Bipolar Complex Intuitionistic Fuzzy Lie Algebras

Abstract

We introduce and study bipolar complex intuitionistic fuzzy Lie algebras (BCIFLAs), a new algebraic framework merging bipolar fuzzy theory, complex-valued membership functions in Cartesian form, and intuitionistic fuzzy Lie algebra theory. Adopting the Cartesian coordinate formulation of bipolar complex intuitionistic fuzzy sets (BCIFSs), we define bipolar complex intuitionistic fuzzy Lie subalgebras (BCIFLSAs) and ideals (BCIFLIs) and establish their fundamental properties, including closure under arbitrary meets. We prove that images and preimages of BCIFLSAs and BCIFLIs are preserved under Lie algebra homomorphisms, characterize them via level sets, and show that the sum of two BCIFLIs is, again, a BCIFLI. Non-degenerate three-level examples on $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{t}(2,\mathbb{R})$ illustrate the gap between the subalgebra and ideal conditions. This framework strictly generalizes fuzzy, intuitionistic fuzzy, complex intuitionistic fuzzy, and bipolar fuzzy Lie algebras.

1. Introduction

The classical theory of Lie algebras, originating with Sophus Lie (1842–1899) [1] in his study of continuous transformation groups, has grown into a cornerstone of modern mathematics and theoretical physics. Lie algebras arise naturally in the infinitesimal description of Lie groups, in the study of symmetries of differential equations, and in quantum mechanics, where the commutator bracket of operators captures fundamental physical relationships [2,3].

Independently, the theory of fuzzy sets, introduced by Zadeh [4], has become a powerful tool for representing uncertainty and imprecision, with recent developments demonstrating its effectiveness in predictive modelling [5].

Several layers of generalization have since been developed:

(i)

Intuitionistic fuzzy sets (Atanassov [6]) extend fuzzy sets by assigning both a membership and a non-membership degree, with the constraint that their sum does not exceed one.

(ii)

Bipolar fuzzy sets (Zhang [7]) introduce a second polarity: a negative membership degree taking values in $[-1,0]$, capturing opposition, repulsion, or contradictory information alongside the standard positive degree in $[0,1]$.

(iii)

Complex fuzzy sets (Ramot et al. [8]) extend the membership function to take values in the unit disk of the complex plane, encoding an amplitude and a phase simultaneously.

(iv)

Complex intuitionistic fuzzy sets (Alkouri and Salleh [9]) combine (i) and (iii), assigning complex-valued membership and non-membership functions subject to a constraint on the sum of their moduli.

(v)

Bipolar complex intuitionistic fuzzy sets (Alkouri and Alshboul [10]) combine (ii) and (iv), using a Cartesian coordinate representation in which all four functions are expressed as complex numbers $a+ib$, satisfying appropriate bipolar intuitionistic constraints.

The fusion of fuzzy theories with Lie algebra theory is well-established. Yehia [11] introduced fuzzy Lie subalgebras and ideals. Akram [12] defined intuitionistic fuzzy Lie subalgebras. Shaqaqha [13] introduced complex fuzzy Lie algebras, while Shaqaqha and Al-Deiakeh [14] studied complex intuitionistic fuzzy Lie algebras and their behavior under homomorphisms. Akram [15] investigated bipolar fuzzy soft Lie algebras.

Before developing the mathematical theory, we explain why the eight-component Cartesian structure of BCIFSs is both natural and necessary in several real-world domains. The key advantage is that BCIFSs simultaneously handle bipolarity (positive and negative poles), complex-valued information (real and imaginary parts encoding amplitude and phase/periodic factor), and intuitionistic uncertainty (membership and non-membership hesitancy) in a single mathematical object [10].

Quantum mechanics and spin systems: Quantum observables often exhibit inherent bipolarity—for instance, spin-up versus spin-down in a two-level system—and complex-phase structure, since quantum states are described by complex-valued wave functions [16,17]. The Lie algebra $\mathfrak{su}\left(2\right)$, the symmetry algebra governing spin- ${\displaystyle \frac{1}{2}}$ systems and angular momentum, is generated by operators satisfying $[{\widehat{J}}_x,{\widehat{J}}_y]=i\hslash {\widehat{J}}_z$ and its cyclic permutations [17]. In a BCIFLA-theoretic model of an element $m\in \mathfrak{su}\left(2\right)$ representing an observable:

• ${\mu }^{+}\left(m\right)=r^{+}\left(m\right)+i{w}_{r^{+}}\left(m\right)$ encodes the degree and phase/energy-level contribution toward the positive pole (e.g., spin-up eigenstates);
• ${\mu }^{-}\left(m\right)=r^{-}\left(m\right)+i{w}_{r^{-}}\left(m\right)$, with $r^{-},w_{r^{-}}\in [-1,0]$, encodes the corresponding opposition strength and phase for the negative pole (spin-down);
• ${\nu }^{\pm }\left(m\right)$ captures the intuitionistic hesitancy (non-membership) in assigning each polarity, modeling the irreducible quantum uncertainty in measurement outcomes [18].

