EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6798 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Innovative Method for Employing Complex Intuitionistic Fuzzy Ideals in BCK/BCI-Algebras Muhammad Jawad1, Sarka Hoskova-Mayerova2,∗, Niat Nigar1, Muhammad Haris Mateen1,∗ 1 School of Mathematics, Minhaj University Lahore, Pakistan 2 Department of Mathematics and Physics, University of Defence, Brno, 66210, Czech Republic Abstract. The complex intuitionistic fuzzy set is a more generalized version of the complex fuzzy set. It is made by including the complex degree of non-grading functions, which are also important in the decision-making process, and studying their basic properties. The complex intuitionistic fuzzy set extends theories like the complex fuzzy set, intuitionistic fuzzy set, and fuzzy set. The goal of this paper is to apply complex intuitionistic fuzzy sets in BCK/BCI-algebras (M), explain what a complex intuitionistic fuzzy ideal is, and explore some of its properties. We introduce the notion of a complex intuitionistic fuzzy sub-algebra in M , and its characteristics are investigated. We also look into the level operators and models of these complex intuitionistic fuzzy sub-algebras and explain their importance in M . Finally, we discuss the laws and operations of a complex intuitionistic fuzzy set in M , such as complement, intersection, union, boundedness, and simple differences of complex intuitionistic fuzzy ideals. 2020 Mathematics Subject Classifications: 03G25, 08A72, 08E35 Key Words and Phrases: Complex intuitionistic fuzzy sub-algebra, fuzzy logic, BCK/BCI- algebras, complex intuitionistic fuzzy environment 1. Introduction The concept of BCK/BCI-algebra developed from two distinct techniques: (1) set theory, and (2) non-classical and classical propositional calculi. Currently, BCK/BCI- algebras are utilized in a variety of mathematical fields, including topology, probability theory, functional analysis, group theory, fuzzy set theory, and others. Zhang [1] pre- sented the BCK and BCI algebra concepts and improved their definitions by providing new equivalent conditions. Liu [2] established the new concepts of fuzzy BCI-implicative ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6798 Email addresses: sarka.mayerova@unob.cz (S. Hoskova-Mayerova), harism.math@gmail.com (M. H. Mateen) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 2 of 22 ideals and fuzzy BCI-positive implicative ideals in BCI-algebras and explored their char- acteristics. The relations between distinct fuzzy ideals demonstrate that a fuzzy set (FS) µ of a BCI-algebra is a fuzzy BCI-implicative ideal if and only if µ is both a fuzzy BCI- commutative ideal and a fuzzy BCI-positive implicative ideal. Meng [3] investigated the idea of fuzzy implicative ideals in BCK-algebras and presented various characterizations of these ideals. Jun [4] utilized the concept of soft sets to the concept of BCK/BCI algebras. The concepts of soft BCK/BCI algebras and soft sub-algebras were presented, and their fundamental characteristics were established. Senapati and Shum [5] introduced the cubic set notion to implicative ideals of BCK-algebras and characterised their basic attributes. Andres-Sanchez [6] developed fuzzy sets in quantities correlated with a set of objects, with the degree of membership of [0, 1] expressing the degree of elements belonging to the set. Mardani et al. [7] established among the available strategies for dealing with uncertainty issues, fuzzy mathematics mainly focuses on items with specifically internal definitions but uncertain outward deployment as membership degree; however, the in- dependency value as a key component of elements was ignored in the frameworks. The intuitionistic fuzzy set (IFS) theory has a broad range of uses in many domains, such as medical diagnosis [8, 9], pattern recognition [10], engineering systems [11], and decision- making [12]. In today’s world, scientists and technologists routinely meet complex processes and phenomena that are beyond full and exact insight. Therefore, it is essential to include accurate mathematical models into systems that exhibit a high level of uncertainty. The reason for developing fuzzy set theory stemmed from the need to broaden traditional set theory in order to efficiently address a certain purpose. The methodology offered herein provides a systematic strategy for developing and evaluating various models that effectively capture and tackle the inherent uncertainties in a particular environment. This theory is critical for the development of such structures. Furthermore, it improves our ability to investigate and adapt to the complex and unpredictable properties of systems within a wide range of scientific and technical disciplines. The uses of fuzzy set concept have been demonstrated across a wide range of scientific areas and natural phenomena. Fuzzy sets (FSs) have proven to be an adaptable strat- egy for dealing with complex and uncertain situations in a variety of contexts. FS relies primarily on membership functions that operate in a single dimension, making it difficult to describe complex relationships and variables over several dimensions. Ordinary FS serves as a valuable mathematical tool in such circumstances. Complex fuzzy sets have the ability to represent uncertainty in a more detailed fashion by adding many