EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6909 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bipolar Fuzzy Commutative Hyper BCK-Ideals in Hyper BCK-Algebras D. Ramesh1, Shake Baji2, Aiyared Iampan3,∗, B. Satyanarayana4 1 Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Educational 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, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur-522 510, Andhra Pradesh, India Abstract. This study introduces the concept of bipolar fuzzy commutative hyper BCK-ideals (BF-CHBCKIs) within the algebraic framework of hyper BCK-algebras, offering a novel approach to modeling dual uncertainty through bipolar fuzzy sets. By defining and classifying BF-CHBCKIs across multiple types and examining their structural relationships with reflexive, strong, and weak hyper BCK-ideals, we establish a comprehensive theoretical foundation supported by formal the- orems and illustrative examples. These findings extend current understandings in hyperstructure theory and fuzzy algebra, contributing to the broader landscape of abstract mathematical reason- ing. Importantly, this research aligns with Sustainable Development Goal 4 (SDG-4) by promoting inclusive and equitable quality education. The formalization of BF-CHBCKIs fosters advanced mathematical thinking and provides meaningful tools for enhancing learning environments, par- ticularly in schools and institutions that emphasize research-oriented instruction. By integrating abstract algebraic structures with uncertainty modeling, this work supports the cultivation of an- alytical skills, mathematical creativity, and deeper engagement with formal logic among students and emerging researchers. 2020 Mathematics Subject Classifications: 03E72, 06F35, 03G25 Key Words and Phrases: Hyper BCK-algebra (HBCKA), commutative hyper BCK-ideal (CHBCKI), fuzzy commutative hyper BCK-ideal (FCHBCKI), bipolar fuzzy commutative hyper BCK-ideal (BF-CHBCKI) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6909 Email addresses: ram.fuzzy@gmail.com (D. Ramesh), shakebaji6@gmail.com (S. Baji), aiyared.ia@up.ac.th (A. Iampan), 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 (4) (2025), 6909 2 of 16 1. Introduction The study of algebraic structures provides a common foundation for understanding a wide range of mathematical concepts. These structures form the mathematical basis for many of the algorithms and protocols that support our digital world. These provide the way to study mathematical operations in their most general form. By concentrating on the essential properties of algebraic structures, we can gain a deeper understanding of the underlying ideas of mathematical systems. In 1966, Imai et al. (see [1–3]) introduced the algebraic structure called a BCK-algebra as an extension of the concepts of propositional calculus and set-theoretic difference. Fol- lowing their introduction, many researchers have extensively studied BCK-algebras, es- pecially on ideals. In 1934, Marty [4] presented the idea of hyperstructure theory, or multi-algebras, during the 8th Congress of Scandinavian Mathematicians. There are nu- merous areas in both applied and pure research where the idea of hyperstructures can be useful. By applying hyperstructure theory to BCK-algebras, Jun et al. [5] developed a hy- per BCK-algebra, expanded the BCK-algebra, and studied its related properties. Building on the foundational structure of hyper BCK-algebras, Borzooei and Bakhshi [6] systemati- cally introduced four distinct types of commutative hyper BCK-ideals (CHBCKIs), laying the groundwork for a series of significant algebraic results. Expanding this framework into the domain of uncertainty modeling, Durga Prasad et al. [7] proposed the concepts of intuitionistic fuzzy positive implicative hyper BCK-ideals, also categorized into types 1, 2, 3, and 4, thereby bridging intuitionistic fuzzy logic with hyperstructure theory. Further advancing this trajectory, Satyanarayana et al. [8] developed