EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6331 ISSN 1307-5543 – ejpam.com Published by New York Business Global Complex Intuitionistic Fuzzy Quasi-Associative Ideals of BCI-algebras T. Ramesh1,2, M. Balamurugan2, Aiyared Iampan3,∗ 1Department of Mathematics, Sri Vidya Mandir Arts & Science College (Autonomous), Uthangarai 636902, Tamil Nadu, India 2Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, Tamil Nadu, India 3Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand Abstract. This paper explores the application of complex intuitionistic fuzzy sets to the study of quasi-associative ideals in BCI-algebras. We introduce a new concept—complex intuitionistic fuzzy quasi-associative ideals in BCI-algebras—and analyze their fundamental properties. The relationships between complex intuitionistic fuzzy ideals and complex intuitionistic fuzzy quasi- associative ideals are thoroughly investigated. Furthermore, we present key characterizations of these quasi-associative ideals, providing deeper insights into their structure and role within BCI- algebraic systems. Finally, we prove that every complex intuitionistic fuzzy b-ideal is a complex intuitionistic fuzzy quasi-associative ideal in BCI-algebras. 2020 Mathematics Subject Classifications: 06F35, 08A72, 03B47, 30E10 Key Words and Phrases: BCI-algebra, complex intuitionistic fuzzy quasi-associative ideal, complex intuitionistic fuzzy b-ideal 1. Introduction The concept of fuzzy sets (FSs), introduced by Zadeh in 1965, revolutionized the mod- eling of uncertainty and vagueness in mathematical structures [1]. By extending classical set theory, FSs enabled the representation of partial membership, leading to widespread applications in control systems, decision-making, and beyond. Rosenfeld (1971) advanced the theory further by introducing fuzzy groups, which integrated group theory with fuzzy logic to study algebraic structures under uncertainty [2]. The foundational work of Imai and Iséki (1966) established the axiom systems for BCK-algebras, bridging logical frameworks with algebraic formalism [3, 4]. Iséki and ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6331 Email addresses: ramesh1000.d@gmail.com (T. Ramesh), drbalamuruganm@veltech.edu.in (M. Balamurugan), aiyared.ia@up.ac.th (Aiyared Iampan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 2 of 14 Tanaka (1978) later provided a comprehensive introduction to BCK-algebra theory, laying the groundwork for exploring their structural properties and mathematical applications [5]. Zhang [6] introduced quasi-associative ideals in BCI-algebras. The integration of fuzzy logic with BCK-algebras was pioneered by Xi (1991), who introduced fuzzy BCK-algebras to study algebraic structures under uncertainty [7]. Ah- mad (1993) extended this framework to fuzzy BCI-algebras, deepening the understanding of fuzzy algebraic systems [8]. Jun and Song (2012) contributed to the field by examin- ing falling fuzzy quasi-associative ideals in BCI-algebras, thereby enriching the study of non-associative fuzzy structures [9]. In contrast, Lele and Moutari (2007) explored n-fold quasi-associative ideals in the same context [10]. Atanassov (1986) generalized FSs by introducing intuitionistic fuzzy sets (IFSs), which incorporate degrees of membership, non-membership, and hesitation to model uncertainty more robustly [11]. Jun and Kim (2000) applied this framework to BCK-algebras by investigating intuitionistic fuzzy ideals (IFIds), adding a new dimension to their algebraic structure [12]. Ramot et al. (2002, 2003) further expanded the scope of fuzzy logic by proposing complex fuzzy sets (CFSs) and complex fuzzy logic, where phase components enable richer representations of