EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3291-3303 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Lie Homomorphisms of Complex Intuitionistic Fuzzy Lie Algebras Shadi M. Shaqaqha1,∗, Mounther Y. Al-Deiakeh1 1 Department of Mathematics, Yarmouk University, Shafiq Irshidat Street, Irbid 21163, Jordan Abstract. In this paper, we investigate the properties of complex intuitionistic fuzzy Lie subal- gebras and ideals under Lie algebra homomorphisms, focusing on both images and inverse images. We provide detailed proofs for several new results concerning the preservation of complex intu- itionistic fuzzy structures through homomorphisms, and we introduce additional homomorphism- related properties for these structures. This work extends known results to the context of complex intuitionistic fuzzy Lie algebras, contributing new insights into their behavior under algebraic mappings. 2020 Mathematics Subject Classifications: 17B99, 08A72, 03E72 Key Words and Phrases: Lie ideal, Lie algebra homomorphism, fuzzy set, fuzzy Lie subalgebra, intuitionistic fuzzy set, intuitionistic fuzzy Lie subalgebra, homogeneous complex intuitionistic fuzzy Lie subalgebra, complex intuitionistic fuzzy Lie ideal 1. Introduction Introduced by Lotfi Zadeh in 1965, fuzzy sets provide a mathematical framework for representing and managing imprecise and uncertain information [22]. Unlike classical crisp sets with binary membership values, fuzzy sets allow for gradual degrees of membership, offering a more flexible approach to dealing with uncertainty. In 1986, Atanassov extended the concept of fuzzy sets by introducing intuitionistic fuzzy sets [6], which account for both membership and non-membership degrees. This concept garnered significant attention, such as in [10, 11], leading to the evolution of intuitionistic fuzzy set theory, which has found utility in diverse domains such as decision- making, control systems, and pattern recognition. Building on these ideas, complex intuitionistic fuzzy sets (CIFS) were introduced by Alkouri and Salleh in 2012 [5]. CIFS extend intuitionistic fuzzy sets by using complex numbers to represent membership and non-membership values, offering a more expressive ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5385 Email address: shadi.s@yu.edu.jo (S. Shaqaqha), mountheraldeiakeh@gmail.com (M. Al-Deiakeh) https://www.ejpam.com 3291 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3292 way to handle uncertainty and ambiguity. This framework has proven useful in decision- making, pattern recognition, and image processing, where complex and conflicting infor- mation needs to be systematically managed. Related work on fuzzy Lie algebras can be found in [3], and studies on bipolar fuzzy soft Lie algebras in [2]. These foundational studies have contributed significantly to the extension of fuzzy algebraic structures and motivate further research in this area. This work aligns with recent research in the field, such as the study of intuitionistic fuzzy ordered subalgebras in ordered BCI-algebras by Roh et al. [12]. In a recent work [18], we introduced the notion of a complex intuitionistic fuzzy Lie algebra, which characterizes a Lie algebra using elements represented as complex intuitionistic fuzzy sets. The Lie bracket operation is formulated based on the principles of complex intuitionistic fuzzy logic. These complex intuitionistic fuzzy Lie algebras can be perceived as a broader generalization encompassing fuzzy Lie algebras [21], intuitionistic fuzzy Lie algebras [1], and complex fuzzy Lie algebras [13]. 