the notions of intuitionistic fuzzy commutative hyper BCK-ideals across the same typological classifications, reinforc- ing the depth and versatility of fuzzy extensions within commutative hyper BCK-algebraic systems. Numerous techniques extend the concept of a set, with Zadeh’s fuzzy sets [9] being the most prominent example. Fuzzy sets allow elements to belong to a set with a degree of membership, known as the membership grade (between 0 and 1). They provide a framework for handling uncertainty. Lee [10] initially proposed the notion of bipolar-valued fuzzy sets as an extension of fuzzy sets. The membership function that characterizes these sets assigns each element a pair of values, one from [0, 1] (indicating positive membership) and one from [−1, 0] (indicating negative membership). This representation is effective when analyzing topics that require examining both positive and negative components. The bipolar fuzzy set (BFS) framework has emerged as a powerful extension of classical fuzzy set theory, offering a dual-valued approach to uncertainty by modeling both degrees of satisfaction and dissatisfaction. This paradigm has been fruitfully applied to a wide range of algebraic structures, demonstrating its versatility and depth. In particular, BFS logic has been explored in the context of near-rings [11], where novel classes of bipolar fuzzy ideals have been introduced. In Γ-semirings, BFSs provide a flexible tool to define bipolar fuzzy ideals that generalize standard multiplicative behaviors under uncertainty [12]. The theory has also been extended to semigroups, where rough bipolar fuzzy ideals offer refined ways to handle incomplete or vague operations [13]. More abstract alge- D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 3 of 16 braic systems have likewise benefited from BFS modeling. For example, TM-algebras [14], prime ideals in lattices [15], and subalgebras and ideals in BCK/BCI-algebras [16] have all been studied using bipolar fuzzy logic. Within the framework of BCK-algebras, BFSs have been used to define commutative ideals [17], as well as bipolar intuitionistic fuzzy implicative [18] and positive implicative ideals [19], incorporating a richer uncertainty se- mantics that captures both belief and disbelief. BFSs have also played a significant role in hyperstructure theory, particularly in the study of hyper BCK-ideals [20] and implicative hyper BCK-ideals [21], where operations yield sets rather than single outcomes. Foun- dational work by Jun and colleagues [22–24] introduced and classified several types of bipolar fuzzy hyper BCK-ideals, developing rigorous definitions and algebraic properties based on cut-level approaches and structural inclusion. These studies laid the theoretical groundwork for further generalizations, including the integration of BFS logic with soft set theory [25], where Muhiuddin et al. introduced the notion of bipolar-valued fuzzy soft hyper BCK-ideals—a hybrid model that enables multi-criteria uncertainty reasoning in hyperalgebraic contexts. Recently, this approach has even been applied to fantastic ideals in BCK/BCI-algebras [26], further emphasizing the capacity of BFSs to capture nuanced algebraic behavior under dual uncertainty. These developments collectively highlight the growing influence of bipolar fuzzy logic in algebraic reasoning, especially in systems char- acterized by non-determinism, duality, and graded membership. In this paper, we apply the concept of BFSs to CHBCKIs in HBCKAs and intro- duce the new notion of BF-CHBCKIs. We then present several theorems characterizing these notions in terms of level subsets. Furthermore, we establish the relationship among these notions, specifically BF-(strong, weak, reflexive)-HBCKIs and BF-CHBCKIs, and investigate some interesting properties. Let H be a non-empty set endowed with a hyperoperation, that is, ◦ is a function from H×H to P∗(H) = P(H)\{∅}. For any two subsets T and J of H, denoted by T ◦ J , the set ∪ a∈T,b∈J a ◦ b. We will utilize ℏ1 ◦ ℏ2 instead of ℏ1 ◦ {ℏ2}, {ℏ1} ◦ ℏ2, or {ℏ1} ◦ {ℏ2}. 