uncertainty [13, 14]. Alkouri and Salleh (2012) introduced complex intuitionistic fuzzy sets (CIF-sets), com- bining the advantages of IFSs and CFSs to model uncertainty with greater precision [15]. Deepika et al. [16] introduced hybrid quasi-ideals in ternary semigroups. Balamurugan et al. [15, 17] investigated complex Linear Diaphantine fuzzy ideals in BCK-algebras. The structure of the paper is organized as follows: • Section 2 reviews fundamental definitions of BCI-algebras and CIF-sets. • Section 3 introduces and analyzes complex intuitionistic fuzzy quasi-associative ide- als in BCI-algebras. • Section 4 investigates complex intuitionistic fuzzy b-ideals and their related proper- ties. • Section 5 concludes with findings and future perspectives. 2. Preliminaries To facilitate the understanding of the main results, this section briefly recalls funda- mental concepts related to BCI-algebras and CIF-sets. We begin by reviewing the ax- ioms and structural properties of BCI-algebras, followed by essential definitions concerning CIF-sets, including their algebraic behavior and membership representations. These pre- liminaries lay the groundwork for the subsequent development of complex intuitionistic fuzzy quasi-associative ideals. Definition 1. [5] A BCI-algebra is an algebra (X ; ∗, 0) of type (2, 0) that obeys the axioms for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X , (BCI-1) ((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (⌜ϱ̃⌝ ∗ ⌜κ̃⌝)) ∗ (⌜κ̃⌝ ∗ ⌜ϑ̃⌝) = 0, T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 3 of 14 (BCI-2) (⌜ϱ̃⌝ ∗ (⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)) ∗ ⌜ϑ̃⌝ = 0, (BCI-3) ⌜ϱ̃⌝ ∗ ⌜ϱ̃⌝ = 0, (BCI-4) ⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝ = 0, ⌜ϑ̃⌝ ∗ ⌜ϱ̃⌝ = 0 ⇒ ⌜ϱ̃⌝ = ⌜ϑ̃⌝. Definition 2. [3, 4] A void subset C is an ideal if the following hold for all ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X , (I-1) 0 ∈ C, (I-2) ⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝ ∈ C ⇒ ⌜ϱ̃⌝ ∈ C. Definition 3. [6] A void subset C is quasi-associative ideal if the following hold for all ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X , (QAI-1) 0 ∈ C, (QAI-2) ⌜ϑ̃⌝ ∈ C, ⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝) ∈ C ⇒ ⌜ϱ̃⌝ ∗ ⌜κ̃⌝ ∈ C. Definition 4. [18] A CIF-set C defined on universal set X is characterized as follows: A = {(⌜ϱ̃⌝, ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝)), ∀⌜ϱ̃⌝ ∈ X}, where ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) is truth function, ∅̃C(⌜ϱ̃⌝) : X → [0, 1] and ω̃C(⌜ϱ̃⌝) ∈ [0, 2π] is a periodic function, Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝) is falsity function, Φ̃C(⌜ϱ̃⌝) : X → [0, 1] and θ̃C(⌜ϱ̃⌝) ∈ [0, 2π] is a periodic function. Finally 0 ≤ ∅̃C(⌜ϱ̃⌝) + Φ̃C(⌜ϱ̃⌝) ≤ 1 for all ⌜ϱ̃⌝ ∈ X . Example 1. Let X = {⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝}. Then, CIF-set C = {(⌜ϱ̃⌝, 0.7e iπ 6 , 0.2e iπ 4 ), (⌜ϑ̃⌝, 0.6e iπ 3 , 0.3e iπ 2 ), (⌜κ̃⌝, 0.5e i2π 3 , 0.4e i4π 5 )}. 90◦ 270◦ 0◦180◦ 135◦ 45◦ 315◦225◦ 0.2 0.4 0.6 0.8 −⌜ϱ̃⌝∅̃C −⌜ϱ̃⌝Φ̃C −⌜ϑ̃⌝∅̃C −⌜ϑ̃⌝Φ̃C −⌜κ̃⌝∅̃C −⌜κ̃⌝Φ̃C 3. Complex Intuitionistic Fuzzy Quasi-Associative Ideals In this section, we introduce the concept of complex intuitionistic fuzzy quasi-associative ideals (CIFQA-ideals) within the framework of BCI-algebras. By extending the classical notion of quasi-associative ideals through the lens of complex intuitionistic fuzzy logic, we T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 4 of 14 establish new algebraic structures equipped to handle both magnitude and phase uncer- tainty. We define these ideals formally and explore their basic properties and examples to illustrate their significance and distinct behavior. Definition 5. A CIF-set