2. Some preliminaries on Lie Algebras and Complex Intuitionistic Fuzzy Lie Ideals 2.1. Fundamentals of Lie Algebras In this section, we delve into the core concepts surrounding Lie algebras, drawing insights from the sources [7] and [9]. We explore the foundational notion of Lie algebras, denoted as (M, [., .]), whereM signifies a vector space over the fieldK, and [., .] : M×M → M represents a bilinear mapping (called Lie product). The Lie product adheres to two key axioms for any elements m1,m2,m3 ∈ M: (i) The condition [m1, m2] = 0M holds, leading to the implication of anti-symmetry: [m1, m2] = −[m2, m1]. (ii) The Jacobi identity, denoted as [m1, [m2,m3]] + [m2, [m3,m1]] + [m3, [m1, m2]] = 0M, is satisfied. It’s clear that [m, 0M] = [0M, m] = 0M for any m ∈ M. Additionally, when Char(K) ̸= 2, the identity [m1, m2] = −[m2, m1] for all m1, m2 ∈ M implies [m, m] = 0M for each m ∈ M. Various illustrative examples highlight diverse instances of Lie algebras: (i) Any vector space V can be viewed as an abelian Lie algebra using the Lie bracket [v1, v2] = 0V for all elements v1, v2 ∈ V . (ii) When a vector space V is endowed with an associative multiplication, it can be transformed into a Lie algebra with the commutator operation [v1, v2] = v1v2−v2v1 S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3293 for all v1, v2 ∈ V . In the realm of finite-dimensional vector spaces, the set of linear transformations from V to itself, denoted gl(V ), constitutes a Lie algebra, with the bracket defined as [f1, f2] = f1 ◦ f2 − f2 ◦ f1 for any f1, f2 ∈ gl(V ). (iii) The Lie algebra structure is evident in the vector space R3 when the cross product operation is employed as [a, b] = a× b. These examples showcase the wide-ranging applications and properties of Lie algebras. In the context of Lie algebras, a subspace Q of M is referred to as a Lie subalgebra if it remains closed under the Lie product operation, implying that [q1, q2] ∈ Q for any q1, q2 ∈ Q. Also, Q is termed a Lie ideal of M if [m, n] ∈ N for any m ∈ M and n ∈ N . Notably, the subspaces 0M and M are considered ideals of M, termed the trivial ideals. Additionally, the center of M, denoted Z(M), consists of elements m in M for which [m, n] = 0M for all n ∈ M, making it a Lie ideal of M. Furthermore, if Q and S are ideals of M, then Q + S = {q + s : q ∈ Q and s ∈ S}, [Q,S] = Span{[q, s] : q ∈ Q and s ∈ S}, and Q ∩ S = {m : m ∈ Q and m ∈ S} are also considered ideals of M. Considering two Lie algebras M1 and M2 over K, a linear transformation T : M1 → M2 is considered a Lie algebra homomorphism if it satisfies T ([m,n]) = [T (m), T (n)] for all m, n ∈ M1. Furthermore, if T is both a Lie algebra homomorphism and a bijection (one-to-one and onto), it’s termed a Lie algebra isomorphism. An illustrative example showcases the adjoint representation of a Lie algebra. For m ∈ M, the function adm : M → M; n 7→ [m,n] is defined, and the set adM = {adm : m ∈ M} is shown to be a Lie subalgebra of gl(M). The function ad : M → adM;m 7→ adm is a Lie algebra homomorphism and is known as the adjoint representation of M. Moreover, for a Lie algebra homomorphism f : M1 → M2, the image of f is charac- terized as im(f) = {f(q) : q ∈ M1}, while the kernel of f is denoted as ker(f) = {s ∈ M1 : f(s) = 0M2}. As a consequence, im(f) emerges as a Lie subalgebra within M2, and ker(f) establishes itself as an ideal within M1. 