2. Preliminaries In this section, we recall fundamental concepts and notation essential to the develop- ment of the main results in this paper. These include basic definitions related to BCK- algebras, hyper BCK-algebras, fuzzy sets, and bipolar fuzzy sets, as well as relevant classes of ideals. Unless stated otherwise, all algebraic structures considered here are assumed to be non-trivial. The notion of BCK-algebras was first introduced by Iséki and Tanaka [1] as an alge- braic counterpart to certain propositional calculi. Subsequently, hyper BCK-algebras were developed to generalize BCK-algebras by allowing hyperoperations, as introduced in [5]. Meanwhile, the concept of fuzzy sets, introduced by Zadeh [9], was extended to bipolar fuzzy sets (BFSs) by Zhang [27] and Lee [10] to model duality in membership functions. For the reader’s convenience, we summarize here the essential definitions and properties that will be used throughout the rest of the paper. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 4 of 16 Definition 1. [5] By a hyper BCK-algebra (HBCKA), we mean a non-empty collection H possessed of a hyperoperation ◦ and a constant 0 fulfilling the principles listed below: (HBCKA-1) (ℏ1 ◦ ℏ3) ◦ (ℏ2 ◦ ℏ3) ≪ ℏ1 ◦ ℏ2, (HBCKA-2) (ℏ1 ◦ ℏ2) ◦ ℏ3 = (ℏ1 ◦ ℏ3) ◦ ℏ2, (HBCKA-3) ℏ1 ◦ H ≪ {ℏ1}, (HBCKA-4) ℏ1 ≪ ℏ2 and ℏ2 ≪ ℏ1 ⇒ ℏ1 = ℏ2, for all ℏ1, ℏ2, ℏ3 ∈ H. We denote a relationship ≪ on H by letting ℏ1 ≪ ℏ2 ⇔ 0 ∈ ℏ1 ◦ ℏ2 and every H1,H2 ⊆ H,H1 ≪ H2 is described by ∀r ∈ H1, ∃j ∈ H2 such that r ≪ j. In such a case, we call ≪ the hyper order in H. Note that the scenario (HBCKA-3) is equal to the condition (P1). In any HBCKA H, the following is true. (P1) ℏ1 ◦ ℏ2 ≪ {ℏ1}, (P2) ℏ1 ◦ 0 ≪ {ℏ1}, 0 ◦ ℏ1 ≪ {ℏ1}, and 0 ◦ 0 ≪ {0}, (P3) (H1 ◦ H2) ◦ H3 = (H1 ◦ H3) ◦ H2, H1 ◦ H2 ≪ H1, and 0 ◦ H1 ≪ {0}, (P4) 0 ◦ 0 = {0}, (P5) 0 ≪ ℏ1, (P6) ℏ1 ≪ ℏ1, (P7) H1 ≪ H1, (P8) H1 ⊆ H2 ⇒ H1 ≪ H2, (P9) {0} = 0 ◦ ℏ1, (P10) ℏ1 ◦ 0 = {ℏ1}, (P11) 0 ◦ H1 = {0}, (P12) ℏ1 ≪ {0} ⇒ ℏ1 = {0}, (P13) H1 ◦ H2 ≪ H1, (P14) ℏ1 ∈ ℏ1 ◦ 0, (P15) ℏ1 ◦ 0 ≪ {ℏ2} ⇒ ℏ1 ≪ ℏ2, (P16) ℏ2 ≪ ℏ3 ⇒ ℏ1 ◦ ℏ3 ≪ ℏ1 ◦ ℏ2, (P17) ℏ1 ◦ ℏ2 = {0} ⇒ (ℏ1 ◦ ℏ3) ◦ (ℏ2 ◦ ℏ3) = {0} and ℏ1 ◦ ℏ3 ≪ ℏ2 ◦ ℏ3, (P18) H1◦0 = {0} ⇒ H1 = {0}, for everyone ℏ1, ℏ2, ℏ3 ∈ H in addition to every non-empty subsets H1,H2, and H3 of H. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 5 of 16 Definition 2. [5] Let I be a non-empty subset of an HBCKA H and 0 ∈ I. Then I is known as • an HBCKSA of H if ℏ1 ◦ ℏ2 ⊆ I, for all ℏ1, ℏ2 ∈ I, • a hyper BCK-ideal of H if for all ℏ1, ℏ2 ∈ H, ℏ1 ◦ ℏ2 ≪ I and ℏ2 ∈ I ⇒ ℏ1 ∈ I, • a weak hyper BCK-ideal of H if for all ℏ1, ℏ2 ∈ H, ℏ1 ◦ ℏ2 ⊆ I and ℏ2 ∈ I ⇒ ℏ1 ∈ I, • a strong hyper BCK-ideal of H if for all ℏ1, ℏ2 ∈ H, (ℏ1 ◦ ℏ2) ∩ I ̸= ∅ and ℏ2 ∈ I ⇒ ℏ1 ∈ I, • reflexive if ℏ1 ◦ ℏ1 ⊆ I, for all ℏ1 ∈ H, • S-reflexive if for all ℏ1, ℏ2 ∈ H, (ℏ1 ◦ ℏ2) ∩ I ̸= ∅ ⇒ ℏ1 ◦ ℏ2 ≪ I, • closed if for all ℏ1, ℏ2 ∈ H, ℏ1 ≪ ℏ2 and ℏ2 ∈ I ⇒ ℏ1 ∈ I. It is calm to see that each S-reflexive subset of H is reflexive. Definition 3. [6] Let I be a non-empty subset of an HBCKA H and 0 ∈ I. Then I is called a CHBCKI of (i) Type-1 if for all ℏ1, ℏ2, ℏ3 ∈ H, (ℏ1◦ℏ2)◦ℏ3 ⊆ I and ℏ3 ∈ I ⇒ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ⊆ I, (ii) Type-2 if for all ℏ1, ℏ2, ℏ3 ∈ H, (ℏ1◦ℏ2)◦ℏ3 ⊆ I and ℏ3 ∈ I ⇒ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ≪ I, (iii) Type-3 if for all ℏ1, ℏ2, ℏ3 ∈ H, (ℏ1◦ℏ2)◦ℏ3 ≪ I and ℏ3 ∈ I ⇒ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ⊆ I, (iv) Type-4 if for all ℏ1, ℏ2, ℏ3 ∈ H, (ℏ1◦ℏ2)◦ℏ3 ≪ I and ℏ3 ∈ I ⇒ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ≪ I. Theorem 1. [28] Let H1 and I be non-empty subsets of an HBCKA H. Then (i) if I is an HBCKI of H and H1 ≪ I, then H1 ⊆ I. (ii) if I is a reflexive HBCKI of H, then (ℏ1 ◦ ℏ2) ∩ I ̸= ∅ ⇒ ℏ1 ◦ ℏ2 ≪ I, for all ℏ1, ℏ2 ∈ H. Definition 4. [10] A bipolar fuzzy set (BFS) A of a set I is defined as A = {(ℏ1, α+ A(ℏ1), β − A (ℏ1)) | ℏ1 ∈ I}, (1) where α+ A : I → [0, 1] and β− A : I → [−1, 0] are mappings. The positive membership degree α+ A denotes the level of satisfaction that the element of I to the property associated with the bipolar fuzzy set A, while the negative membership degree β− A denotes the level of satisfaction that the element of I to some implicit counter property of A. We shall use the symbol (α+ A, β − A ) to denote a bipolar fuzzy set A (see (1)). Definition 5. [20] A BFS (α+ A, β − A ) in H is called a bipolar fuzzy hyper BCK-ideal (BFH- BCKI) of H if it satisfies: D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 6 of 16 (i) ℏ1 ≪ ℏ2 ⇒ α+ A(ℏ1) ≥ α+ A(ℏ2) and β− A (ℏ1) ≤ β− A (ℏ2), (ii) α+ A(ℏ1) ≥ min { inf α+ A(a) a∈ℏ1◦ℏ2 , α+ A(ℏ2) } , (iii) β− A (ℏ1) ≤ max { supβ− A (a) a∈ℏ1◦ℏ2 , β− A (ℏ2) } , for all ℏ1, ℏ2 ∈ H. Definition 6. [20] A BFS (α+ A, β − A ) in H is known as a bipolar fuzzy strong hyper BCK- ideal (BFSHBCKI) of H if it satisfies: (i) inf α+ A(a) a∈ℏ1◦ℏ1 ≥ α+ A(ℏ1) ≥ min { inf α+ A(b) b∈ℏ1◦ℏ2 , α+ A(ℏ2) } , (ii) supβ− A (c) c∈ℏ1◦ℏ1 ≤ β− A (ℏ1) ≤ max { supβ− A (d) d∈ℏ1◦ℏ2 , β− A (ℏ2) } , for all ℏ1, ℏ2 ∈ H. Definition 7. [20] A BFS (α+ A, β − A ) in H is known as a bipolar fuzzy s-weak hyper BCK- ideal (BFsWHBCKI) of H if it satisfies: (i) α+ A(0) ≥ α+ A(ℏ1) and β− A (0) ≤ β− A (ℏ1), for all ℏ1 ∈ H, (ii) for every ℏ1, ℏ2 ∈ H, there exist a, b ∈ ℏ1◦ℏ2 such that α+ A(ℏ1) ≥ min{α+ A(a), α + A(ℏ2)} and β− A (ℏ1) ≤ max{β− A (b), β − A (ℏ2)}. Definition 8. [20] A BFS (α+ A, β − A ) in H is known as a BF-weak HBCKI of H if it satisfies: (i) α+ A(0) ≥ α+ A(ℏ1) ≥ min { inf α+ A(a) a∈ℏ1◦ℏ2 , α+ A(ℏ2) } , (ii) β− A (0) ≤ β− A (ℏ1) ≤ max { supβ− A (a) a∈ℏ1◦ℏ2 , β− A (ℏ2) } , for all ℏ1, ℏ2 ∈ H. Definition 9. [20] A BFS (α+ A, β − A ) in H is known as a BFHBCK-subalgebra of H if it satisfies: (i) inf α+ A(a) a∈ℏ1◦ℏ2 ≥ min{α+ A(ℏ1), α + A(ℏ2)} (ii) supβ− A (a) a∈ℏ1◦ℏ2 ≤ max{β− A (ℏ1), β − A (ℏ2)}, for all ℏ1, ℏ2 ∈ H. Definition 10. A BFS (α+ A, β − A ) in H is said to satisfy sup-inf property if for any subset H1 of H, there exist ℏ10, ℏ20 ∈ H1 such that α+ A(ℏ10) = supα+ A(ℏ1) ℏ1∈H1 and β− A (ℏ20) = inf β− A (ℏ2) ℏ2∈H1 . D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 7 of 16 3. Bipolar fuzzy commutative hyper BCK-ideals In this section, we introduce and investigate a new class of ideals in the framework of hyper BCK-algebras, namely bipolar fuzzy commutative hyper BCK-ideals. This concept is formulated by combining the structural flexibility of hyperoperations with the dual- valued semantics of bipolar fuzzy sets. Our motivation stems from the growing demand to model algebraic uncertainty in a way that simultaneously captures both supportive and opposing evidence — a task well-suited to the bipolar fuzzy paradigm. Building upon the foundational definitions outlined in Section 2, we first define bipolar fuzzy commutative hyper BCK-ideals and examine their basic properties. Special atten- tion is given to the interplay between commutativity and bipolarity, which provides a nuanced perspective on membership under hyperoperations. Several illustrative examples are provided to clarify the behavior of such ideals in various algebraic settings. We also explore necessary and sufficient conditions under which these ideals exhibit structural regularity, as well as their relationships to existing classes of bipolar fuzzy and hyper BCK-ideals. These results not only generalize previous work on commutative ideals and bipolar fuzzy structures, but also pave the way for further developments in multi- valued algebraic logic. Definition 11. Let (α+ A, β − A ) be a BFS in H with α+ A(0) ≥ α+ A(ℏ1) and β− A (0) ≤ β− A (ℏ1), for all ℏ1 ∈ H. Then (α+ A, β − A ) is said to be a bipolar fuzzy commutative hyper BCK-ideal (BF-CHBCKI) of (i) Type-1 if for all t ∈ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), α+ A(t) ≥ min { inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } , β− A (t) ≤ max { supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } (ii) Type-2 