dimen- sions or membership attributes. This allows for a more thorough and effective analysis of circumstances with complex physical characteristics, as well as an individual’s ability to make informed decisions in complicated situations and challenging. In today’s society, the advancement of computer technology, the availability of high- speed processors, and the widespread use of programming languages have provided re- searchers with new opportunities to investigate and develop algorithms that specifically address intricate physical phenomena in a variety of scientific fields. The field of general operator theory presents a theoretical structure for understanding the mathematical prin- M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 3 of 22 ciples that serve as the foundation for numerous technical approaches utilized in a variety of fields. The complex intuitionistic fuzzy environment displays mathematical patterns that can be easily understood within the framework of general operator theory. By em- bracing this expansion, software applications that have the capacity to solve a broad range of problems and advance a number of academic disciplines can be developed. The vague and uncertain are essential parts of mankind’s existence. Accurate estimates or hypotheses are unattainable and significantly detrimental to human intelligence. Several mathemati- cal concepts, including fuzzy sets (FSs), have emerged as effective strategies to tackle this challenge. To address this uncertainty in the data, Zadeh [13] established the concept of a FS, and defined as µ → A : {(s, µ(s)); s ∈ A}, where µ(s) ∈ [0, 1] known as membership value. Rosenfeld [14] introduced the fuzzy subgroup (FSG) and investigated the algebraic characteristics of the service. Das [15] initiated the concept of “level subgroups” of a FSG. Meng [16] developed a fuzzy concept using a FS in a BCI-algebra. Specifically, certain concepts of Noether BCK/BCI-algebras employ fuzzy ideals. Atanassov [17] developed a new notion of IFS and described as A = {(s, µ(s), ν(s)) : s ∈ H}, where µ(s) and ν(s) are the membership degree (MD) and non-MD of an element, as well as 0 < µ(s)+ν(s) ≤ 1. Additionally, it demonstrated various attributes associated with relations and operations over sets, as well as the definition of topological operators and modal over the set of IFSs were discussed. Akram [18] proposed the notion of an intuitionistic fuzzy (IF) closed ideal of a BCI-algebra, applied the idea of IFS to closed ideals in BCI-algebras, and various attached characteristics were examined. Muhiuddin et al. [19] established the (α, β)-IF soft ideal of the BCK/BCI algebras, where α and β represent the membership values of an IF soft point, IFS and their related characteris- tics were examined. Senapati et al. [20] introduced the concepts of intuitionistic fuzzy translation to intuitionistic fuzzy sub-algebras and ideals in BCK/BCI-algebras. Also, the relationships between intuitionistic fuzzy translations and intuitionistic fuzzy exten- sions of intuitionistic fuzzy sub-algebras and ideals were investigated. Senapati et al. [21] investigated the cubic intuitionistic implicative ideals in BCK-algebras and the relation- ship between a cubic intuitionistic sub-algebra, a cubic intuitionistic ideal and a cubic intuitionistic implicative ideal were investigated. Furthermore, the conditions for a cubic intuitionistic ideal to be a cubic intuitionistic implicative ideal were described. A complex fuzzy set (CFS) is more efficient and flexible than FSs. Ramote et al. [22] presented the concept of complex fuzzy logic (CFL). CFL is a generalization of traditional fuzzy logic, based on CFSs. Ramote et al. [22] established the novel concept of CFSs, which was defined as {s, ν(s) = µ(s)eιθ(s) : s ∈ H}, where µ(s) → [0, 1] is the MD of the real part and θ(p) → [0, 2π] is a MD of the imaginary part of a complex number. The range of membership functions (MF) in a CFS is extended from a unit interval to the complex plane (CP) with the unit disc. The CFS helps equally in the process of evaluating the system because it considers magnitude term as well as phase term, where phase term presents the orientation of data item in complex unit disk plan. The primary set theoretic operations, like intersection, union, and complement, were discussed in detail under the influence of CFS. Jun and Xin [23] presented the principle of CFSs to BCK/BCI-algebras. In a BCK/BCI algebra, the concepts of complex left (right) reduced ideal and complex M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 4 of 22 subalgebra were presented, and their linked properties were discussed. Balamurugan et al. [24] established the notion of complex fuzzy sub-algebras (CFSA) in BCK/BCI-algebra and their characteristics were discussed. Also, numerous laws and operations of a complex fuzzy system, such as bounded differences, union, simple differences, intersection, and complement of complex fuzzy (CF) ideals within BCK/BCI-algebras were investigated. Alolaiyan et al. [25] proposed (α, β)-CFSs and subgroups, indicating that all complex fuzzy subgroups were (α, β)-CFSGs. Furthermore, (α, β)-CF cosets, (α, β)-CF normal subgroup, and (α, β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’s theorem of classical group theory were investigated. Zhang [1] established essential ideas about fuzzy complex numbers (FCNs) such as fuzzy distance and fuzzy limit. Furthermore, some fundamental properties of fuzzy limits, fuzzy complex numbers, and fuzzy distance were provided. Also, several essential theorems of FCNs, such as the nested closed rect- angles theorem, the accumulation principle, and Cauchy’s criterion for convergence were discussed. Alkouri and Salleh [? ] established a new notion of complex intutionistic fuzzy set (CIFS) defined as {s, µ(s) = γ(s)eιθ(s), ν(s) = γ(s)eιθ(s) : s ∈ H}, which is extended by adding a non-MD term to the basic notion of a CFS, where γ(s) → [0, 1] is the MD of the real part , γ(s) → [0, 1] is the non-MD of the real part , θ(s) → [0, 2π] is the MD of the imaginary part and θ(s) → [0, 2π] is a non-MD of the imaginary part of a complex number such that 0 < µ(s) + ν(s) = 1 and 0 < θ(s) + θ = 2π, for all complex numbers s ∈ H. The novelty of CIFS lies in its capabilities to achieve a wider range of values for both MF and non-MF. Gong and Wang [26] proposed a range of operation characteristic of CIFS were examined under the condition that both the non-membership phase and membership phase were limited to the interval [0, 2π]. Generally, membership and non-membership values have little practical significance, and there was no study of proximity and equality measures for CIFS. The distance measure (DM) was used to explain the (α, β)-equalities of CIFS. The (α, β)-equal describes two CIFS in which the difference between their non- membership degrees (MND) and membership degrees (MD) is less than β and 1 − α, respectively. Gulzar et al. [27] developed the notion of direct product between two complex IF subrings and the level sub-sets of the direct product of two complex IF subsets were defined. The complex intuitionistic fuzzy sub-algebra has a broader conceptual range compared to previously existing theories. The complex intuitionistic fuzzy sub-algebra is a generalization of the complex fuzzy sub-algebra, which does not deal with the degree of non-membership. Also, complex intuitionistic fuzzy sub-algebra is a generalization of intuitionistic fuzzy sub-algebra, which does not deal with phase terms. The complex intuitionistic fuzzy sub-algebra deals both degree of membership and degree of non-membership, as well as phase term and amplitude terms. Motivation and contribution for proposed concept: • Ramot et al. [22] initiated the concept of a CFS by extending the MF from real to complex numbers with the unit disc. Because the CFS only evaluated the MD rather than the non-MD element of data components, which also performs an equal M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 5 of 22 interest in the decision-making method for system evaluation, it provided weight to the MD. • Latif and Shuaib [28] developed the idea of t-IF conjugate element and discovered the t-IF conjugacy classes of t-IF sub-group. The idea of t-IF p sub-group, the t-IF Sylow p sub-group, and the t-intuitionistic fuzzification of Sylow’s Theorems were also explained. Gulzar et al. [29] proposed the t-IF centralizer and normalizer for t-IF subgroups. Additionally, the concept of t-IF cyclic and Abelian subgroups were introduced. • Salleh [30] proposed the notion of CF space and CF sub-groups. The idea of fuzzy space extended beyond the real range of MFs to the complex range of MFs, which was represented by the unit disc in the complex plane. Ali et al. [31] introduced the idea of (ϵ, δ)-CAFSs, which give a more thorough representation of the ambiguity of information than CAFSs by incorporating both the magnitude and phase of the MFs. Also, the (ϵ, δ)-complex anti-fuzzy subgroups (CAFSG) in the environment of CAFS were explained. Gulzar et al. [32] introduced the idea of complex IF subgroups and showed that each complex IF subgroup can be split into two IF subgroups. • Jawad et al. [33] investigated into group isomorphism under the influence of CIFS, a more general form of the CFS that included the non-MF degree. The complex algebraic structure offered useful tools for understanding complex techniques. • The idea of complex fuzzy ideals in a BCK/BCI-algebra was first put forwarded by Balamurugan et al. [24]. Our work is motivated by the reality that a CFS considers only the MD but does not weigh the non-MD of data elements. However, we apply the CIFS in BCK/BCI-algebra, which is the most vital component of algebraic structure. • The idea of complex IF ideals is not yet applied to the basic algebraic structure of BCK/BCI-algebra. In this proposed work, we discuss the various basic characteris- tics of BCK/BCI-algebra under the influence of CIFS. Table 1: List of abbreviations and symbols. Symbols Abbreviations M BCK/BCI-algebra IFI Intuitionistic Fuzzy Ideal IFS Intuitionistic Fuzzy Set IFSA Intuitionistic Fuzzy Sub-algebra CIFI Complex Intuitionistic Fuzzy Ideal CIFS Complex Intuitionistic Fuzzy Set CIFSA Complex Intuitionistic Fuzzy Sub-algebra M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 6 of 22 2. Preliminaries We start by analyzing the basic concepts of CFSA and CIFSA, both are necessary for study. Definition 1. [34] Let U be a non-empty set has an identity element denoted as 0 and a binary operation “⋆” then U is called BCI-algebra if the following axioms holds. (i) ((s ⋆ l) ⋆ (s ⋆ z)) ⋆ (z ⋆ l) = 0, ∀s, l, z ∈ M , (ii) (s ⋆ (s ⋆ l) ⋆ l) = 0, ∀s, l,∈ M , (iii) (s ⋆ s) = 0, ∀s ∈ M , (iv) (s ⋆ l = 0, l ⋆ s = 0 ⇒ s = l), ∀s, l,∈ M . Then, we describe M as a BCI-algebra. Moreover, a BCI-algebra M also fulfills: (v) (0 ⋆ s = 0), ∀s ∈ M , then M as a BCK-algebra. Definition 2. [17] A IFS L defined on M is given by: L = {(s, µL(s), νL(s)) : s ∈ M}, where µ(s) and ν(s) represent membership degree (MD) and non-MD of element of universe set M which belong to [0, 1] such that 0 < µ(s) + ν(s) ≤ 1, ∀s ∈ M . Definition 3. [35] An IFS L of M is known as an IFSA of M if it meets (i) µL(0) ≥ µL(s), (ii) νL(0) ≤ νL(s), (iii) µL(s ⋆ l) ≥ µL(s) ∧ µL(l), ∀s, l ∈ M , (iv) νL(s ⋆ l) ≤ νL(s) ∨ νL(l), ∀s, l ∈ M . Definition 4. [36] An IFS L of M is known as an IFI of M if it meets (i) µL(s) ≥ µL(s ⋆ l) ∧ µL(l), ∀s, l ∈ M , (ii) νL(s) ≤ νL(s ⋆ l) ∨ νL(l), ∀s, l ∈ M . Definition 5. [37] A CIFS, defined on M is described by a complex-valued grade of the non-membership function, membership function νL(s), µL(s) that assigns any element in L. The CIFS may be expressed by the set of ordered pairs L = {(s, µL(s), νL(s)) : s ∈ M} where µL(s) = γL(s)e ιθL(s), ι = √ −1, γL(s) ∈ [0, 1] and θL(s) ∈ [0, 2π]. νL(s) = γL(s)e ιθL(s), ι = √ −1, γL(s) ∈ [0, 1] and θL(s) ∈ [0, 2π]. Definition 6. [38] Let L = {(s, µL(s), νL(s)) : s ∈ M} and B = {(s, µB(s), νB(s)) : s ∈ M} be complex sub-sets of a non-void set M with membership functions µL(s) = γL(s)e ιθL(s), µB(s) = γB(s)e ιθB(s) respectively and non-membership functions νL(s) = γL(s)e ιθL(s), νB(s) = γB(s)e ιθB(s) respectively. By µL(s) ≤ µB(s), this implies that γL(s) ≤ γB(s), θL(s) ≤ θB(s) and νL(s) ≤ νB(s), we mean that γL(s) ≤ γB(s), θL(s) ≤ θB(s). M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 7 of 22 3. Complex Intuitionistic Fuzzy Sub-algebras(CIFSAs) of BCK/BCI-Algebras In this part, we explore fundamental notions related to CIFSs, including CIFS on the universal set M and modal operators are defined. Definition 7. Let L be an IFS of M . Then, modal operators and level operators (i), (ii), (iii), (iv) are defined by (i) ⊕L = {(s, µL(s) 2 ), νL(s)2 ) : ∀s ∈ M}, (ii) ⊗L = {(s, µL(s)+1 2 , νL(s)+1 2 ) : ∀s ∈ M}, (iii) †L = {(s, 12 ∨ µL(s), 1 2 ∧ νL(s)) : ∀s ∈ M}, (iv) ‡L = {(s, 12 ∧ µL(s), 1 2 ∨ νL(s)) : ∀s ∈ M}. Definition 8. A CIFS L = (s, µL(s), νL(s)) is considered a CIFSA of M if s, l ∈ M , and it satisfies following: (i) µL(0)e ιθL(0) ≥ µL(s)e ιθL(s), (ii) µL(s ⋆ l)e ιθL(s⋆l) ≥ µL(s)e ιθL(s) ∧ µL(l)e ιθL(l), (iii) νL(0)e ιθL(0) ≤ νL(s)e ιθL(s), (iv) νL(s ⋆ l)e ιθL(s⋆l) ≤ νL(s)e ιθL(s) ∨ νL(l)e ιθL(l). Example 1. Take a BCK algebra M = {0, s, l, z}, where the binary operation is defined by the Caley Table 2. Now explain a CIFS on M as: L = {(0, 0.8eι0.5π, 0.6eι0.25π), (s, 0.8eι0.5π, 0.6eι0.25π), (l, 0.5eι0.2π, 0.3eι0.1π), (z, 0.8eι0.5π, 0.6eι0.25π)}. It is simple to demonstrate that L is a CIFS of M . Table 2: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z s 0 0 0 0 0 s s s 0 0 s s l l l 0 l s z z z z 0 s p q r s r s Example 2. Take a BCK-algebra M = {0, s, l, z, w}, where the binary operation is defined by the Caley Table 3. Now explain a CIFS on M as: L = {(0, 0.9eι0.6π, 0.7eι0.4π), (s, 0.7eι0.5π, 0.5eι0.3π), (l, 0.4eι0.3π, 0.2eι0.1π), (z, 0.4eι0.3π, 0.2eι0.1π), (w, 0.4eι0.3π, 0.2eι0.1π)}. It is simple to demonstrate that L is a CIFS of M . M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 8 of 22 Table 3: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z w 0 0 0 w z l s s 0 w z l l l l 0 l z z z z l 0 w w w w z l 0 The following result indicates membership degree of identity element of CIFS is greater than all other elements. Also, the non-membership degree of identity element is less than the non-membership of remaining elements of CIFS. Theorem 1. If L is a CIFS of M , then µL(0) ≥ µL(s) and νL(0) ≤ νL(s). Proof. Let L be a CIFS of M . Then µL(0) = γL(0)e ιθL(0) = γL(s ⋆ s)eιθL(s⋆s) ≥ (γL(s) ∧ γL(s))e ι(θL(s)∧θL(s)) = γL(s)e ιθL(s) ≥ µL(s). And νL(0) = γL(0)e ιθL(0) = γL(s ⋆ s)eιθL(s⋆s) ≤ (γL(s) ∨ γL(s))e ι(θL(s)∨θL(s)) = γL(s)e ιθL(s) ≤ νL(s). Thus, we conclude the required result. Definition 9. Let L be a CIFS of M . Then modal operator ⊕L is defined as µ⊕L(s) = γL(s) 2 eι( θL(s) 2 ), ν⊕L(s) = γL(s) 2 eι( θL(s) 2 ). Example 3. Let (µL(s), νL(s)) = {(0, 0.4eι0.5π, 0.2eι0.3π), (s, 0.8eι0.1π, 0.6eι0.01π), (l, 0.6eι0.3π, 0.4eι0.1π)} be a CIFS of M . Then (µ⊕L(s), ν⊕L(s)) = {(0, 0.2eι0.25π, 0.1eι0.15π), (s, 0.4eι0.05π, 0.3eι0.005π), (l, 0.3eι0.15π, 0.2eι0.05π)} is a CIFS of M . The following result shows that the modal operator ⊕ of CIFI of M is also CIFI. Theorem 2. If L is a CIFS of M . Then ⊕L is a CIFS L of M . Proof. For each s ∈ M , we have µ⊕L(0) = γL(0) 2 eι( θL(0) 2 ) ≥ γL(s) 2 eι( θL(s) 2 ) = µ⊕L(s). Let s, l ∈ M , then µ⊕L(s⋆l) = γL(s⋆l) 2 eι( θL(s⋆l) 2 ) ≥ (γL(s)2 ∧ γL(l) 2 )eι( θL(s) 2 ∧ θL(l) 2 ) = µ⊕L(s)∧µ⊕L(l). For each s ∈ M , we have ν⊕L(0) = γL(0) 2 eι( θL(0) 2 ) ≤ γL(s) 2 eι( θL(s) 2 ) = ν⊕L(s). Suppose that s, l ∈ M , Then ν⊕L(s⋆l) = γL(s⋆l) 2 eι( θL(s⋆l) 2 ) ≤ (γL(s) 2 ∨ γL(l) 2 )eι( θL(s) 2 ∨ θL(l) 2 ) = ν⊕L(s)∨ν⊕L(l). This concludes the proof. Definition 10. Let L be a CIFS of M . Then modal operator ⊗L is describe as µ⊗L(s) = γL(s)+1 2 eι( θL(s)+1 2 ), ν⊗L(s) = γL(s)+1 2 eι( θL(s)+1 2 ). Example 4. Let (µL(s), νL(s)) = {(s, 0.3eι0.5π, 0.1eι0.3π), (l, 0.7eι0.2π, 0.5eι0.1π), (z, 0.5eι0.4π, 0.3eι0.2π)} be a CIFS of M . Then (µ⊗L(s), ν⊗L(s)) = {(s, 0.65eι0.75π, 0.55eι0.65π), (l, 0.85eι0.6π, 0.75eι0.55π), (z, 0.75eι0.7π, 