2.2. Fundamentals of Complex Intuitionistic Fuzzy Lie Ideals A complex intuitionistic fuzzy set defined on a non-empty set X can be denoted as E = (ϕE , ψE) = {(x, ϕE(x), ψE(x)) : x ∈ X}. In this representation, ϕE(x) and ψE(x) are complex numbers situated within the unit circle, ensuring that their absolute values satisfy the condition | ϕE(x) | + | ψE(x) |≤ 1. In this context, i = √ −1, and the expressions for ϕE(x) and ψE(x) take the form ρE(x)e iζE(x) and ρ̂E(x)e iζ̂E(x) respectively. Here, ρE(x) and ρ̂A(x) are real numbers ranging from 0 to 1, while ζM(x) and ζ̂M(x) are real numbers within the interval of [0, 2π]. The notion of complex intuitionistic fuzzy sets (CIFS) can be seen as an expansion of intuitionistic fuzzy sets. When both ζE(x) and ζ̂E(x) are set to 0, it reverts to the classical S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3294 fuzzy set. Furthermore, when ψE(x) = (1−ρE(x))ei(2π−ζE(x)), it produces a complex fuzzy set. A homogeneous complex intuitionistic fuzzy set is defined as a CIFS that fulfills two conditions for any x and y belonging to the set X: (i) ρE(x) ≤ ρE(y) if and only if ζE(x) ≤ ζE(y), and (ii) ρ̂E(x) ≤ ρ̂E(y) if and only if ζ̂E(x) ≤ ζ̂E(y). In the article, it is assumed that all complex intuitionistic fuzzy sets are of this homogeneous type. Also, for z1 = ρ1e iζ1 and z2ρ2e iζ2 (ρ1, ρ2 ∈ [0, 1] and ζ1, ζ2 ∈ [0, 2π]) are two complex numbers, we say z1 ≤ z2 if and only if ρ1 ≤ ρ2 and ζ1 ≤ ζ2. Definition 1. ([18]) A complex intuitionistic fuzzy set E = (ϕE , ψA) defined on a Lie algebra M is categorized as a complex intuitionistic fuzzy Lie subalgebra when it meets three conditions for all m1,m2 ∈ M, and k ∈ K: (i) ϕE(m1 +m2) ≥ ϕE(m1) ∧ ϕE(m2) and ψM(m1 +m2) ≤ ψE(m1) ∨ ψE(m2), (ii) ϕE(km1) ≥ ϕE(x) and ψE(km1) ≤ ψE(m1), and (iii) ϕE([m1,m2]) ≥ ϕE(m1) ∧ ϕE(m2) and ψE([m1,m2]) ≤ ψE(m1) ∨ ψE(m2). If condition (iii) is substituted with ϕE([m1,m2]) ≥ ϕE(m1)∨ϕE(m2), and ψE([m1,m2]) ≤ ψE(m1) ∧ ψE(m2), then E is denoted as a complex intuitionistic fuzzy Lie ideal within the context of M. 3. Mappings and Inverse Mappings of Complex Intuitionistic Fuzzy Lie Subalgebras (Ideals) Through Lie Morphisms. Consider Lie algebras M1 and M2, a complex intuitionistic subset E = (ϕE , ψE) of M1, and a function f : M1 → M2. The complex intuitionistic fuzzy subset f(E) of f(M1) is defined as f(E) = (ϕf(E), ψf(E)). Here (m2 ∈ f(M1)), ϕf(E)(m2) = supm1∈f−1({m2}){ϕE(m1)}, and ψf(E)(m2) = infm1∈f−1({m2}){ψE(m1)}. This subset is termed as the mapping of E through f . In a similar manner, for a complex intuitionistic fuzzy subset P = (ϕP , ψP) of M2, the inverse mapping of P under f , f−1(P), is defined as (ϕf−1(P), ψf−1(P)), where (m1 ∈ M1) ϕf−1(P)(m1) = ϕP(f(m1)) and ψf−1(P)(m1) = ψP(f(m1)). Theorem 1. Assume f : M1 → M2 is a homomorphism between Lie algebras. If P = (ϕP , ψP) is a complex intuitionistic fuzzy Lie subalgebra of M2, then the complex intuitionistic fuzzy set f−1(P) is a complex intuitionistic fuzzy Lie subalgebra of M1. S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3295 Proof. To show that f−1(P) is a complex intuitionistic fuzzy Lie subalgebra in M1, we need to verify that it satisfies the properties of a complex intuitionistic fuzzy set and maintains the closure properties under Lie operations within M1. We start by checking the homogeneity property. For any m1 ∈ M1, by definition of the inverse image under the homomorphism f , we have: ϕf−1(P)(m1) = ϕP(f(m1)) = ρP(f(m1))e iζP (f(m1)), and similarly, ψf−1(P)(m1) = ψP(f(m1)) = ρ̂P(f(m1))e iζ̂P (f(m1)). Now, consider two elements