Standard complex intuitionistic fuzzy models [9] lose bipolarity. BCIFLAs capture all three aspects simultaneously, providing a more faithful algebraic framework for modeling uncertainty in quantum spin systems described by $\mathfrak{su}\left(2\right)$.

Equivariant neural networks and Lie algebra symmetries: Modern deep learning architectures increasingly exploit Lie group and Lie algebra symmetries to improve generalization and data efficiency [19,20]. For example, $\mathrm{SO}\left(3\right)$-equivariant networks for 3D point cloud analysis use the Lie algebra $\mathfrak{so}\left(3\right)$, ensuring that learned features transform predictably under 3D rotations [20,21]. Finzi et al. [22] extended equivariance to arbitrary Lie groups on continuous data. In these settings, the symmetry features extracted from data are often bipolar (encoding both the presence and violation of a symmetry), complex-valued (carrying frequency-domain information), and uncertain (subject to detection hesitancy). Modeling such symmetry features as BCIFLAs provides an eight-component algebraic structure that respects the Lie bracket interaction between generators while encoding all four uncertainty modalities simultaneously.

Multi-criteria decision making with periodic criteria: Fuzzy set theory and its generalizations have been extensively applied to multi-criteria decision making (MCDM), with intuitionistic fuzzy approaches handling membership hesitancy [23], bipolar fuzzy approaches handling conflicting evaluations [24], and complex fuzzy approaches handling periodic or time-varying information [8,25]. Alkouri and Alshboul [10] demonstrated that BCIFSs are particularly suitable for environmental impact assessment (EIA), where experts must simultaneously evaluate positive and negative environmental impacts (bipolarity), impact factors that vary across time or seasons (imaginary part, complex periodicity), and uncertainty in both positive and negative assessments (intuitionistic non-membership). In a Lie-algebraic MCDM extension, the set of criteria forms a Lie algebra M over $\mathbb{R}$, and the Lie bracket $[m_1,m_2]$ models the non-commutative interaction between criteria. A BCIFLI of M then provides a fuzzy classification of criteria that is stable under these interactions, giving a principled algebraic framework for grouping criteria that interact consistently.

Signal processing and the Heisenberg–Weyl Lie algebra: The Heisenberg–Weyl algebra, generated by position $\widehat{q}$, momentum $\widehat{p}$, and identity I with $[\widehat{q},\widehat{p}]=i\hslash I$, is the foundational Lie algebra underlying the time-frequency uncertainty principle [26,27]. Complex fuzzy sets were introduced by Ramot et al. [8] for signal detection [28] and time-series analysis [25]. BCIFLAs extend this to the bipolar and intuitionistic settings simultaneously: the real parts $r^{\pm }$ encode signal amplitudes, the imaginary parts $w_{r^{\pm }}$ encode frequency/phase, and the non-membership components $k^{\pm },w_{k^{\pm }}$ capture ambiguity in both polarities. The Lie bracket structure among generators of the Heisenberg–Weyl algebra encodes the fundamental non-commutativity of time and frequency localization, providing a natural algebraic arena for uncertainty-aware signal analysis.

These four settings share a common feature: the phenomena being modeled are simultaneously oppositional (bipolar), phase- or time-dependent (complex-valued), and uncertain in a two-sided, membership/non-membership sense (intuitionistic); no existing fuzzy Lie-algebraic framework captures all three simultaneously, which is the gap BCIFLAs are designed to fill. A fully worked computational case study in any one of these domains is identified as a future direction in Section 8.

Despite this rich activity and the real-world motivations described above, no study has introduced a bipolar complex intuitionistic fuzzy Lie algebra. This structure simultaneously incorporates bipolarity, Cartesian complex-valued membership and non-membership functions, and intuitionistic uncertainty into the Lie-algebraic setting. The present paper fills this gap.

The main contributions of this paper are as follows:

• We introduce the concepts of a bipolar complex intuitionistic fuzzy Lie subalgebra (BCIFLSA) and a bipolar complex intuitionistic fuzzy Lie ideal (BCIFLI) of a Lie algebra, grounded in the Cartesian BCIFS framework of [10].
• We provide non-trivial three-level examples and establish elementary properties.
• We prove that the image and preimage of a BCIFLSA (resp. BCIFLI) under a Lie algebra homomorphism is, again, a BCIFLSA (resp. BCIFLI).
• We characterize BCIFLSAs via level subsets.
• We study the sum of two BCIFLIs and show that it is, again, a BCIFLI.
• We exhibit the hierarchy of special cases, showing how all known fuzzy Lie algebra concepts are recovered.
• We show that BCIFLSA(M) and BCIFLI(M) are closed under arbitrary meets, forming meet-semilattices under the natural order on BCIFSs (Proposition 4).
• Unlike a purely componentwise translation of known single-polarity theories, several of these results represent genuinely new technical content, not automatic restatements of the single-polarity theory. These include the interaction between bipolarity and the sup/inf image construction (Lemma 1), the four-term bilinear expansion needed for closure under sums (Theorem 4), and the opposite monotonicity directions of $r^{-}$ and $k^{-}$ required (Example 2).