if for all t ∈ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), α+ A(t) ≥ min { inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } , β− A (t) ≤ max { supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } (iii) Type-3 if for all t ∈ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), α+ A(t) ≥ min { supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } , β− A (t) ≤ max { inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 8 of 16 (iv) Type-4 if for all t ∈ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), α+ A(t) ≥ min { supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } , β− A (t) ≤ max { inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } , for all ℏ1, ℏ2, ℏ3 ∈ H. Example 1. Let H = {0, ℏ1, ℏ2}. Consider the following Cayley table: ◦ 0 ℏ1 ℏ2 0 {0} {0} {0} ℏ1 {ℏ1} {0, ℏ1} {0, ℏ1} ℏ2 {ℏ2} {ℏ1, ℏ2} {0, ℏ1, ℏ2} Then (H, ◦) is an HBCKA. We define a BFS (α+ A, β − A ) in H as follows: α+ A(0) = α+ A(ℏ2) = 1, α+ A(ℏ1) = 0.5, β− A (0) = β− A (ℏ2) = −0.6, β− A (ℏ1) = −0.2. Then (α+ A, β − A ) is a BF-CHBCKI of Type-1 and, consequently, of Type-2. Example 2. Let H = {0, ℏ1, ℏ2}. Consider the following Cayley table: ◦ 0 ℏ1 ℏ2 0 {0} {0} {0} ℏ1 {ℏ1} {0} {ℏ1} ℏ2 {ℏ2} {ℏ2} {0, ℏ2} Then (H, ◦) is an HBCKA. We define a BFS (α+ A, β − A ) in H as follows: α+ A(0) = 0.9, α+ A(ℏ1) = 0.6, α+ A(ℏ2) = 0.3, β− A (0) = −0.29, β− A (ℏ1) = −0.23, β− A (ℏ2) = −0.13. Then (α+ A, β − A ) is a BF-CHBCKI of Type-3 and, consequently, of Type-4. Theorem 2. Let (α+ A, β − A ) be a BFS in H. Then the following statements are valid. (i) If (α+ A, β − A ) is a BF-CHBCKI of Type-3, then it is a BF-CHBCKI of Type-1 and Type-4. (ii) If (α+ A, β − A ) is a BF-CHBCKI of Type-4 (or) 1, then it is a BF-CHBCKI of Type-2. Proof. The proof is straightforward. The converse of Theorem 2 is not necessarily true in general. This fact is illustrated in the Example 3. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 9 of 16 Example 3. Let H = {0, ℏ1, ℏ2, ℏ3}. Consider the following Cayley table: ◦ 0 ℏ1 ℏ2 ℏ3 0 {0} {0} {0} {0} ℏ1 {ℏ1} {0} {0} {0} ℏ2 {ℏ2} {ℏ2} {0} {0} ℏ3 {ℏ3} {ℏ3} {ℏ2, ℏ3} {0, ℏ2, ℏ3} Then (H, ◦) is an HBCKA. We define a BFS (α+ A, β − A ) in H as follows: α+ A(0) = α+ A(ℏ1) = 1, α+ A(ℏ2) = 0.5, α+ A(ℏ3) = 0.45, β− A (0) = β− A (ℏ1) = −0.7, β− A (ℏ2) = −0.3, β− A (ℏ3) = −0.1. Then (α+ A, β − A ) is a BF-CHBCKI of Type-1. However, it does not follow the conditions for Type-3, as ℏ3 ∈ ℏ3 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ3)) = {ℏ2, ℏ3}, α+ A(ℏ3) = 0.45 < 0.6 = α+ A(ℏ2) = min { supα+ A(a) a∈(ℏ3◦ℏ2)◦0 , α+ A(0) } , β− A (ℏ3) = −0.1 > −0.3 = β− A (ℏ2) = max { inf β− A (b) b∈(ℏ3◦ℏ2)◦0 , β− A (0) } . Theorem 3. If (α+ A, β − A ) is a BF-CHBCKI of Type-1 of H, then it is a BF-weak HBCKI. Proof. The proof is straightforward. The converse of Theorem 3 is not necessarily true in general. This fact is illustrated in Example 4. Example 4. Consider the Cayley table given in Example 3.2. Then (H, ◦) is an HBCKA. We define a BFS (α+ A, β − A ) in H as follows: α+ A(0) = 0.8, α+ A(ℏ1) = 0.4, α+ A(ℏ2) = 0.25, β− A (0) = −0.7, β− A (ℏ1) = −0.5, β− A (ℏ2) = 0. Then (α+ A, β − A ) is a BF-weak HBCKI of H. However, it does not follow the conditions for BF-CHBCKI of Type-1, because ℏ2 ∈ ℏ2 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)) = {0, ℏ1}, α+ A(ℏ1) = 0.35 < 0.8 = α+ A(0) = min { inf α+ A(a) a∈(ℏ1◦ℏ2)◦0 , α+ A(0) } , β− A (ℏ1) = −0.5 > −0.7 = β− A (0) = max { supβ− A (b) b∈(ℏ1◦ℏ2)◦0 , β− A (0) } . Theorem 4. If (α+ A, β − A ) is a BF-CHBCKI of Type-3 of H, then it is a BFSHBCKI. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 10 of 16 Proof. Suppose (α+ A, β − A ) is a BF-CHBCKI of Type-3 of H. Setting ℏ2 = 0 in Definition 11 (iii), we obtain t ∈ ℏ1 ◦ (0 ◦ (0 ◦ ℏ1)) = ℏ1, α+ A(ℏ1) ≥ min { supα+ A(a) a∈(ℏ1◦0)◦ℏ3 , α+ A(ℏ3) } = min { supα+ A(a) a∈ℏ1◦ℏ3 , α+ A(ℏ3) } β− A (ℏ1) ≤ max { inf β− A (b) b∈(ℏ1◦0)◦ℏ3 , β− A (ℏ3) } = max { inf β− A (b) b∈ℏ1◦ℏ3 , β− A (ℏ3) }  (2) for all ℏ1, ℏ3 ∈ H. First, we show that for ℏ1, ℏ2 ∈ H, if ℏ1 ≪ ℏ2, then α+ A(ℏ1) ≥ α+ A(ℏ2) and β− A (ℏ1) ≤ β− A (ℏ2). To show this, let ℏ1, ℏ2 ∈ H be such that ℏ1 ≪ ℏ2. Then 0 ∈ ℏ1 ◦ ℏ2, and by (2), we have α+ A(ℏ1) ≥ min { supα+ A(a) a∈ℏ1◦ℏ2 , α+ A(ℏ2) } = min{α+ A(0), α + A(ℏ2)} = α+ A(ℏ2) β− A (ℏ1) ≤ max { inf β− A (b) b∈ℏ1◦ℏ2 , β− A (ℏ2) } = max{β− A (0), β − A (ℏ2)} = β− A (ℏ2)  (3) Let ℏ1 ∈ H and a ∈ ℏ1 ◦ℏ1. Since ℏ1 ◦ℏ1 ≪ ℏ1, we have a ≪ ℏ1, for all a ∈ ℏ1 ◦ℏ1. Hence, by (3), we deduce α+ A(a) ≥ α+ A(ℏ1) and β− A (a) ≤ β− A (ℏ1), for all a ∈ ℏ1 ◦ ℏ1. Hence, inf α+ A(a) a∈ℏ1◦ℏ1 ≥ α+ A(ℏ1) and supβ− A (c) c∈ℏ1◦ℏ1 ≤ β− A (ℏ1). (4) From the combination of (2) and (4), we obtain inf α+ A(a) a∈ℏ1◦ℏ1 ≥ α+ A(ℏ1) ≥ min { supα+ A(b) b∈ℏ1◦ℏ2 , α+ A(ℏ2) } , supβ− A (c) c∈ℏ1◦ℏ1 ≤ β− A (ℏ1) ≤ max { inf β− A (d) d∈ℏ1◦ℏ2 , β− A (ℏ2) } . Thus, (α+ A, β − A ) is a BFSHBCKI of H. The converse of Theorem 4 is not necessarily true in general. This fact is illustrated in Example 5. Example 5. Consider the Cayley table given in Example 2. Then (H, ◦) is an HBCKA. We define a BFS (α+ A, β − A ) in H as follows: α+ A(0) = 0.8, α+ A(ℏ1) = 0.6, α+ A(ℏ2) = 0.3, β− A (0) = −0.23, β− A (ℏ1) = −0.19, β− A (ℏ2) = −0.13. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 11 of 16 Then (α+ A, β − A ) is a BFSHBCKI of H. However, it does not follow the conditions for BF-CHBCKI of Type-3, because ℏ2 ∈ ℏ2 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ2)) = {0, ℏ2}, α+ A(ℏ2) = 0.3 < 0.8 = α+ A(0) = min { inf α+ A(a) a∈(ℏ2◦ℏ2)◦0 , α+ A(0) } , β− A (ℏ2) = −0.13 > −0.23 = β− A (0) = max { supβ− A (b) b∈(ℏ2◦ℏ2)◦0 , β− A (0) } . Corollary 1. Let (α+ A, β − A ) be a BF-CHBCKI of Type-3, and let ℏ1, ℏ2 ∈ H. Then (i) ℏ1 ≪ ℏ2 ⇒ α+ A(ℏ1) ≥ α+ A(ℏ2) and β− A (ℏ1) ≤ β− A (ℏ2), (ii) α+ A(ℏ1) ≥ min{α+ A(a), α + A(ℏ2)}, (iii) β− A (ℏ1) ≤ max{β− A (b), β − A (ℏ2)}, for all a, b ∈ ℏ1 ◦ ℏ2. Corollary 2. If (α+ A, β − A ) is a BF-CHBCKI of Type-3, then (i) α+ A(ℏ1) ≥ min { inf α+ A(a) a∈ℏ1◦ℏ2 , α+ A(ℏ2) } , (ii) β− A (ℏ1) ≤ max { supβ− A (b) b∈ℏ1◦ℏ2 , β− A (ℏ2) } , for all ℏ1, ℏ2 ∈ H. Proposition 1. Let (α+ A, β − A ) be a BF-CHBCKI of Type-3. If (α+ A, β − A ) satisfies the inf-sup property, then it is a BFsWHBCKI of H. Proof. The proof is straightforward. Definition 12. Let (α+ A, β − A ) be a BFS in H. Then (α+ A, β − A ) is closed if for all ℏ1, ℏ2 ∈ H such that ℏ1 ≪ ℏ2, we have α+ A(ℏ1) ≥ α+ A(ℏ2) and β− A (ℏ1) ≤ β− A (ℏ2). Definition 13. [16] For a BFS (α+ A, β − A ), the positive p-cut (where p ∈ [0, 1]) and the negative n-cut (where n ∈ [−1, 0]) are defined as follows: U(α+ A; p) = {ℏ1 ∈ H | α+ A(ℏ1) ≥ p}, L(β− A ; n) = {ℏ1 ∈ H | β− A (ℏ1) ≤ n}. Theorem 5. Let (α+ A, β − A ) be a BFS in H. Then the following statements are hold: (i) A BFS (α+ A, β − A ) is a BF-CHBCKI of Type-1 of H if and only if for all (p, n) ∈ [0, 1]× [−1, 0], the non-empty cut sets U(α+ A; p) and L(β− A ; n) are CHBCKIs Type-1 of H. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 12 of 16 (ii) If (α+ A, β − A ) is a BF-CHBCKI of Type-3 of H, then for all (p, n) ∈ [0, 1]× [−1, 0], the non-empty cut sets U(α+ A; p) and L(β− A ; n) are CHBCKIs Type-3 of H. (iii) If (α+ A, β − A ) satisfies the sup-inf property and for all (p, n) ∈ [0, 1]× [−1, 0], the non- empty cut sets U(α+ A; p) and L(β− A ; n) are reflexive-CHBCKIs of Type-3 of H, then (α+ A, β − A ) is a BF-CHBCKI of Type-3 of H. Proof. (i) Suppose (α+ A, β − A ) is a BF-CHBCKI of Type-1 of H and for all (p, n) ∈ [0, 1] × [−1, 0], U(α+ A; p) and L(β− A ; n) are non-empty. By definition, 0 ∈ U(α+ A; p) and 0 ∈ L(β− A ; n). Therefore, 0 ∈ U(α+ A; p) ∩ L(β− A ; n). Let ℏ1, ℏ2, ℏ3 be elements of H such that (ℏ1 ◦ ℏ2) ◦ ℏ3 ⊆ U(α+ A; p) and ℏ3 ∈ U(α+ A; p). Then, for all a ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3, we have a ∈ U(α+ A; p) and ℏ3 ∈ U(α+ A; p). This implies α+ A(a) ≥ p, for all a ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and α+ A(ℏ3) ≥ p. Consequently, inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 ≥ p and α+ A(ℏ3) ≥ p. Thus, for all k ∈ ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)), we have α+ A(k) ≥ min { inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } ≥ min{p, p} = p. This implies k ∈ U(α+ A; p), for all k ∈ ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)). Therefore, ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ℏ1)) ⊆ U(α+ A; p). Let (ℏ1 ◦ ℏ2) ◦ ℏ3 ⊆ L(β− A ; n) and ℏ3 ∈ L(β− A ; n). Then, for all b ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3, we have b ∈ L(β− A ; n) and ℏ3 ∈ L(β− A ; n). This implies β− A (b) ≤ n, for all b ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and β− A (ℏ3) ≤ n. Consequently, supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 ≤ n and β− A (ℏ3) ≤ n. Thus, for all k ∈ ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)), we have β− A (k) ≤ max { supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } = max{n, n} = n. This implies k ∈ L(β− A ; n), for all t ∈ ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)). Therefore, ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ⊆ L(βN A ; n). Thus, for all (p, n) ∈ [0, 1] × [−1, 0], the cut sets U(α− A; p) and (β− A ; n) are CHBCKIs of Type-1 of H. Conversely, let us assume that for all (p, n) ∈ [0, 1] × [−1, 0], the cut sets U(α+ A; p) and L(β− A ; n) are CHBCKIs of Type-1 of H. Let ℏ1, ℏ2, ℏ3 be elements of H. We define p = min { inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } . Then inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 ≥ p and α+ A(ℏ3) ≥ p. ⇒ α+ A(a) ≥ p for all a ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and α+ A(ℏ3) ≥ p ⇒ a ∈ U(α+ A; p) for all a ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and ℏ3 ∈ U(α+ A; p) ⇒ (ℏ1 ◦ ℏ2) ◦ ℏ3 ⊆ U(α+ A; p) and ℏ3 ∈ U(α+ A; p). D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 13 of 16 By hypothesis, we have ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ⊆ U(α+ A; p). Thus, for all k1 ∈ (ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), α+ A(k1) ≥ p = min { inf α+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } . Define n = max { supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } . Then supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 ≤ n and β− A (ℏ3) ≤ n. ⇒ β− A (b) ≤ n for all b ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and β− A (ℏ3) ≤ n ⇒ b ∈ L(β− A ; n) for all b ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and ℏ3 ∈ L(β− A ; n) ⇒ (ℏ1 ◦ ℏ2) ◦ ℏ3 ⊆ L(β− A ; n) and ℏ3 ∈ L(β− A ; n). By hypothesis, we have ℏ1◦(ℏ2◦(ℏ2◦ℏ1)) ⊆ L(β− A ; n). Thus, for all k2 ∈ (ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), β− A (k2) ≤ n = max { supβ− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } . Consider the case where α+ A(ℏ1) = p and β− A (ℏ2) = n for some ℏ1, ℏ2 ∈ H. Since 0 ∈ U(α+ A; p) ∩ L(β− A ; n), we obtain α+ A(0) ≥ p = α+ A(ℏ1) and β− A (0) ≤ n = β− A (ℏ2), for all ℏ1, ℏ2 ∈ H. Therefore, (α+ A, β − A ) is a BF-CHBCKI of Type-1 of H. (ii) Assume (α+ A, β − A ) is a BF-CHBCKI of Type-3 of H, and for all (p, n) ∈ [0, 1] × [−1, 0], the cut sets U(α+ A; p) and L(β− A ; n) are non-empty. By definition, we have 0 ∈ U(α+ A; p) ∩ L(β− A ; n). Let ℏ1, ℏ2, ℏ3 be elements of H such that (ℏ1 ◦ ℏ2) ◦ ℏ3 ≪ U(α+ A; p) and ℏ3 ∈ U(α+ A; p). Then, for every a ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3, we can find k ∈ U(α+ A; p) such that a ≪ k. By Corollary 1, we obtain α+ A(a) ≥ α+ A(k) ≥ p. This implies α+ A(a) ≥ p, for all a ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and ℏ3 ∈ U(α+ A; p), Consequently, supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 ≥ p and α+ A(ℏ3) ≥ p. Thus, by hypothesis, for all k ∈ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), α+ A(k) ≥ min { supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } ≥ min{p, p} = p. Therefore, it follows that ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)) ⊆ U(α+ A; p). Let ℏ1, ℏ2, ℏ3 ∈ H be such that (ℏ1 ◦ ℏ2) ◦ ℏ3 ≪ L(β− A ; n) and ℏ3 ∈ L(β− A ; n). Then, for every b ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3, we