m1,m2 ∈ M1. If ρP(f(m1)) ≤ ρP(f(m2)), then by the homo- geneity property of P, we know that ζP(f(m1)) ≤ ζP(f(m2)). Similarly, if ρ̂P(f(m1)) ≤ ρ̂P(f(m2)), then ζ̂P(f(m1)) ≤ ζ̂P(f(m2)). Therefore, the homogeneity property holds for f−1(P) in M1. Next, we verify the algebraic properties under addition, scalar multiplication, and the Lie bracket. For m1,m2 ∈ M1 and k ∈ K, we need to show the following: 1. **Addition**: ϕf−1(P)(m1 +m2) = ϕP(f(m1 +m2)) = ϕP(f(m1) + f(m2)), since f is linear. By Definition 1 (the property of complex intuitionistic fuzzy Lie subal- gebras), we have: ϕP(f(m1) + f(m2)) ≥ ϕP(f(m1)) ∧ ϕP(f(m2)). Thus, ϕf−1(P)(m1 +m2) ≥ ϕf−1(P)(m1) ∧ ϕf−1(P)(m2). For ψ, we proceed similarly: ψf−1(P)(m1 +m2) = ψP(f(m1 +m2)) = ψP(f(m1) + f(m2)), and by Definition 1: ψP(f(m1) + f(m2)) ≤ ψP(f(m1)) ∨ ψP(f(m2)). Hence, ψf−1(P)(m1 +m2) ≤ ψf−1(P)(m1) ∨ ψf−1(P)(m2). 2. **Scalar Multiplication**: For any k ∈ K, we have: ϕf−1(P)(km1) = ϕP(f(km1)) = ϕP(kf(m1)), and by Definition 1: ϕP(kf(m1)) ≥ ϕP(f(m1)). Thus, ϕf−1(P)(km1) ≥ ϕf−1(P)(m1). S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3296 Similarly, for ψ: ψf−1(P)(km1) = ψP(f(km1)) = ψP(kf(m1)), and by Definition 1: ψP(kf(m1)) ≤ ψP(f(m1)), so ψf−1(P)(km1) ≤ ψf−1(P)(m1). 3. **Lie Bracket**: Finally, for the Lie bracket, we have: ϕf−1(P)([m1,m2]) = ϕP(f([m1,m2])) = ϕP([f(m1), f(m2)]), and by Definition 1: ϕP([f(m1), f(m2)]) ≥ ϕP(f(m1)) ∧ ϕP(f(m2)), so ϕf−1(P)([m1,m2]) ≥ ϕf−1(P)(m1) ∧ ϕf−1(P)(m2). Similarly, for ψ: ψf−1(P)([m1,m2]) = ψP(f([m1,m2])) = ψP([f(m1), f(m2)]), and by Definition 1: ψP([f(m1), f(m2)]) ≤ ψP(f(m1)) ∨ ψP(f(m2)), so ψf−1(P)([m1,m2]) ≤ ψf−1(P)(m1) ∨ ψf−1(P)(m2). Thus, f−1(P) satisfies the necessary properties and constitutes a complex intuitionistic fuzzy Lie subalgebra within M1. Corollary 1. Assume f : M1 → M2 is a homomorphism between Lie algebras. If P = (ϕP , ψP) is a complex intuitionistic fuzzy Lie ideal within M2, then the complex intuitionistic fuzzy set f−1(P) is also a complex intuitionistic fuzzy Lie ideal in M1. Proof. We will follow the same general approach used in the proof of Theorem 1. The only key difference lies in the third requirement of Definition 1, which pertains to the behavior under the Lie bracket, as the structure involved is now a fuzzy Lie ideal rather than a fuzzy subalgebra. We begin by considering two elements m1,m2 ∈ M1. Our goal is to show that the inverse image f−1(P) satisfies the condition for being a complex intuitionistic fuzzy Lie ideal. For the membership function ϕf−1(P) under the Lie bracket, we have: ϕf−1(P)([m1,m2]) = ϕP(f([m1,m2])) = ϕP([f(m1), f(m2)]), S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3297 where the second equality follows from the fact that f is a homomorphism. Since P is a fuzzy Lie ideal in M2, by Definition 1, we know: ϕP([f(m1), f(m2)]) ≥ ϕP(f(m1)) ∨ ϕP(f(m2)). Thus, we obtain: ϕf−1(P)([m1,m2]) ≥ ϕf−1(P)(m1) ∨ ϕf−1(P)(m2). This demonstrates that ϕf−1(P) satisfies the required condition under the Lie bracket. For the non-membership function ψf−1(P), we similarly have: ψf−1(P)([m1,m2]) = ψP(f([m1,m2])) = ψP([f(m1), f(m2)]). Since P is a fuzzy Lie ideal, it follows that: ψP([f(m1), f(m2)]) ≤ ψP(f(m1)) ∧ ψP(f(m2)). Therefore, we have: ψf−1(P)([m1,m2]) ≤ ψf−1(P)(m1) ∧ ψf−1(P)(m2). Thus, the set f−1(P) satisfies the conditions for being a complex intuitionistic fuzzy Lie ideal in M1, completing the proof. Lemma 1. ([1]) If f : M1 → M2 represents a Lie algebra homomorphism, and E = (ϕE , ψE) constitutes an intuitionistic fuzzy