C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) forms a complex intuitionistic fuzzy subal- gebra (CIF-subalgebra) if it satisfies the following: (CIFS-1) ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}, (CIFS-2) Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝), Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝)}, for all ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X . Definition 6. A CIF-set C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) forms a complex intuitionistic fuzzy ideal (CIF-ideal) if it satisfies the following: (CIFI-1) ∅̃C(0)eiω̃C(0) ≥ ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), Φ̃C(0)e iθ̃C(0) ≤ Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝), (CIFI-2) ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}, (CIFI-3) Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝)}, for all ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X . Lemma 1. Let C be a CIF-ideal of X . For ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X with ⌜ϱ̃⌝ ≤ ⌜ϑ̃⌝ holds in X . The following are equivalent: (1) ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝), Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝) ≤ Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝), (2) ∅̃C(0)eiω̃C(0) = ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), Φ̃C(0)e iθ̃C(0) = Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝). Proof. Straightforward. Definition 7. A CIF-set C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) forms a complex intuitionistic fuzzy quasi- associative ideal (CIFQA-ideal) of X if it satisfies the following: (CIFQAI-1) ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}, (CIFQAI-2) Φ̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝)}, for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X . Example 2. Let X = {0, τ̃ , υ̃, ζ̃} be a BCI-algebra with cayley Table 1. Let C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) Table 1: Cayley Table for (X , ∗) * 0 τ̃ υ̃ ζ̃ 0 0 τ̃ υ̃ ζ̃ τ̃ τ̃ 0 ζ̃ υ̃ υ̃ υ̃ ζ̃ 0 τ̃ ζ̃ ζ̃ υ̃ τ̃ 0 be a CIF-set is given by Table 2. T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 5 of 14 Table 2: The values of (∅̃Ceiω̃C , Φ̃Ce iθ̃C) X ∅̃Ceiω̃C Φ̃Ce iθ̃C 0 0.7e iπ 3 0.2e iπ 2 τ̃ 0.5e iπ 4 0.3e iπ 3 υ̃ 0.3e iπ 3 0.5e iπ 4 ζ̃ 0.3e iπ 5 0.3e iπ 3 Our calculations demonstrate that C is a CIFQA-ideal in X . Theorem 1. In a BCI-algebra, every CIFQA-ideal is necessarily a CIF-ideal. Proof. Let C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) be a CIFQA-ideal of X . Put ⌜κ̃⌝ = 0 in Definition 7, CIFQAI-2 and CIFQAI-3, we have ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) = ∅̃C(⌜ϱ̃⌝ ∗ 0)eiω̃C(⌜ϱ̃⌝∗0) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ 0))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗0)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}, and Φ̃C(⌜ϱ̃⌝)e iθ̃C(⌜ϱ̃⌝) = Φ̃C(⌜ϱ̃⌝ ∗ 0)eiθ̃C(⌜ϱ̃⌝∗0) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ 0))eiθ̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗0)), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} Φ̃C(⌜ϱ̃⌝)e iθ̃C(⌜ϱ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)}, for all ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X . Hence, A = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) is a CIF-ideal of X . Remark 1. The converse of Theorem 1 fails to hold in general. Example 3. Let X = {0, τ̃ , υ̃, ζ̃, σ̃} be a BCI-algebra in Table 3. Let C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) Table 3: (X , ∗) * 0 τ̃ υ̃ ζ̃ σ̃ 0 0 0 σ̃ ζ̃ υ̃ τ̃ τ̃ 0 σ̃ ζ̃ υ̃ υ̃ υ̃ υ̃ 0 σ̃ ζ̃ ζ̃ ζ̃ ζ̃ υ̃ 0 σ̃ σ̃ σ̃ σ̃ ζ̃ υ̃ 0 be a CIF-set in X which is given by Table 4. Using the routine calculation, we obtained that C is a CIF-ideal of X , but not a CIFQA-ideal