0.65eι0.6π)} is a CIFS of M . M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 9 of 22 The subsequent result demonstrates that the modal operator ⊗ of CIFI of M is also CIFI. Theorem 3. Suppose that L is a CIFS of M . Then ⊗L is a CIFS L of M . Proof. For each s ∈ M , we have µ⊗L(0) = γL(0)+1 2 eι( θL(0)+1 2 ) ≥ γL(s)+1 2 eι( θL(s)+1 2 ) = µ⊕L(s). Let s, l ∈ M , then µ⊗L(s⋆l) = γL(s⋆l)+1 2 eι( θL(s⋆l)+1 2 ) ≥ (γL(s)+1 2 ∧γL(l)+1 2 )eι( θL(s)+1 2 ∧ θL(l)+1 2 ) = µ⊗L(s)∧µ⊗L(l). For each s ∈ M , we have ν⊗L(0) = γL(0)+1 2 eι( θL(0)+1 2 ) ≤ γL(s)+1 2 eι( θL(s)+1 2 ) = ν⊕L(s). Suppose that s, l ∈ M , then ν⊗L(s⋆ l) = γL(s⋆l)+1 2 eι( θL(s⋆l)+1 2 ) ≤ (γL(s)+1 2 ∨ γL(l)+1 2 ) eι( θL(s)+1 2 ∨ θL(l)+1 2 ) = ν⊗L(s) ∨ ν⊗L(l). Therefore, ⊗L is a CIFS L of M . Definition 11. Suppose that L is a CIFS of M . Then level operator †L is describe as µ†L(s) = (12 ∨ γL(s))e ι( 1 2 ∨θL(s)), ν†L(s) = (12 ∧ γL(s))e ι( 1 2 ∧θL(s)). Example 5. Let (µL(s), νL(s)) = {(s, 0.4eι0.6π, 0.2eι0.4π), (l, 0.6eι0.2π, 0.4eι0.1π), (z, 0.5eι0.7π, 0.3eι0.5π)} be a CIFS of M . Then µ†L(s), ν†L(s)) = {(s, 0.5eι0.6π, 0.2eι0.4π), (l, 0.6eι0.5π, 0.4eι0.1π), (z, 0.5eι0.6π, 0.3eι0.5π)} is a CIFS of M . The following outcome shows that the level operator † of the CIFI of M is also CIFI. Theorem 4. Suppose that L is a CIFS of M . Then †L is a CIFS L of M . Proof. For each s ∈ M , we have µ†L(0) = (12∨γL(0))e ι( 1 2 ∨θL(0)) ≥ (12∨γL(s))e ι( 1 2 ∨θL(s)) = µ†L(s). Let s, l ∈ M , then µ†L(s ⋆ l) = (12 ∨ γL(s ⋆ l))e ι( 1 2 ∨θL(s⋆l)) ≥ 1 2 ∨ (γL(s) ∧ γL(l)) eι( 1 2 ∨(γL(s)∧γL(l))) = (12 ∨ (γL(s))e ι( 1 2 ∨(γL(s)))) ∧ (12 ∨ (γL(l))e ι( 1 2 ∨(γL(l)))) = µ†L(s) ∧ µ†L(l). For each s ∈ M , we have ν†L(0) = (12 ∧ γL(0))e ι( 1 2 ∧θL(0)) ≤ (12 ∧ γL(s))e ι( 1 2 ∧θL(s)) = ν†L(s). Suppose that s, l ∈ M , then ν†L(s ⋆ l) = (12 ∧ γL(s ⋆ l))eι( 1 2 ∧θL(s⋆l)) ≤ 1 2 ∧ (γL(s) ∨ γL(l))e ι( 1 2 ∧(γL(s)∨γL(l))) = (12 ∧ (γL(s))e ι( 1 2 ∧(γL(s)))) ∨ (12 ∧ (γL(l))e ι( 1 2 ∧(γL(l)))) = ν†L(s) ∨ ν†L(l). Therefore, †L is a CIFS L of M . Definition 12. Let L be a CIFS of M . Then level operator ‡L is describe as µ‡L(s) = (12 ∧ γL(s))e ι( 1 2 ∧θL(s)), ν‡L(s) = (12 ∨ γL(s))e ι( 1 2 ∨θL(s)). Example 6. Let (µL(s), νL(s)) = {(s, 0.4eι0.6π, 0.2eι0.4π), (l, 0.6eι0.2π, 0.4eι0.1π), (z, 0.5eι0.7π, 0.3eι0.5π)} be a CIFS of M . Then (µ‡L(s), ν‡L(s)) = {(s, 0.4eι0.5π, 0.5eι0.5π), (l, 0.5eι0.2π, 0.5eι0.5π), (z, 0.5eι0.5π, 0.5eι0.5π)} is a CIFS of M . The following result shows that the level operator ‡ of the CIFI of M is also CIFI. Theorem 5. Suppose that L is a CIFS of M . Then ‡L is a CIFS L of M . Proof. For each s ∈ M , we have µ‡L(0) = (12∧γL(0))e ι( 1 2 ∧θL(0)) ≥ (12∧γL(s))e ι( 1 2 ∧θL(s)) = µ‡L(s). Let s, l ∈ M , then µ‡L(s ⋆ l) = (12 ∧ γL(s ⋆ l))e ι( 1 2 ∧θL(s⋆l)) ≥ 1 2 ∧ (γL(s) ∧ γL(l)) M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 10 of 22 eι( 1 2 ∧(γL(s)∧γL(l))) = (12 ∧ (γL(s))e ι( 1 2 ∧(γL(s)))) ∧ (12 ∧ (γL(l))e ι( 1 2 ∧(γL(l)))) = µ‡L(s) ∧ µ‡L(l). For each s ∈ M , we have ν‡L(0) = (12 ∨ γL(0))e ι( 1 2 ∨θL(0)) ≤ (12 ∨ γL(s))e ι( 1 2 ∨θL(s)) = ν‡L(s). Suppose that s, l ∈ M , then ν‡L(s ⋆ l) = (12 ∨ γL(s ⋆ l))eι( 1 2 ∨θL(s⋆l)) ≤ 1 2 ∨ (γL(s) ∨ γL(l))e ι( 1 2 ∨(γL(s)∨γL(l))) = (12 ∨ (γL(s))e ι( 1 2 ∨(γL(s)))) ∨ (12 ∨ (γL(l))e ι( 1 2 ∨(γL(l)))) = ν‡L(s) ∨ ν‡L(l). Therefore, ‡L is a CIFS L of M . 4. Complex Intuitionistic Fuzzy Ideals (CIFIs) of BCK/BCI-Algebras In the following part, we will examine fundamental concepts about the CIFIs, and CIFI over the universal set M . Definition 13. A CIFS L = (s, µL(s), νL(s)) is considered a CIFSA of M , where s, l ∈ M , and the following hold: µL(s)e ιθL(s) ≥ µL(s ⋆ l)eιθL(s⋆l) ∧ µL(l)e ιθL(l), ∀s, l ∈ M , νL(s)e ιθL(s) ≤ νL(s ⋆ l)e ιθL(s⋆l) ∨ νL(l)e ιθL(l), ∀s, l ∈ M . Example 7. Take a BCK-algebra M = {0, s, l, z}, where the binary operation is defined by the Caley Table 4. Now describe a CIFS on M as: L = {(0, 0.67eι0.5π, 0.47eι0.25π), (s, 0.34eι0.43π, 0.14eι0.23π), (l, 0.67eι0.05π, 0.47eι0.025π), (z, 0.34eι0.43π, 0.14eι0.23π), (w, 0.34eι0.43π, 0.14eι0.23π)}. It is straightforward to prove that L is a CIFI of M . Table 4: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z w 0 0 0 0 0 0 s s 0 s 0 0 l l l 0 0 0 z z z z 0 0 w w z w s 0 The next theorem shows that every CIFI of M is also order preserving. Theorem 6. Every CIFI of M is order-preserving. Proof. Assume that L is a CIFSA of M and assume that s, l ∈ M are such that s ≤ l. Then µL(s) = γL(s)e ιθL(s) ≥ γL(s ⋆ l)e ιθL(s⋆l) ∧ γL(l)e ιθL(l) = (γL(s ⋆ l) ∧ γL(l))e ι(θL(s⋆l)∧θL(l)) = (γL(0) ∧ γL(0))e ι(θL(0)∧θL(0)) = γL(l)e ιθL(l) ≥ µL(l). And νL(s) = γL(s)e ιθL(s) M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 11 of 22 ≤ γL(s ⋆ l)e ιθL(s⋆l) ∨ γL(l)e ιθL(l) = (γL(s ⋆ l) ∨ γL(l))e ι(θL(s⋆l)∨θL(l)) = (γL(0) ∨ γL(0))e ι(θL(0)∨θL(0)) = γL(l)e ιθL(l) ≤ νL(l). This concludes the Proof. The next theorem shows that every CIFI of the set is equal to a CIFSA. Theorem 7. Every CIFI of M is a CIFSA of M . Proof. Since s ⋆ l ≤ s, it follows from Property 2 that µL(s ⋆ l) ≥ µL(s) and νL(s ⋆ l) ≤ νL(s). Hence by Definition, µL(s ⋆ l) ≥ γL(s)e ιθL(s) = (γL(s ⋆ l)e ιθL(s⋆l)) ∧ γL(l)e ιθL(l) = (γL(s ⋆ l) ∧ γL(l))e ι(θL(s⋆l)∧θL(l)) = (γL(s) ∧ γL(l))e ι(θL(s)∧θL(l)) ≥ µL(s) ∧ µL(l). Moreover νL(s ⋆ l) ≤ γL(s)e ιθL(s) = (γL(s ⋆ l)e ιθL(s⋆l)) ∨ γL(l)e ιθL(l) = (γL(s ⋆ l) ∨ γL(l))e ι(θL(s⋆l)∨θL(l)) = (γL(s) ∨ γL(l))e ι(θL(s)∨θL(l)) ≤ νL(s) ∨ νL(l). So L is a CIFS L of M . The