Lie subalgebra within M1, then the intuitionistic fuzzy set f(E) transforms into an intuitionistic fuzzy Lie subalgebra over the domain im(f). Lemma 2. ([14]) If E = (ϕE , ψE) represents a complex intuitionistic fuzzy set of a Lie algebra M, then E qualifies as a complex intuitionistic fuzzy Lie ideal (or subalgebra) of M if and only if the associated intuitionistic fuzzy subset E = {(m, ρE(m), ρ̂E(m)) : m ∈ M} emerges as an intuitionistic fuzzy Lie ideal (or subalgebra) of M. Theorem 2. Assume f : M1 → M2 is a homomorphism between Lie algebras. If E = (ϕE , ψE) constitutes a complex intuitionistic fuzzy Lie subalgebra within M1, then the complex intuitionistic fuzzy set f(E) forms a complex intuitionistic fuzzy Lie subalgebra in the context of the image of f . Proof. Initially, we demonstrate the homogeneity of f(E) as follows: ϕf(E)(n) = sup n=f(m) {ϕE(m)} = sup n=f(m) {ρE(m)eiζE(m)} S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3298 = sup n=f(m) {ρE(m)}ei(supn=f(m){ζE(m)}) (since E is homogeneous). In a similar manner, we can derive ψf(E)(n) = inf n=f(m) {ρ̂E(m)}ei(infn=f(m){ζ̂E(m)}). Next, we analyze the case where n1 and n2 belong to the image of f , denoted as im(f), and where supn1=f(m){ρE(m)} ≤ supn2=f(m){ρE(m)}. Suppose, for the sake of contradiction, that supn2=f(m){ζE(m)} < supn1=f(m){ζE(m)}. This implies the existence of m1 ∈ M1 such that f(m1) = n1 and supn2=φ(m){ζE(m)} < ζE(m1). Consider the case where f(m) = n2. This leads to ζE(m) < ζE(m1), and due to the homogeneity of E , we deduce that ρE(m) < ρE(m1). Consequently, we find supn2=f(m){ρE(m)} < ρE(m1). This directly contradicts our assumption supn1=f(m){ρE(m)} ≤ supn2=f(m){ρE(m)}. By employing a similar reasoning, we can establish that if infn1=f(m){ρ̂E(m)} ≤ infn2=f(m){ρ̂E(m)}, then infn1=f(m){ζ̂E(m)} ≤ infn2=f(m){ζ̂E(m)}. As a result of the above analysis, we conclude that f(E) exhibits homogeneity within im(f). Given that E is a complex intuitionistic fuzzy Lie subalgebra, we can deduce from Lemma 2 that E = {(m, ρE(m), ρ̂E(m)) : m ∈ M1} constitutes an intuitionistic fuzzy Lie subalgebra. According to Lemma 1, when considering the transformation of an intuition- istic fuzzy Lie subalgebra, it maintains its nature as an intuitionistic fuzzy Lie subalgebra. Consequently, the transformed set f(E) = {(n, ρEf(E) (n), ρ̂Ef(E) (n)) : n ∈ im(f)} represents an intuitionistic fuzzy Lie subalgebra within the domain of im(f). For any elements n1 and n2 belonging to im(f), as well as for scalar k ∈ K, we can ascertain the following assertions concerning Ef(E): (i) ρEf(E) (n1 +n2) ≥ ρEf(E) (n1)∧ ρEf(E) (n2) and ρ̂Ef(E) (n1 +n2) ≤ ρ̂Ef(E) (n1)∨ ρ̂Ef(E) (n2), (ii) ρEf(E) (kn1) ≥ ρEf(E) (n1) and ρ̂Ef(E) (kn1) ≤ ρ̂Ef(E) (n1), (iii) ρEf(E) ([n1, n2]) ≥ ρEf(E) (n1)∧ρEf(E) (n2) and ρ̂Ef(E) ([n1, n2]) ≤ ρ̂Ef(E) (n1)∨ ρ̂Ef(E) (n2). Given that f(E) is endowed with homogeneity, we can thereby conclude that f(E) qualifies as a complex intuitionistic fuzzy Lie subalgebra situated within the realm of im(f). Here, we present the proof of the subsequent outcome, originally established in [8] for Lie superalgebras, adapted to the context of Lie algebra homomorphisms. This outcome demonstrates that the transformation of an intuitionistic fuzzy Lie ideal through a Lie algebra homomorphism maintains its nature as an intuitionistic fuzzy Lie ideal as well. Corollary 2. Consider a Lie algebra homomorphism f : M1 → M2. If E = (ϕE , ψE) constitutes an intuitionistic fuzzy Lie ideal within M1, then the intuitionistic fuzzy