as ∅̃C(σ̃ ∗ υ̃)eiω̃C(σ̃∗υ̃) = ∅̃C(ζ̃)eiω̃C(ζ̃) T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 6 of 14 Table 4: The values of (∅̃Ceiω̃C , Φ̃Ce iθ̃C) X ∅̃Ceiω̃C Φ̃Ce iθ̃C 0 0.7e iπ 2 0.2e iπ 7 τ̃ 0.5e iπ 3 0.3e iπ 5 υ̃ 0.3e iπ 5 0.4e iπ 4 ζ̃ 0.3e iπ 4 0.4e iπ 3 σ̃ 0.3e iπ 3 0.4e iπ 2 = 0.3e iπ 4 ≱ 0.7e iπ 2 = ∅̃C(0)eiω̃C(0) = min{∅̃C(σ̃ ∗ (0 ∗ υ̃))eiω̃C(σ̃∗(0∗υ̃)), ∅̃C(0)eiω̃C(0)} and Φ̃C(σ̃ ∗ υ̃)eiθ̃C(d∗υ̃) = Φ̃C(ζ̃)e iθ̃C(ζ̃) = 0.4e iπ 3 ≰ 0.2e iπ 7 = Φ̃C(0)e iθ̃C(0) = max{Φ̃C(σ̃ ∗ (0 ∗ υ̃))eiθ̃C(σ̃∗(0∗υ̃)), Φ̃C(0)e iθ̃C(0)}. Remark 2. Since Example 3 shows that CIF-ideals need not be CIFQA-ideals, we de- termine the exact conditions under which this inclusion holds in Theorem 2. Theorem 2. If the relations (⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝ ≤ ⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝) is holds for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ Xand X is quasi-associative in BCI-algebra, then every CIF-ideal is a CIFQA-ideal. Proof. Let A = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) be CIF-ideal of X , and X be the quasi-associative (⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝ ≤ ⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝) is valid for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X . By Lemma 1, for every CIF-ideal C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C). Then ∅̃Ceiω̃C is order-reversing and Φ̃Ce iθ̃C is order-preserving, we have ∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) ≥ ∅A(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) ≤ Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝). By CIFI-2 and CIFI-3 in Definition 6, we have ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 7 of 14 and Φ̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≤ max{Φ̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} ≤ max{Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)}. Hence, C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) is a CIFQA-ideal of X . Corollary 1. A BCI-algebra X has the property that all CIF-ideals are CIFQA-ideals if and only if it satisfies 0 ∗ (0 ∗ ⌜ϱ̃⌝) = 0 ∗ ⌜ϱ̃⌝. Proof. Straightforward. Proposition 1. If C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) is a CIFQA-ideal of X , then the following hold: (1) ⌜ϱ̃⌝ ≤ ⌜ϑ̃⌝ ⇒ Φ̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝), Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝) ≤ Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝), (2) ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) = ∅̃C(0)eiω̃C(0) ⇒ ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝), Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) = Φ̃C(0)e iθ̃C(0) ⇒ Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝) ≤ Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝), (3) ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}, Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝)} (4) ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝), ∅̃C(⌜κ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜κ̃⌝∗⌜ϑ̃⌝)}, Φ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝∗⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝), Φ̃C(⌜κ̃⌝∗⌜ϑ̃⌝)eiθ̃C(⌜κ̃⌝∗⌜ϑ̃⌝)} (5) ∅̃C((0 ∗ ⌜ϱ̃⌝) ∗ ⌜ϱ̃⌝)eiω̃C((0∗⌜ϱ̃⌝)∗l) = ∅̃C(0)eiω̃C(0), Φ̃C((0 ∗ ⌜ϱ̃⌝) ∗ ⌜ϱ̃⌝)eiθ̃C((0∗⌜ϱ̃⌝)∗⌜ϱ̃⌝) = Φ̃C(0)e iθ̃C(0), for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X . Proof. (1) If ⌜ϱ̃⌝ ≤ ⌜ϑ̃⌝, then ⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝ = 0. Then ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) = ∅̃C(⌜ϱ̃⌝ ∗ 0)eiω̃C(⌜ϱ̃⌝∗0) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ 0))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗0)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C(0)eiω̃C(0), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝) and