following result shows that if s ⋆ l ≤ z then µL(s) ≥ µL(l) ∧ µL(z) and νL(s) ≤ νL(l) ∨ νL(z). Theorem 8. Let L be a CIFI of M . If the inequality s ⋆ l ≤ z holds in M , then µL(s) ≥ µL(l) ∧ µL(z) and νL(s) ≤ νL(l) ∨ νL(z). Proof. Suppose that L is a CFSA of M and let s ⋆ l ≤ z holds in M . Then µL(s ⋆ l) = γL(s ⋆ l)e ιθL(s⋆l) ≥ (γL((s ⋆ l) ⋆ z) ∧ γL(z))e ι(θL((s⋆l)⋆z)∧θL(z)) = (γL(0) ∧ γL(z))e ι(θL(0)∧θL(z)) = γL(z)e ιθL(z) ≥ µL(z). It follows that µL(s) ≥ µL(l) ∧ µL(z). Moreover νL(s ⋆ l) = γL(s ⋆ l)e ιθL(s⋆l) M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 12 of 22 ≤ (γL((s ⋆ l) ⋆ z) ∨ γL(z))e ι(θL((s⋆l)⋆z)∨θL(z)) = (γL(0) ∨ γL(z))e ι(θL(0)∨θL(z)) = γL(z)e ιθL(z) ≤ νL(z). It follows that νL(s) ≤ νL(l) ∨ νL(z). Definition 14. Let L be a CIFS of M . Then, the complement L is described as: C(µL(s)) = (1− γL(s))e ι(2π−θL(s)), C(νL(s)) = (1− γL(s))e ι(2π−θL(s)). Example 8. Let (µL(s), νL(s)) = {(s1, 0.3eι0.4π, 0.1eι0.24π), (s2, 0.6eι0.2π, 0.4eι0.12π), (s3, 0.8eι0.1π, 0.6eι0.01π), (s4, 0.2e ι0.3π, 0.1eι0.13π), (s5, 0.5e ιπ, 0.3eι0.8π), (s6, 0.9e ι0.1π, 0.7eι0.01π)} be a CIFS of M . Then C(µL(s), νL(s)) = {(s1, 0.7eι1.6π, 0.9eι1.76π), (s2, 0.4eι1.8π, 0.6eι1.88π), (s3, 0.2eι1.9π, 0.4eι1.99π), (s4, 0.8e ι1.7π, 0.9eι1.87π), (s5, 0.5e ιπ, 0.7eι1.2π), (s6, 0.1e ι1.9π, 0.3eι1.99π)} is a CIFI of M . The following outcome shows that the complement of CIFI of M is also CIFI. Theorem 9. A CIFS of M is a CIFI of M iff C(µL(s)) and C(νL(s)) is a CIFI of M . Proof. Suppose that L is a CIFSA of M and let s, l ∈ M . Then C(µL(0)) = 1− µL(0) = (1− γL(0))e ι(2π−θL(0)) ≥ (1− γL(s))e ι(2π−θL(s)) = C(µL(s)). And C(µL(s)) = 1− µL(s) ≥ 1− (γL(s ⋆ l)e ι(2π−θL(s⋆l)) ∧ γL(l)e ι(2π−θL(l))) = (1− γL(s ⋆ l))e ι(2π−θL(s⋆l)) ∧ (1− γL(l))e ι(2π−θL(l)) ≥ C(µL(s ⋆ l)) ∧ C(µL(l)). Suppose that L is a CIFSA of M and let s, l ∈ M . Then C(νL(0)) = 1− νL(0) = (1− γL(0))e ι(2π−θL(0)) ≤ (1− γL(s))e ι(2π−θL(s)) = C(νL(s)). And C(νL(s)) = 1− νL(s) ≤ 1− (γL(s ⋆ l)e ι(2π−θL(s⋆l)) ∨ γL(l)e ι(2π−θL(l))) = (1− γL(s ⋆ l))e ι(2π−θL(s⋆l)) ∨ (1− γL(l))e ι(2π−θL(l)) ≤ C(νL(s ⋆ l)) ∨ C(νL(l)). Thus, the complement of membership and non-membership L is a CIFI of M . M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 13 of 22 Definition 15. Suppose that L1 and L2 are two CIFSs of M . Then, the union L1 ∪ L2 is defined as µL1∪L2(s) = (γL1(s)∨γL2(s))e ι(θL1 (s)∨θL2 (s)), νL1∪L2(s) = (γL1 (s)∧γL2 (s))eι(θL1 (s)∧θL2 (s)). Example 9. Let (µL1(s), νL1(s)) = {(s1, 0.6eι0.5π, 0.4eι0.25π), (s2, 1eι0.5π, 0.8eι0.25π), (s3, 0.8eι2π, 0.6eι1.5π), (s4, 0.9e ι0.4π, 0.7eι0.24π), (s5, 0.7e ιπ, 0.5eι0.8π), (s6, 0.5e ι0.4π, 0.3eι0.24π)} and (µL2(s), νL2(s)) = {(s1, 0.2eιπ, 0.1eι0.88π), (s2, 0.1eι0.8π, 0.01eι0.6π), (s3, 0.8eι0.8π, 0.6eι0.68π), (s4, 0.2eιπ, 0.1eι0.88π), (s5, 0.9e ι0.9π, 0.7eι0.7π), (s6, 0.3e ι2π, 0.1eι1.88π)} be a CIFSs. Then, (µL1∪L2(s), νL1∪L2(s)) = {(s1, 0.6eιπ, 0.1eι0.25π), (s2, 1eι0.8π, 0.01eι0.25π), (s3, 0.8eι2π, 0.6eι0.68π), (s4, 0.9eιπ, 0.1eι0.24π), (s5, 0.9e ιπ, 0.5eι0.7π), (s6, 0.5e ι2π, 0.1eι0.24π)}. The subsequent theorem demonstrates that the union ∪ of two CIFIs of M is also CIFI. Theorem 10. Assume that L1 and L2 are two CIFIs of M . Then, L1 ∪ L2 is a CIFI of M . Proof. Let L1 and L2 be two CIFIs of M and let s, l ∈ M . Then µL1∪L2(0) = (γL1(0) ∨ γL2(0))e ι(θL1 (0)∨θL2 (0)) ≥ (γL1(s) ∨ γL2(s))e ι(θL1 (s)∨θL2 (s)) = µL1∪L2(s). Moreover µL1∪L2(s) = (γL1(s) ∨ γL2(s))e ι(θL1 (s)∨θL2 (s)) ≥ ((γL1(s ⋆ l) ∧ γL1(l)) ∨ (γL2(s ⋆ l) ∧ γL2(l))) eι((θL1 (s⋆l)∧θL1 (l))∨(θL2 (s⋆l)∧θL2 (l))) ≥ ((γL1(s ⋆ l) ∨ γL2(s ⋆ l)) ∧ (γL1(l) ∨ γL2(l))) eι((θL1 (s⋆l)∨θL2 (s⋆l))∧(θL1 (l)∨θL2 (l))) = (γL1(s ⋆ l) ∨ γL2(s ⋆ l))e ι(θL1 (s⋆l)∨θL2 (s⋆l)) ∧ (γL1(l) ∨ γL2(l))e ι(θL1 (l)∨θL2 (l)) ≥ µL1∪L2(s ⋆ l) ∧ µL1∪L2(l). Suppose that L1 and L2 are two CIFIs of M and let s, l ∈ M . Then νL1∪L2(0) = (γL1 (0) ∧ γL2 (0))eι(θL1 (0)∧θL2 (0)) ≤ (γL1 (s) ∧ γL2 (s))eι(θL1 (s)∧θL2 (s)) = νL1∪L2(s). Moreover νL1∪L2(s) = (γL1 (s) ∧ γL2 (s))eι(θL1 (s)∧θL2 (s)) ≤ ((γL1 (s ⋆ l) ∨ γL1 (l)) ∧ (γL2 (s ⋆ l) ∨ γL2 (l))) M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 14 of 22 eι((θL1 (s⋆l)∨θL1 (l))∧(θL2 (s⋆l)∨θL2 (l))) ≤ ((γL1 (s ⋆ l) ∧ γL2 (s ⋆ l)) ∨ (γL1 (l) ∧ γL2 (l))) eι((θL1 (s⋆l)∧θL2 (s⋆l))∨(θL1 (l)∧θL2 (l))) = (γL1 (s ⋆ l) ∧ γL2 (s ⋆ l))eι(θL1 (s⋆l)∧θL2 (s⋆l)) ∨ (γL1 (l) ∧ γL2 (l))eι(θL1 (l)∧θL2 (l)) ≤ νL1∪L2(s ⋆ l) ∨ νL1∪L2(l). Therefore, L1 ∪ L2 is a CIFI of M . Example 10. Take a BCK-algebra M = {0, s, l, z, w}, where the binary operation is de- fined by the Caley Table 5. Now define a CIFS L1 on M as: L1 = {(0, 0.9eι0.7π, 0.7eι0.5π), (s, 0.7eι0.5π, 0.5eι0.3π), (l, 0.5eι0.3π, 0.3eι0.1π), (z, 0.3eι0.1π, 0.1eι0.01π), (w, 0.3eι0.1π, 0.1eι0.01π)}. It is easy to show that L1 is a CIFI of M . Now define a CIFS L2 on M as: L2 = {(0, 0.6eι0.5π, 0.4eι0.3π), (s, 0.4eι0.6π, 0.2eι0.4π), (l, 0.6eι0.5π, 0.4eι0.3π), (z, 0.4eι0.6π, 0.2eι0.4π), (w, 0.4eι0.6π, 0.2eι0.4π)}. It is easy to show that L2 is a CIFI of M . Now define a CIFS L1∪L2 on M as: L1∪L2 = {(0, 0.9eι0.7π, 0.4eι0.3π), (s, 0.7eι0.6π, 0.2eι0.3π), (l, 0.6eι0.5π, 0.3eι0.1π), (z, 0.4eι0.6π, 0.1eι0.01π), (w, 0.4eι0.6π, 0.1eι0.01π)}. It is straightforward to prove that L1 ∪L2 is a CIFI of M . Table 5: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z w 0 0 0 0 0 0 s s 0 s 0 0 l l l 0 0 0 z z z z 0 0 w w z w z 0 Definition 16. Assume that L1 and L2 are two CIFSs of M . Then, the intersection L1 ∩ L2 is defined as