Throughout this paper, M denotes a Lie algebra over a field K with Lie bracket $[·,·]:M\times M\to M$. We use $i=\sqrt{-1}$.

2. Preliminaries

We recall the necessary background material and foundational definitions.

Definition 1

([2,3]). A Lie algebra $(M,[·,·\left]\right)$ over a field K is a vector space M together with a bilinear map $[·,·]:M\times M\to M$ satisfying:

(i) 

$[m,m]=0_M$ for all $m\in M$ (implying $[m_1,m_2]=-[m_2,m_1]$);

(ii) 

$[m_1,[m_2,m_3]]+[m_2,[m_3,m_1]]+[m_3,[m_1,m_2]]=0_M$ for all $m_1,m_2,m_3\in M$ (Jacobi identity).

A subspace $Q\subseteq M$ is a Lie subalgebra if $[q_1,q_2]\in Q$ for all $q_1,q_2\in Q$. It is a Lie ideal of M if $[m,q]\in Q$ for all $m\in M$, $q\in Q$. A linear map $f:M_1\to M_2$ between Lie algebras is a Lie algebra homomorphism if $f\left(\right[m,n\left]\right)=\left[f\right(m),f(n\left)\right]$ for all $m,n\in M_1$.

Definition 2

(Bipolar Fuzzy Set [7]). A bipolar fuzzy set on a nonempty set X is a pair $B=(B^{+},B^{-})$ where $B^{+}:X\to [0,1]$ and $B^{-}:X\to [-1,0]$.

Definition 3

(Complex Intuitionistic Fuzzy Set [9]). A complex intuitionistic fuzzy set (CIFS) on X is a pair $E=({\varphi }_E,{\psi }_E)$ where ${\varphi }_E\left(x\right)={\rho }_E\left(x\right)e^{i{\zeta }_E\left(x\right)}$, ${\psi }_E\left(x\right)={\widehat{\rho }}_E\left(x\right)e^{i{\widehat{\zeta }}_E\left(x\right)}$, with ${\rho }_E\left(x\right),{\widehat{\rho }}_E\left(x\right)\in [0,1]$ and ${\rho }_E\left(x\right)+{\widehat{\rho }}_E\left(x\right)\le 1$ for all $x\in X$.

Definition 4

(Bipolar Complex Intuitionistic Fuzzy Set [10]). A bipolar complex intuitionistic fuzzy set (BCIFS) on a nonempty set X is a 4-tuple:

$$\mathcal{E}=\left({\mu }_{\mathcal{E}}^{+},{\nu }_{\mathcal{E}}^{+},{\mu }_{\mathcal{E}}^{-},{\nu }_{\mathcal{E}}^{-}\right),$$

where

• The positive membership function is ${\mu }_{\mathcal{E}}^{+}\left(x\right)=r_{\mathcal{E}}^{+}\left(x\right)+i{w}_{r_{\mathcal{E}}^{+}}\left(x\right)$, with $r_{\mathcal{E}}^{+}\left(x\right),w_{r_{\mathcal{E}}^{+}}\left(x\right)\in [0,1]$.
• The positive non-membership function is ${\nu }_{\mathcal{E}}^{+}\left(x\right)=k_{\mathcal{E}}^{+}\left(x\right)+i{w}_{k_{\mathcal{E}}^{+}}\left(x\right)$, with $k_{\mathcal{E}}^{+}\left(x\right),w_{k_{\mathcal{E}}^{+}}\left(x\right)\in [0,1]$.
• The negative membership function is ${\mu }_{\mathcal{E}}^{-}\left(x\right)=r_{\mathcal{E}}^{-}\left(x\right)+i{w}_{r_{\mathcal{E}}^{-}}\left(x\right)$, with $r_{\mathcal{E}}^{-}\left(x\right),w_{r_{\mathcal{E}}^{-}}\left(x\right)\in [-1,0]$.
• The negative non-membership function is ${\nu }_{\mathcal{E}}^{-}\left(x\right)=k_{\mathcal{E}}^{-}\left(x\right)+i{w}_{k_{\mathcal{E}}^{-}}\left(x\right)$, with $k_{\mathcal{E}}^{-}\left(x\right),w_{k_{\mathcal{E}}^{-}}\left(x\right)\in [-1,0]$.