Φ̃C(⌜ϱ̃⌝)e iθ̃C(⌜ϱ̃⌝) = Φ̃C(⌜ϱ̃⌝ ∗ 0)eiθ̃C(⌜ϱ̃⌝∗0) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ 0))eiθ̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗0)), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 8 of 14 ≤ max{Φ̃C(0)e iθ̃C(0), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} Φ̃C(⌜ϱ̃⌝)e iθ̃C(⌜ϱ̃⌝) ≤ Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝). (2) This is analogous to (1). (3) Let ⌜ϱ̃⌝, ⌜ϑ̃⌝ ∈ X . Then ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜ϑ̃⌝))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜ϑ̃⌝)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C(⌜ϱ̃⌝ ∗ 0)eiω̃C(⌜ϱ̃⌝∗0), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} and Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜ϑ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜ϑ̃⌝)), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ 0)eiθ̃C(⌜ϱ̃⌝∗0), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} ≤ max{Φ̃C(⌜ϱ̃⌝)e iθ̃C(⌜ϱ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)}. (4) Using BCI-1 and (1) above, we have ∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (⌜ϱ̃⌝ ∗ ⌜κ̃⌝))eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) and Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (⌜ϱ̃⌝ ∗ ⌜κ̃⌝))eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗(⌜ϱ̃⌝∗⌜κ̃⌝) ≤ Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝). By using Theorem 1, we have ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (⌜ϱ̃⌝ ∗ ⌜κ̃⌝))eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗(⌜ϱ̃⌝∗⌜κ̃⌝), ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)} ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝)} and Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ max{Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗(⌜ϱ̃⌝∗⌜κ̃⌝), Φ̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝)} ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), Φ̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝)}. (5) Put ⌜ϱ̃⌝ = 0 ∗ ⌜ϱ̃⌝, ⌜ϑ̃⌝ = 0, ⌜κ̃⌝ = ⌜ϱ̃⌝ in CIFQAI-2 and CIFQAI-3, we have ∅̃C((0 ∗ ⌜ϱ̃⌝) ∗ ⌜ϱ̃⌝)eiω̃C((0∗⌜ϱ̃⌝)∗⌜ϱ̃⌝) T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 9 of 14 ≥ min{∅̃C((0 ∗ ⌜ϱ̃⌝) ∗ (0 ∗ ⌜ϱ̃⌝))eiω̃C(0∗⌜ϱ̃⌝)∗(0∗⌜ϱ̃⌝), ∅̃C(0)eiω̃C(0)} = ∅̃C(0)eiω̃C(0) and Φ̃C((0 ∗ ⌜ϱ̃⌝) ∗ ⌜ϱ̃⌝)eiθ̃C((0∗⌜ϱ̃⌝)∗⌜ϱ̃⌝) ≤ max{Φ̃C((0 ∗ ⌜ϱ̃⌝) ∗ (0 ∗ ⌜ϱ̃⌝))eiθ̃C(0∗⌜ϱ̃⌝)∗(0∗⌜ϱ̃⌝), Φ̃C(0)e iθ̃C(0)} = Φ̃C(0)e iθ̃C(0). It follows from CIFI-1 that ∅̃C((0 ∗ ⌜ϱ̃⌝) ∗ ⌜ϱ̃⌝)eiω̃C((0∗⌜ϱ̃⌝)∗⌜ϱ̃⌝) = ∅̃C(0)eiω̃C(0) and Φ̃C((0 ∗ ⌜ϱ̃⌝) ∗ ⌜ϱ̃⌝)eiθ̃C((0∗⌜ϱ̃⌝)∗⌜ϱ̃⌝) = Φ̃C(0)e iθ̃C(0). Remark 3. In Proposition 1, every CIFQA-ideal is a CIF-subalgebra. Proposition 2. Let C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) be a CIF-ideal. Then the following statements are equivalent: (1) C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) is a CIFQA-ideal of X . (2) ∅̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ ∅̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝))eiω̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝)), Φ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ Φ̃C(⌜ϱ̃⌝ ∗ (0 ∗ ⌜ϑ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝)). (3) ∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) ≥ ∅̃C(l ∗ (m ∗ n))eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝)), Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) ≤ Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝))eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝)), for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X . Proof. (1) ⇒ (2) Assume that C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) is a CIFQA-ideal. Then by using CIFI-1, CIFQAI-1 and CIFQAI-2, we have ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (0 ∗ ⌜ϑ̃⌝))eiω̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝)), ∅̃C(0)eiω̃C(0)} ≥ ∅̃C(⌜ϱ̃⌝ ∗ (0 ∗ ⌜ϑ̃⌝))eiω̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝)) and Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝) ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (0 ∗ ⌜ϑ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝)), Φ̃C(0)e iθ̃C(0)} ≤ Φ̃C(⌜ϱ̃⌝ ∗ (0 ∗ ⌜ϑ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(0∗⌜ϑ̃⌝)). T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 10 of 14 Hence, (2) is proved. (2) ⇒ (3) Assume that (2) is satisfied. By using (⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝ = (⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝ and BCI-1, BCI-3, we have ((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (0 ∗ ⌜κ̃⌝)) ∗ (⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝)) = ((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))) ∗ (0 ∗ ⌜κ̃⌝) ≤ ((⌜ϑ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝) ∗ (0 ∗ ⌜κ̃⌝) ≤ ((⌜ϑ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝) ∗ (0 ∗ ⌜κ̃⌝) ≤ (0 ∗ ⌜κ̃⌝) ∗ (0 ∗ ⌜κ̃⌝) ≤ 0. Hence, (⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ (0 ∗ ⌜κ̃⌝) ≤ ⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝). By Lemma 1, we get ∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) ≥ ∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝))eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝)) and Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝) ≤ Φ̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝))eiθ̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝)). Hence, (3) is proved. (3) ⇒ (1) Assume that (3) is satisfied. By using CIFI-2 and CIFI-3 in Definition 6, (⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝ = (⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝ and (3), we have ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)} and Φ̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≤ max{Φ̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), Φ̃C(m)eiθ̃C(m)} ≤ max{Φ̃C((⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝) ∗ ⌜κ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜ϑ̃⌝)∗⌜κ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)} ≤ max{Φ̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiθ̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)}. Thus, CIFQAI-1 and CIFQAI-2 are satisfied. Hence, C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) is a CIFQA- ideal. T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 11 of 14 4. Complex Intuitionistic Fuzzy b-Ideals This section focuses on the formulation and analysis of complex intuitionistic fuzzy b-ideals (CIFB-ideals) in BCI-algebras. These ideals represent a refined class of CIF- ideals, characterized by stronger absorption properties under fuzzy composition. We define CIFB-ideals rigorously and examine their interrelationships with previously established ideal types, highlighting their role in the broader structure of fuzzy algebraic systems. Definition 8. A CIF-set C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) forms a complex intuitionistic fuzzy b-ideal (CIFB-ideal) of X if it satisfies the following: (CIFBI-1) ∅̃C(0)eiω̃C(0) ≥ ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝), Φ̃C(0)e iθ̃C(0) ≤ Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝), (CIFBI-2) ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}, (CIFBI-3) Φ̃C(⌜ϱ̃⌝)eiθ̃C(⌜ϱ̃⌝) ≤ max{Φ̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝)}, for all ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X . Theorem 3. Let C = (∅̃Ceiω̃C , Φ̃Ce iθ̃C) be a CIF-set. Then the following conditions are equivalent: (1) C is a