µL1∩L2(s) = (γL1(s)∧γL2(s))e ι(θL1 (s)∧θL2 (s)), νL1∩L2(s) = (γL1 (s)∨γL2 (s))eι(θL1 (s)∨θL2 (s)). Example 11. Let (µL1(s), νL1(s)) = {(s1, 0.6eι0.5π, 0.4eι0.25π), (s2, 1eι0.5π, 0.8eι0.25π), (s3, 0.8eι2π, 0.6eι1.5π), (s4, 0.9e ι0.4π, 0.7eι0.24π), (s5, 0.7e ιπ, 0.5eι0.8π), (s6, 0.5e ι0.4π, 0.3eι0.24π)} and (µL2(s), νL2(s)) = {(s1, 0.2eιπ, 0.1eι0.88π), (s2, 0.1eι0.8π, 0.01eι0.6π), (s3, 0.8eι0.8π, 0.6eι0.68π), (s4, 0.2eιπ, 0.1eι0.88π), (s5, 0.9e ι0.9π, 0.7eι0.7π), (s6, 0.3e ι2π, 0.1eι1.88π)} be a CIFSs. Then, µL1∩L2(s) = {(s1, 0.2eι0.5π, 0.4eι0.88π), (s2, 0.1eι0.5π, 0.8eι0.6π), (s3, 0.8eι0.8π, 0.6eι1.5π), (s4, 0.2eι0.4π, 0.7eι0.88π), (s5, 0.7e ι0.9π, 0.7eι0.8π), (s6, 0.3e ι0.4π, 0.3eι1.88π)}. The subsequent theorem demonstrates that the intersection ∩ of two CIFIs of M is also a CIFI. Theorem 11. Assume that L1 and L2 are two CIFIs of M . Then L1 ∩ L2 is a CIFI of M . M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 15 of 22 Proof. Let L1 and L2 be two CIFIs of M and let s, l ∈ M . Then µL1∩L2(0) = (γL1(0) ∧ γL2(0))e ι(θL1 (0)∧θL2 (0)) ≥ (γL1(s) ∧ γL2(s))e ι(θL1 (s)∧θL2 (s)) = µL1∩L2(s). Moreover µL1∩L2(s) = (γL1(s) ∧ γL2(s))e ι(θL1 (s)∧θL2 (s)) ≥ ((γL1(s ⋆ l) ∧ γL1(l)) ∧ (γL2(s ⋆ l) ∧ γL2(l))) eι((θL1 (s⋆l)∧θL1 (l))∧(θL2 (s⋆l)∧θL2 (l))) ≥ ((γL1(s ⋆ l) ∧ γL2(s ⋆ l)) ∧ (γL1(l) ∧ γL2(l))) eι((θL1 (s⋆l)∧θL2 (s⋆l))∧(θL1 (l)∧θL2 (l))) = (γL1(s ⋆ l) ∧ γL2(s ⋆ l))e ι(θL1 (s⋆l)∧θL2 (s⋆l)) ∧ (γL1(l) ∧ γL2(l))e ι(θL1 (l)∧θL2 (l)) ≥ µL1∩L2(s ⋆ l) ∧ µL1∩L2(l). Suppose that L1 and L2 are two CIFIs of M and let s, l ∈ M . Then νL1∩L2(0) = (γL1 (0) ∨ γL2 (0))eι(θL1 (0)∨θL2 (0)) ≤ (γL1 (s) ∨ γL2 (s))eι(θL1 (s)∨θL2 (s)) = νL1∩L2(s). And νL1∩L2(s) = (γL1 (s) ∨ γL2 (s))eι(θL1 (s)∨θL2 (s)) ≤ ((γL1 (s ⋆ l) ∨ γL1 (l)) ∨ (γL2 (s ⋆ l) ∨ γL2 (l))) eι((θL1 (s⋆l)∨θL1 (l))∨(θL2 (s⋆l)∨θL2 (l))) ≤ ((γL1 (s ⋆ l) ∨ γL2 (s ⋆ l)) ∨ (γL1 (l) ∨ γL2 (l))) eι((θL1 (s⋆l)∨θL2 (s⋆l))∨(θL1 (l)∨θL2 (l))) = (γL1 (s ⋆ l) ∨ γL2 (s ⋆ l))eι(θL1 (s⋆l)∨θL2 (s⋆l)) ∨ (γL1 (l) ∨ γL2 (l))eι(θL1 (l)∨θL2 (l)) ≤ νL1∩L2(s ⋆ l) ∨ νL1∩L2(l). Therefore, L1 ∩ L2 is a CIFI of M . Example 12. Take a BCK-algebra M = {0, s, l, z, w}, where the binary operation is de- fined by the Caley Table 6. Now, define a CIFS L1 on M as: L1 = {(0, 0.9eι0.7π, 0.7eι0.5π), (s, 0.7eι0.5π, 0.5eι0.3π), (l, 0.5eι0.3π, 0.3eι0.1π), (z, 0.3eι0.1π, 0.1eι0.01π), (w, 0.3eι0.1π, 0.1eι0.01π)}. It is easy to show that L1 is a CIFI of M . Now, define a CIFS L2 on M as: L2 = M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 16 of 22 {(0, 0.6eι0.5π, 0.4eι0.3π), (s, 0.4eι0.6π, 0.2eι0.4π), (l, 0.6eι0.5π, 0.4eι0.3π), (z, 0.4eι0.6π, 0.2eι0.4π), (w, 0.4eι0.6π, 0.2eι0.4π)}. It is easy to show that L2 is a CIFI of M . Now, define a CIFS L1∩L2 on M as: L1∩L2 = {(0, 0.6eι0.5π, 0.7eι0.5π), (s, 0.4eι0.5π, 0.5eι0.4π), (l, 0.5eι0.3π, 0.4eι0.3π), (z, 0.3eι0.1π, 0.2eι0.4π), (w, 0.3eι0.1π, 0.2eι0.4π)}. It is straightforward to prove that L1 ∩L2 is a CIFI of M . Table 6: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z w 0 0 0 0 0 0 s s 0 s 0 0 l l l 0 0 0 z z z z 0 0 w w w z l 0 Definition 17. Let L1 and L2 be two CIFSs of M . Then, the simple difference L1\L2 is defined as µL1\L2 (s) = (γL1(s)∧γL2(s))e ι(θL1 (s)∧θL2 (s)), νL1\L2 (s) = (γL1 (s)∨γL2 (s))eι(θL1 (s)∨θL2 (s)). Example 13. Let (µL1(s), νL1(s)) = {(s1, 0.6eι0.5π, 0.4eι0.25π), (s2, 1eι0.5π, 0.8eι0.25π), (s3, 0.8eι2π, 0.6eι1.5π), (s4, 0.9e ι0.4π, 0.7eι0.24π), (s5, 0.7e ιπ, 0.5eι0.8π), (s6, 0.5e ι0.4π, 0.3eι0.24π)} and (µL2(s), νL2(s)) = {(s1, 0.2eιπ, 0.1eι0.88π), (s2, 0.1eι0.8π, 0.01eι0.6π), (s3, 0.8eι0.8π, 0.6eι0.68π), (s4, 0.2eιπ, 0.1eι0.88π), (s5, 0.9e ι0.9π, 0.7eι0.7π), (s6, 0.3e ι2π, 0.1eι1.88π)} be a CIFSs. Then, µL1\L2 (s) = {(s1, 0.2eι0.5π, 0.4eι0.88π), (s2, 0.1eι0.5π, 0.8eι0.6π), (s3, 0.8eι0.8π, 0.6eι1.5π), (s4, 0.2eι0.4π, 0.7eι0.88π), (s5, 0.7e ι0.9π, 0.7eι0.8π), (s6, 0.3e ι0.4π, 0.3eι1.88π)}. It is straightforward to prove that L1\L2 is a CIFI of M . The subsequent theorem demonstrates that the simple difference \ of two CIFIs of M is also CIFI. Theorem 12. Assume that L1 and L2 are two CIFIs of M . Then, L1\L2 is a CIFI of M . Proof. Let L1 and L2 be two CIFIs of M and let s, l ∈ M . Then µL1\L2 (0) = (γL1(0) ∧ γL2(0))e ι(θL1 (0)∧θL2 (0)) ≥ (γL1(s) ∧ γL2(s))e ι(θL1 (s)∧θL2 (s)) = µL1\L2 (s). Moreover µL1\L2 (s) = (γL1(s) ∧ γL2(s))e ι(θL1 (s)∧θL2 (s)) ≥ ((γL1(s ⋆ l) ∧ γL1(l)) ∧ (γL2(s ⋆ l) ∧ γL2(l))) eι((θL1 (s⋆l)∧θL1 (l))∧(θL2 (s⋆l)∧θL2 (l))) M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 17 of 22 ≥ ((γL1(s ⋆ l) ∧ γL2(s ⋆ l)) ∧ (γL1(l) ∧ γL2(l))) eι((θL1 (s⋆l)∧θL2 (s⋆l))∧(θL1 (l)∧θL2 (l))) = (γL1(s ⋆ l) ∧ γL2(s ⋆ l))e ι(θL1 (s⋆l)∧θL2 (s⋆l)) ∧ (γL1(l) ∧ γL2(l))e ι(θL1 (l)∧θL2 (l)) ≥ µL1\L2 (s ⋆ l) ∧ µL1\L2 (l). Suppose that L1 and L2 are two CIFIs of M and let s, l ∈ M . Then νL1\L2 (0) = (γL1 (0) ∨ γL2 (0))eι(θL1 (0)∨θL2 (0)) ≤ (γL1 (s) ∨ γL2 (s))eι(θL1 (s)∨θL2 (s)) = νL1\L2 (s). And νL1\L2 (s) = (γL1 (s) ∨ γL2 (s))eι(θL1 (s)∨θL2 (s)) ≤ ((γL1 (s ⋆ l) ∨ γL1 (l)) ∨ (γL2 (s ⋆ l) ∨ γL2 (l))) eι((θL1 (s⋆l)∨θL1 (l))∨(θL2 (s⋆l)∨θL2 (l))) ≤ ((γL1 (s ⋆ l) ∨ γL2 (s ⋆ l)) ∨ (γL1 (l) ∨ γL2 (l))) eι((θL1 (s⋆l)∨θL2 (s⋆l))∨(θL1 (l)∨θL2 (l))) = (γL1 (s ⋆ l) ∨ γL2 (s ⋆ l))eι(θL1 (s⋆l)∨θL2 (s⋆l)) ∨ (γL1 (l) ∨ γL2 (l))eι(θL1 (l)∨θL2 (l)) ≤ νL1\L2 (s ⋆ l) ∨ νL1\L2 (l). Therefore, L1\L2 is a CIFI of M . Example 14. Take a BCK-algebra M = {0, s, l, z, w}, where the binary operation is de- fined by the Caley Table 7. Now, define a CIFS L1 on M as: L1 = {(0, 0.9eι0.7π, 0.7eι0.5π), (s, 0.7eι0.5π, 0.5eι0.3π), (l, 0.5eι0.3π, 0.3eι0.1π), (z, 0.3eι0.1π, 0.1eι0.01π), (w, 0.3eι0.1π, 0.1eι0.01π)}. It is easy to show that L1 is a CIFI of M . Now define a CIFS L2 on M as: L2 = {(0, 0.6eι0.5π, 0.4eι0.3π), (s, 0.4eι0.6π, 0.2eι0.4π), (l, 0.6eι0.5π, 0.4eι0.3π), (z, 0.4eι0.6π, 0.2eι0.4π), (w, 0.4eι0.6π, 0.2eι0.4π)}. It is easy to show that L2 is a CIFI of M . Now define a CIFS L1\L2 on M as: L1\L2 = {(0, 0.6eι0.5π, 0.7eι0.5π), (s, 0.4eι0.5π, 0.5eι0.4π), (l, 0.5eι0.3π, 0.4eι0.3π), (z, 0.3eι0.1π, 0.2eι0.4π), (w, 0.3eι0.1π, 0.2eι0.4π)}. It is straightforward to prove that L1\L2 is a CIFI of M . M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 18 of 22 Table 7: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z w 0 0 0 0 0 0 s s 0 s 0 0 l l l 0 0 0 z z z z 0 0 w w w z l 0 Definition 18. Let L1 and L2 be two CIFSs of M . Then, the bounded difference L1⊖L2 is defined as µL1⊖L2(s) = (0∨(γL1(s)−γL2(s)))e ι(θL1 (s)∨θL2 (s)), νL1⊖L2(s) = (0∧(γL1 (s)− γL2 (s)))eι(θL1 (s)∧θL2 (s)). Example 15. Let (µL1(s), νL1(s)) = {(s1, 0.6eι0.5π, 0.4eι0.25π), (s2, 1eι0.5π, 0.8eι0.25π), (s3, 0.8eι2π, 0.6eι1.5π), (s4, 0.9e ι0.4π, 0.7eι0.24π), (s5, 0.7e ιπ, 0.5eι0.8π), (s6, 0.5e ι0.4π, 0.3eι0.24π)} and (µL2(s), νL2(s)) = {(s1, 0.2eιπ, 0.1eι0.88π), (s2, 0.1eι0.8π, 0.01eι0.6π), (s3, 0.8eι0.8π, 0.6eι0.68π), (s4, 0.2eιπ, 0.1eι0.88π), (s5, 0.5e ι0.9π, 0.3eι0.7π), (s6, 0.3e ι2π, 0.1eι1.88π)} be a CIFSs. Then, µL1⊖L2(s) = {(s1, 0.4eιπ, 0.3eι0.25π), (s2, 0.9eι0.8π, 0.79eι0.25π), (s3, 0eι2π, 0eι0.68π), (s4, 0.7eιπ, 0.6eι0.24π), (s5, 0.5e ιπ, 0.2eι0.7π), (s6, 0.2e ι2π, 0.2eι0.24π)}. The following outcome shows that the bounded difference ⊖ of two CIFIs of M is also CIFI. Theorem 13. Assume that L1 and L2 are two CIFIs of M . Then L1 ⊖ L2 is a CIFI of M . Proof. Let L1 and L2 be two CIFIs of M and let s, l ∈ M . Then µL1⊖L2(0) = (0 ∨ (γL1(0)− γL2(0)))e ι(θL1 (0)∨θL2 (0)) ≥ (0 ∨ (γL1(s)− γL2(s)))e ι(θL1 (s)∨θL2 (s)) = µL1⊖L2(s). Moreover µL1⊖L2(s) = (0 ∨ (γL1(s)− γL2(s)))e ι(θL1 (s)∨θL2 (s)) ≥ (0 ∨ (γL1(s ⋆ l)− γL1(l))) ∨ (γL2(s ⋆ l)− γL2(l))) eι((θL1 (s⋆l)∧θL1 (l))∨(θL2 (s⋆l)∧θL2 (l))) = ((0 ∨ (γL1(s ⋆ l)− γL2(s ⋆ l))) ∧ (0 ∨ γL1(l)− γL2(l))) eι((θL1 (s⋆l)∨θL2 (s⋆l))∧(θL1 (l)∨θL2 (l))) = (0 ∨ (γL1(s ⋆ l)− γL2(s ⋆ l)))e ι(θL1 (s⋆l)∨θL2 (s⋆l)) ∧ (0 ∨ (γL1(l)− γL2(l)))e ι(θL1 (l)∨θL2 (l)) ≥ µL1⊖L2(s ⋆ l) ∧ µL1⊖L2(l). M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 19 of 22 Suppose that L1 and L2 are two CIFIs of M and let s, l ∈ M . Then νL1⊖L2(0) = (0 ∧ (γL1 (0)− γL2 (0)))eι(θL1 (0)∧θL2 (0)) ≤ (0 ∧ (γL1 (s)− γL2 (s)))eι(θL1 (s)∧θL2 (s)) = νL1⊖L2(s). And νL1⊖L2(s) = (0 ∧ (γL1 (s)− γL2 (s)))eι(θL1 (s)∧θL2 (s)) ≤ (0 ∧ (γL1 (s ⋆ l)− γL1 (l))) ∨ (γL2 (s ⋆ l)− γL2 (l))) eι((θL1 (s⋆l)∧θL1 (l))∧(θL2 (s⋆l)∧θL2 (l))) = ((0 ∧ (γL1 (s ⋆ l)− γL2 (s ⋆ l))) ∨ (0 ∧ γL1 (l)− γL2 (l))) eι((θL1 (s⋆l)∧θL2 (s⋆l))∧(θL1 (l)∧θL2 (l))) = (0 ∧ (γL1 (s ⋆ l)− γL2 (s ⋆ l)))eι(θL1 (s⋆l)∧θL2 (s⋆l)) ∨ (0 ∧ (γL1 (l)− γL2 (l)))eι(θL1 (l)∧θL2 (l)) ≤ νL1⊖L2(s ⋆ l) ∨ νL1⊖L2(l). Therefore, L1 ⊖ L2 is a CFI of M . Example 16. Take a BCK-algebra M = {0, s, l, z, w} with Table 8. Now, define a CIFS L1 on M as: L1 = {(0, 0.9eι0.7π, 0.7eι0.5π), (s, 0.7eι0.5π, 0.5eι0.3π), (l, 0.7eι0.3π, 0.5eι0.1π), (z, 0.5eι0.1π, 0.4eι0.01π), (w, 0.7eι0.1π, 0.5eι0.01π)}. It is easy to show that L1 is a CIFI of M . Now, define a CIFS L2 on M as: L2 = {(0, 0.6eι0.5π, 0.4eι0.3π), (s, 0.4eι0.6π, 0.2eι0.4π), (l, 0.6eι0.5π, 0.4eι0.3π), (z, 0.4eι0.6π, 0.2eι0.4π), (w, 0.4eι0.6π, 0.2eι0.4π)}. It is easy to show that L2 is a CIFI of M . Now define a CIFS L1⊖L2 on M as: L1⊖L2 = {(0, 0.3eι0.7π, 0.3eι0.3π), (s, 0.3eι0.6π, 0.3eι0.3π), (l, 0.1eι0.5π, 0.1eι0.1π), (z, 0.1eι0.6π, 0.2eι0.01π), (w, 0.3eι0.6π, 0.3eι0.01π)}. It is straightforward to prove that L1 ⊖ L2 is a CIFI of M . Table 8: Cayley’s table describing the binary operation expressed by “⋆”. ⋆ 0 s l z w 0 0 0 0 0 0 s s 0 s 0 0 l l l 0 0 0 z z z z 0 0 w w z w z 0 Conclusion We have used CIFSs in the context of BCK/BCI-algebras in this paper; also, the CIFI have been defined, and its features have been investigated. This adds a lot to the field of M. Jawad et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6798 20 of 22 classical fuzzy set theory. In CIFS, both non-membership and membership functions with complex degrees have been used to improve the algebraic structure and decision-making processes within BCK/BCI-algebras. In BCK/BCI-algebras, level operators and model operators have been used to explain what a CIFSA is, and then its basic properties have been looked at. We have also studied various operations, including complement, inter- section, union, and differences. We plan to investigate the more complex features of the BCK/BCI-algebras under the influence of CIFS. Furthermore, we want to apply complex spherical fuzzy set and complex linear Diophantine fuzzy set on BCK/BCI-algebras. In the future, we focus on characterising CIFSAs under algebraic homomorphisms and isomor- phisms. This will provide a better understanding of how complex intuitionistic fuzziness interacts with structure preservation between algebras. Also, complex intuitionistic fuzzy sub-algebra provide richer modelling tools for uncertainty; future research will explore their applications in multi-criteria decision-making, artificial intelligence, and information systems. 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