set f(E) preserves its character as an intuitionistic fuzzy Lie ideal even within the domain im(f). S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3299 Proof. It suffices to demonstrate that for any n1 and n2 in im(f), the conditions ϕf(E)([n1, n2]) ≥ ϕf(E)(n1) ∨ ϕf(E)(n2) and ψf(E)([n1, n2]) ≤ ψf(E)(n1) ∧ ψf(E)(n2) hold. Let n1 and n2 be elements of im(f). Suppose, for the sake of contradiction, that ϕψ(E)([n1, n2]) < ϕf(E)(n1)∨ϕf(E)(n2). Let t be chosen from the interval [0, 1] such that ϕf(E)([n1, n2]) < t < ϕf(E)(n1) ∨ ϕf(E)(n2). Without any loss of generality, we can assume ϕf(E)(n1) ≥ ϕf(E)(n2). Consequently, it follows that ϕf(E)([n1, n2]) < t < supn1=f(m){ϕE(m)}. Hence, there exists an q within M1 such that f(q) = n1 and ϕf(E)([n1, n2]) < t < ϕE(q). For any s in M1 with f(s) = n2, it holds that f([q, s]) = [f(q), f(s)] = [n1, n2]. As a result, we find that ϕf(E)([n1, n2]) = sup [n1, n2]=f([m1, m2]) ϕE([m1, m2]) ≥ ϕE([q, s]) ≥ ϕE(q) ∨ ϕE(s) > t > ϕf(E)([n1, n2]), leading to a contradiction. Additionally, if ψf(E)([n1, n2]) > ψf(E)(n1)∧ψf(E)(n2), then a value r within the inter- val [0, 1] can be selected such that ψf(E)([n1, n2]) > r > ψf(E)(n1) ∧ ψf(E)(n2). Without loss of generality, let’s assume ψf(E)(n1) ≤ ψf(E)(n2). This leads to ψf(E)([n1, n2]) > r > infn1=f(m) ψE(m), which enables the identification of an q in M1 such that f(q) = n1 and ψf(E)([n1, n2]) > r > ψE(q). By choosing s ∈ M1 with f(s) = n2, it becomes evident that f([q, s]) = [f(q), f(s)] = [n1, n2]. Consequently, ψf(E)([n1, n2]) = inf [n1, n2]=f([m1, m2])ψE([m1, m2]) ≤ ψE([q, s]) ≤ ψE(q) ∧ ψE(s) < r < ψf(E)([n1, n2]), leading to a contradiction. Therefore, it can be concluded that f(E) indeed constitutes an intuitionistic fuzzy Lie ideal within im(f). In Corollary 2, we establish that if f : M1 → M2 serves as a Lie algebra homomor- phism, and E = (ϕE , ψE) constitutes an intuitionistic fuzzy Lie ideal within M1, then the intuitionistic fuzzy set f(E) transforms into an intuitionistic fuzzy Lie ideal within im(f). This result, combined with the insights from Theorem 2, allows us to expand this understanding into the realm of complex intuitionistic fuzzy Lie algebras. Corollary 3. Let f : M1 → M2 represent a Lie algebra homomorphism. If E = (ϕE , ψE) denotes a complex intuitionistic fuzzy Lie ideal of M1, then the complex intuitionistic fuzzy set f(E) evolves into a complex intuitionistic fuzzy Lie ideal within the domain im(f). S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3300 4. More Homomorphism Properties of Complex Intuitionistic Fuzzy Lie Algebras Let E = (ϕE , ψE) and P = (ϕP , ψP) be two complex intuitionistic fuzzy sets on a Lie algebra M over a field K. The sum of E and P, which was defined by Chen and Zhang [8] in the case of intuitionistic fuzzy Lie superalgebras, is defined as follows: E + P = (ϕE+P , ψE+P), where ϕE+P(m) = sup m1+m2 {ϕE(m1) ∧ ϕP(m2)}, and ψE+P(m) = inf m=m1+m2 {ψE(m1) ∨ ψP(m2)}. Let E = (ϕE , ψE) and Q = (ϕQ, ψQ) be two complex intuitionistic fuzzy sets on the same setM , where ϕE = ρEe iζE , ψE = ρ̂Ee iζ̂E , ϕQ = ρQe iζQ , and ψQ = ρ̂Qe iζ̂Q . We say that E is homogeneous with Q if the following hold for all m, n ∈M : (i) ρE(m) ≤ ρQ(n) if and only if ζE(m) ≤ ζQ(n) (in this case ϕE(m) ≤ ϕQ(n)), (ii) ρ̂E(m) ≤ ρ̂Q(n) if and only if ζ̂E(m) ≤ ζ̂Q(n) (in this case ψE(m) ≤ ψQ(n)). Theorem 3. ([14]) Let E = (ϕE , ψE) and P = (ϕP , ψP) be two complex intuitionistic fuzzy Lie ideals on M such that E is homogeneous with P and E + P is homogeneous. Then E + P is a complex intuitionistic fuzzy Lie ideal of M. Consider a surjective Lie algebra homomorphism f : M1 → M2. Suppose we have two complex intuitionistic fuzzy Lie ideals, denoted as E and P, on M1. Assume that E is homogeneous with respect to P, and the sum E + P is also homogeneous. In accordance with Corollary 3, we can conclude that f(E+P) constitutes a complex intuitionistic fuzzy Lie ideal of the image of f , denoted as im(f). The subsequent theorem establishes the relationship between the sets f(E + P), f(E), and f(P). Theorem 4. Consider a surjective Lie algebra homomorphism f : M1 → M2. Let E = (ϕE , ψE) and P = (ϕP , ψP) be complex intuitionistic fuzzy Lie ideals on M1 such that E is homogeneous with respect to P. Then, f(E + P) = f(E) + f(P). Proof. Let n ∈ M2. Then ϕf(E+P)(n) = sup n=f(m) {ϕE+P(m)} = sup n=f(m) { sup m=m1+m2 {ϕE(m1) ∧ ϕP(m2)}} = sup n=q+s { sup q=f(m1) {ϕE(m1)} ∧ sup s=f(m2) {ϕP(m2)}} =ϕf(E)+f(P)(n). S. Shaqaqha, M. Y. Al-Deiakeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3291-3303 3301 In addition, ψf(E+P)(n) = inf n=f(m) {ψE+P(m)} = inf n=f(m) { inf m=m1+m2 {ψE(m1) ∨ ψP(m2)}} = inf n=q+s { inf q=f(m1) {ψE(m1)} ∨ inf s=f(m2) {ψP(m2)}} =ψf(E)+f(P)(n). Therefore, f(E + P) = f(E) + f(P). Let X be a nonempty set. Let ϕE(x) = ρE(x)e iζE(x), ψE(x) = ρ̂E(x)e iζ̂E(x), and E = {(x, ϕE(x), ψE(x)) : x ∈ X} be a complex intuitionistic fuzzy set within X. For α, α̂ ∈ [0, 1] and β, β̂ ∈ [0, 2π], the set E(α̂,β̂) (α,β) = {x : ρE(x) ≥ α, ζE(x) ≥ β, ρ̂E(x) ≤ α̂, ζ̂E(x) ≤ β̂} is called the upper level subset of the complex intuitionistic fuzzy subset E . The subsets E(α̂<,β̂) (α>,β) = {x : ρE(x) > α, ζE(x) ≥ β, ρ̂E(x) < α̂, ζ̂E(x) ≤ β̂}, E(α̂,β̂<) (α,β>) = {x : ρE(x) ≥ α, ζE(x) > β, ρ̂E(x) ≤ α̂, ζ̂E(x) < β̂}, and E(α̂<,β̂<) (α>,β>) = {x : ρE(x) > α, ζE(x) > β, ρ̂E(x) < α̂, ζ̂E(x) < β̂} are called strong upper level subsets of the complex intuitionistic fuzzy subset A. The following theorem was obtained by S. Shaqaqha in the setting of complex fuzzy Lie sub- algebras [13]. We extend it to the case of complex intuitionistic fuzzy Lie subalgebras. Theorem 5. Let f : M1 → M2 be a Lie algebra homomorphism. If P = (ϕP , ψP) is a complex intuitionistic fuzzy set of M2. For α, α̂ ∈ [0, 1] and β, β̂ ∈ [0, 2π], we have (i) f−1 ( P(α̂,β̂) (α,β) ) = ( f−1(P) )(α̂,β̂) (α,β) , (ii) f−1 ( P(α̂<,β̂) (α>,β) ) = ( f−1(P) )(α̂<,β̂) (α>,β) , (iii) f−1 ( P(α̂,β̂<) (α,β>) ) = ( f−1(P) )(α̂,β̂<) (α,β>) , (iv) f−1 ( P(α̂<,β̂<) (α>,β>) ) = ( f−1(P) )(α̂<,β̂<) (α>,β>) . Proof. (i) m ∈ f−1 ( P(α̂,β̂) (α,β) ) if and only if f(m) ∈ P(α̂,β̂) (α,β) if and only if ϕP(f(m)) = ϕf−1(P)(m) ≥ αeiβ and ψP(f(m)) = ψf−1(P)(m) ≤ α̂eiβ̂ if and only if m ∈ (f−1(P)) (α̂,β̂) (α,β). The proofs of (ii), (iii) and (iv) are same. REFERENCES 3302 5. Conclusions In this paper, we explored the interaction between complex intuitionistic fuzzy sets and Lie algebra homomorphisms, deriving several significant results. We showed that the images and preimages of complex intuitionistic fuzzy Lie subalgebras and ideals under homomorphisms maintain their structural properties, extending known results in fuzzy and intuitionistic fuzzy algebra to the complex intuitionistic fuzzy setting. The method used focused on analyzing the effects of homomorphisms on membership and non-membership functions, providing a deeper understanding of the preservation of these fuzzy structures. Our results generalize existing findings on fuzzy Lie algebras [21], intuitionistic fuzzy Lie algebras [1], and complex fuzzy Lie