can find l ∈ L(β− A ; n) such that b ≪ l. By Corollary 1, we obtain β− A (b) ≤ β− A (l) ≤ n. This implies β− A (b) ≤ n, for all b ∈ (ℏ1 ◦ ℏ2) ◦ ℏ3 and ℏ3 ∈ L(β− A ; n). Consequently, inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 ≤ n and β− A (ℏ3) ≤ n. Thus, by hypothesis, for all k ∈ ℏ1◦(ℏ2◦(ℏ2◦ℏ1)), β− A (k) ≤ max { inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } ≤ max{n, n} = n. Therefore, it follows that ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)) ⊆ L(β− A ; n). Therefore, we conclude that for all (p, n) ∈ [0, 1]× [−1, 0], the cut sets U(α+ A; p) and L(β− A ; n) are CHBCKIs of Type-3 of H. D. Ramesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6909 14 of 16 (iii) Assume that H satisfies the sup-inf property, and for all (p, n) ∈ [0, 1]× [−1, 0], the cut sets U(α+ A; p) and L(β− A ; n) are reflexive CHBCKIs of Type-3 of H. Let ℏ1, ℏ2, ℏ3 ∈ H. Define p = min { supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } . Then supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 ≥ p and α+ A(ℏ3) ≥ p. Since α+ A satisfies the sup property, there exists a0 ∈ (ℏ1◦ℏ2)◦ℏ3 such that α+ A(a0) = supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 ≥ p. This implies α+ A(a0) ≥ p. That is, the set ((ℏ1 ◦ℏ2)◦ℏ3)∩U(α+ A; p) is non-empty. We know that every CHBCKI of Type-3 is an HBCKI of H (see Theorem 4.4 in [6]). Therefore, U(α+ A; p) is a reflexive-HBCKI of H and ((ℏ1 ◦ℏ2) ◦ℏ3)∩U(α+ A; p) ̸= ∅. By Theorem 1 (ii), it follows that (ℏ1 ◦ ℏ2) ◦ ℏ3 ≪ U(α+ A; p). Since (ℏ1 ◦ ℏ2) ◦ ℏ3 ≪ U(α+ A; p), ℏ3 ∈ U(α+ A; p), and U(α+ A; p) is a CHBCKI of Type-3 of H, it follows that ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)) ⊆ U(α+ A; p). This implies that for all k ∈ ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ ℏ1)), we have k ∈ U(α+ A; p). Thus, α+ A(a) ≥ p, where p = min { supα+ A(a) a∈(ℏ1◦ℏ2)◦ℏ3 , α+ A(ℏ3) } . Define n = max { inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } . Then, inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 ≤ n and β− A (ℏ3) ≤ n. Since β− A satisfies the inf property, there exists b0 ∈ (ℏ1◦ℏ2)◦ℏ3 such that β− A (b0) = inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 ≤ n. This implies β− A (b0) ≤ n. That is, the set ((ℏ1 ◦ ℏ2) ◦ ℏ3) ∩ L(β− A ; n) is non-empty. We know that every CHBCKI of Type-3 is an HBCKI of H (see Theorem 4.4 in [6]). Therefore, L(β− A ; n) is a reflexive-HBCKI of H and ((ℏ1 ◦ℏ2) ◦ℏ3)∩L(β− A ; n) ̸= ∅. By Theorem 1 (ii), it follows that (ℏ1 ◦ℏ2)◦ℏ3 ≪ L(β− A ; n). Since (ℏ1 ◦ℏ2)◦ℏ3 ≪ L(β− A ; n), ℏ3 ∈ L(β− A ; n), and L(β− A ; n) is a CHBCKI of Type-3 of H, it follows that ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ℏ1)) ⊆ L(β− A ; n). This implies that for all k ∈ ℏ1 ◦ (ℏ2 ◦ (ℏ2 ◦ℏ1)), we have k ∈ L(β− A ; n). Thus, β− A (t2) ≤ n, where n = max { inf β− A (b) b∈(ℏ1◦ℏ2)◦ℏ3 , β− A (ℏ3) } . Therefore, we conclude that (α+ A, β − A ) is a BF-CHBCKI of Type-3 of H. 4. Conclusion In this paper, we propose a comprehensive framework for the bipolar fuzzification of commutative hyper BCK-ideals (CHBCKIs) within the context of hyper BCK-algebras (HBCKAs). By formalizing bipolar fuzzy commutative hyper BCK-ideals (BF-CHBCKIs), we introduced new definitions, classifications, and theorems that elucidate their structural properties and their interrelations with various forms of hyper BCK-ideals, including re- flexive, strong, and weak types. The use of bipolar fuzzy subsets as analytical tools not only enhances the flexibility of algebraic reasoning under uncertainty but also strength- ens the foundation for advanced research in fuzzy hyperstructure theory. Moreover, the findings of this study contribute to the development of research-oriented mathematical pedagogy, offering students in inquiry-driven learning environments access to complex yet structured models of logic and abstraction. In alignment with Sustainable Development D. Ramesh et al. / Eur. J. 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