CIF-ideal of X . (2) C is a CIFQA-ideal of X . (3) C is a CIFB-ideal of X . Proof. The diagram below illustrates the equivalence between the three types of ideals. CIF-ideal CIFQA-ideal CIFB-ideal We prove the equivalences by showing the implications (1) ⇒ (2) ⇒ (3) ⇒ (1). (1) ⇒ (2) Assume that C is a CIF-ideal. We show it is a CIFQA-ideal. For any ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X , by Definition 6, we have ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}. Setting ⌜ϑ̃⌝ = 0, we get ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ 0)eiω̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗0, ∅̃C(0)eiω̃C(0)}. Since (⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ 0 = ⌜ϱ̃⌝ ∗ ⌜κ̃⌝ and ∅̃C(0)eiω̃C(0) ≥ ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝) for any ⌜ϑ̃⌝ ∈ X , this simplifies to ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝), ∅̃C(0)eiω̃C(0)} T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6331 12 of 14 = ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝). Thus, CIFQAI-1 is satisfied. Similarly, for the non-membership function, Φ̃C(⌜ϱ̃⌝∗⌜κ̃⌝)eiθ̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≤ max{Φ̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝)eiθ̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), Φ̃C(⌜ϑ̃⌝)e iθ̃C(⌜ϑ̃⌝)}. Setting ⌜ϑ̃⌝ = 0 and using Φ̃C(0)e iθ̃C(0) ≤ Φ̃C(⌜ϑ̃⌝)eiθ̃C(⌜ϑ̃⌝), we obtain the required condi- tion. (2) ⇒ (3) Assume that C is a CIFQA-ideal. We show it is a CIFB-ideal. For any ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X , by using Definition 7, we have ∅̃C(⌜ϱ̃⌝ ∗ ⌜κ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜κ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ ⌜κ̃⌝))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗⌜κ̃⌝)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}. Setting ⌜κ̃⌝ = 0, we get ∅̃C(⌜ϱ̃⌝ ∗ 0)eiω̃C(⌜ϱ̃⌝∗0) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ (⌜ϑ̃⌝ ∗ 0))eiω̃C(⌜ϱ̃⌝∗(⌜ϑ̃⌝∗0)), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}. Since ⌜ϱ̃⌝ ∗ 0 = ⌜ϱ̃⌝ and ⌜ϑ̃⌝ ∗ 0 = ⌜ϑ̃⌝, this simplifies to ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ min{∅̃C(⌜ϱ̃⌝ ∗ ⌜ϑ̃⌝)eiω̃C(⌜ϱ̃⌝∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}. This is the condition for CIFB-ideal when ⌜κ̃⌝ = 0. For general ⌜κ̃⌝, the proof follows similarly by expanding the terms. The non-membership condition is analogous. (3) ⇒ (1) Assume that C is a CIFB-ideal. We show it is a CIF-ideal. For any ⌜ϱ̃⌝, ⌜ϑ̃⌝, ⌜κ̃⌝ ∈ X , by Definition 8, we have ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ ⌜κ̃⌝) ∗ ⌜ϑ̃⌝)eiω̃C((⌜ϱ̃⌝∗⌜κ̃⌝)∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}. Setting ⌜κ̃⌝ = 0, we get ∅̃C(⌜ϱ̃⌝)eiω̃C(⌜ϱ̃⌝) ≥ min{∅̃C((⌜ϱ̃⌝ ∗ 0) ∗ ⌜ϑ̃⌝)eiω̃C((⌜ϱ̃⌝∗0)∗⌜ϑ̃⌝), ∅̃C(⌜ϑ̃⌝)eiω̃C(⌜ϑ̃⌝)}. This is the condition for CIF-ideal. The non-membership condition is similarly verified. Thus, the three conditions are equivalent. 5. Conclusion In this study, we established a comprehensive framework connecting complex intu- itionistic fuzzy ideals (CIF-ideals), quasi-associative ideals (CIFQA-ideals), and b-ideals (CIFB-ideals) within BCI-algebras. We formally introduced the notion of CIFQA-ideals and demonstrated that every CIFQA-ideal is inherently a CIF-ideal. However, through counterexamples, we showed that the converse does not generally hold. We further de- rived necessary and sufficient conditions under which all three classes of ideals coincide, and established that these conditions depend on the quasi-associativity of the underly- ing algebraic structure. These findings reveal a hierarchical lattice among the ideals and T. Ramesh, M. Balamurugan, A. Iampan / Eur. J. Pure Appl. 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