 3(1):88–96, 2020. [2] T. Alsuraiheed, U. U. Rehman, M. A. Khan, and T. Mahmood. Bipolar complex fuzzy submodules. Physica Scripta, 99(6):065225, 2024. [3] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87–96, 1986. [4] D. Ezhilmaran and K. Sankar. Morphism of bipolar intuitionistic fuzzy graphs. 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