algebras [13], offering new insights into their behavior in more expressive contexts. These findings pave the way for further research, including extensions to gamma rings [14, 15], n-Lie algebras [16], and Hom-Lie algebras [17]. Additionally, investigating the application of these methods to complex Pythagorean Lie algebras [19, 20] could reveal new structural insights. Acknowledgements This manuscript builds upon the thesis work of Mounther Al-Deiakeh [4], whose re- search significantly contributed to the development of the techniques and results presented here. His foundational work laid the groundwork for the ideas and methodologies explored in this paper, and his contributions are gratefully acknowledged. The authors would also like to express gratitude to the anonymous reviewers for their valuable comments and suggestions, which have greatly improved the quality and clarity of this manuscript. References [1] M. Akram. Intuitionistic fuzzy Lie subalgebras. Southeast Asian Bulletin of Mathe- matics, 31:843–855, 2007. [2] M. Akram. Bipolar fuzzy soft Lie algebras. Quasigroups and Related Systems, 21:1–10, 2013. [3] M. Akram. Fuzzy Lie Algebras. Infosys Science Foundation Series in Mathematical Sciences. Springer, 2018. [4] M. Al-Deiakeh. On intuitionistic fuzzy Lie algebras. Master Thesis, Yarmouk Uni- versity, Jordan, 2019. [5] A. S. Alkouri and A. Salleh. Complex intuitionistic fuzzy sets. Proceedings of the In- ternational Conference on Fundamental and Applied Sciences (ICFAS ’12), 1482:464– 470, 2012. REFERENCES 3303 [6] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20:87–96, 1986. [7] Y. Bahturin. Identical relations in Lie algebras. VNU Science Press, Utrecht, 1987. [8] W. Chen and S. Zhang. Intuitionistic fuzzy Lie sub-superalgebras and intuitionistic fuzzy ideals. Computers and Mathematics with Applications, 58:1645–1661, 2009. [9] N. Jacobson. Lie algebras. Wiley, New York, 1962. [10] L. Platil and T. Tanaka. Multi-criteria evaluation for intuitionistic fuzzy sets based on set-relations. Nihonkai Mathematical Journal, 34:1–18, 2023. [11] L. C. Platil and G. C. Petalcorin. Fuzzy Γ-semimodules over Γ-semirings. Journal of Analysis & Applications, 15:71–83, 2017. [12] E. H. Roh, E. Yang, and Y. B. Jun. Intuitionistic fuzzy ordered subalgebras in ordered BCI-algebras. European Journal of Pure and Applied Mathematics, 16(3):1342–1358, 2023. [13] S. Shaqaqha. Complex fuzzy Lie algebras. Jordan J. Math. Stat., 13(2):231–247, 2020. [14] S. Shaqaqha. Characterizations of Artinian and Noetherian gamma rings in terms of homogeneous complex fuzzy ideals. Palestine Journal of Mathematics, 11(4):167–171, 2022. [15] S. Shaqaqha. Isomorphism theorems for complex fuzzy gamma rings. Missouri Jour- nal of Mathematical Sciences, 34(2):196–207, 2022. [16] S. Shaqaqha. On fuzzification of n-Lie algebra. Jordan J. Math. Stat., 15(3A):523– 540, 2022. [17] S. Shaqaqha. Fuzzy Hom–Lie ideals of Hom–Lie algebras. Axioms, 12(7):630, 2023. [18] S. Shaqaqha and M. Al-Deiakeh. Towards studying complex intuitionistic fuzzy Lie algebras. Submitted manuscript, 2023. [19] R. R. Yager. Pythagorean fuzzy subsets. In Proc. Joint IFSA World Congr. NAFIPS Annu. Meeting (IFSA/NAFIPS), pages 57–61, Edmonton, AB, Canada, 2013. [20] R. R. Yager. Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst., 22(4):958–965, 2014. [21] S. E. Yehia. Fuzzy ideals and fuzzy subalgebras of Lie algebra. Fuzzy Sets and Systems, 80:237–244, 1996. [22] L. Zadeh. Fuzzy sets. Inform. Control, 8:338–358, 1965.