These eight real-valued component functions satisfy the following constraints for all $x\in X$:

$$\begin{array}{cc}\hfill & 0\le r_{\mathcal{E}}^{+}\left(x\right)+k_{\mathcal{E}}^{+}\left(x\right)\le 1,0\le w_{r_{\mathcal{E}}^{+}}\left(x\right)+w_{k_{\mathcal{E}}^{+}}\left(x\right)\le 1,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & -1\le r_{\mathcal{E}}^{-}\left(x\right)+k_{\mathcal{E}}^{-}\left(x\right)\le 0,-1\le w_{r_{\mathcal{E}}^{-}}\left(x\right)+w_{k_{\mathcal{E}}^{-}}\left(x\right)\le 0.\hfill \end{array}$$

(2)

The set of all BCIFSs on X is denoted by $\mathcal{BCIF}\left(X\right)$.

Remark 1.

As emphasized in [10], the Cartesian form $a+ib$ is used instead of the polar form $r{e}^{i\theta }$ because polar coordinates cannot naturally represent negative real values. Setting ${\mu }_{\mathcal{E}}^{-}\equiv {\nu }_{\mathcal{E}}^{-}\equiv 0$ reduces to a CIFS [9]; setting all imaginary parts to zero further reduces to a bipolar intuitionistic fuzzy set.

Definition 5

(Order of Components). For two BCIFSs $\mathcal{E}$ and $\mathcal{F}$ on X, we write $\mathcal{E}\le \mathcal{F}$ if and only if for all $x\in X$:

$$\begin{array}{cc}\hfill r_{\mathcal{E}}^{+}\left(x\right)\le r_{\mathcal{F}}^{+}\left(x\right),& w_{r_{\mathcal{E}}^{+}}\left(x\right)\le w_{r_{\mathcal{F}}^{+}}\left(x\right),\hfill \\ \hfill k_{\mathcal{E}}^{+}\left(x\right)\ge k_{\mathcal{F}}^{+}\left(x\right),& w_{k_{\mathcal{E}}^{+}}\left(x\right)\ge w_{k_{\mathcal{F}}^{+}}\left(x\right),\hfill \\ \hfill r_{\mathcal{E}}^{-}\left(x\right)\ge r_{\mathcal{F}}^{-}\left(x\right),& w_{r_{\mathcal{E}}^{-}}\left(x\right)\ge w_{r_{\mathcal{F}}^{-}}\left(x\right),\hfill \\ \hfill k_{\mathcal{E}}^{-}\left(x\right)\le k_{\mathcal{F}}^{-}\left(x\right),& w_{k_{\mathcal{E}}^{-}}\left(x\right)\le w_{k_{\mathcal{F}}^{-}}\left(x\right).\hfill \end{array}$$

The component-wise inequalities defining this partial order are consistent with the general inner product and norm inequality frameworks studied in the literature [29].

Definition 6

(Fuzzy Lie Subalgebra [11]). Let M be a Lie algebra over K. A fuzzy subset $\mu :M\to [0,1]$ is called a fuzzy Lie subalgebra of M if, for all $m_1,m_2\in M$ and $k\in K$:

(i) 

$\mu (m_1+m_2)\ge \min \{\mu \left(m_1\right),\mu \left(m_2\right)\}$;

(ii) 

$\mu \left(km\right)\ge \mu \left(m\right)$;

(iii) 

$\mu \left([m_1,m_2]\right)\ge \min \{\mu \left(m_1\right),\mu \left(m_2\right)\}$.

If condition (iii) is strengthened to $\mu \left([m_1,m_2]\right)\ge \max \{\mu \left(m_1\right),\mu \left(m_2\right)\}$, then μ is called a fuzzy Lie ideal of M.

Definition 7

(Intuitionistic Fuzzy Lie Subalgebra [12]). An intuitionistic fuzzy Lie subalgebra of M is a pair $A=({\mu }_A,{\nu }_A)$ with ${\mu }_A,{\nu }_A:M\to [0,1]$, ${\mu }_A\left(m\right)+{\nu }_A\left(m\right)\le 1$, satisfying for all $m_1,m_2\in M$ and $k\in K$:

$$\begin{array}{cccc}\hfill {\mu }_A(m_1+m_2)& \ge \min \{{\mu }_A\left(m_1\right),{\mu }_A\left(m_2\right)\},\hfill & \hfill {\nu }_A(m_1+m_2)& \le \max \{{\nu }_A\left(m_1\right),{\nu }_A\left(m_2\right)\},\hfill \\ \hfill {\mu }_A\left(km\right)& \ge {\mu }_A\left(m\right),\hfill & \hfill {\nu }_A\left(km\right)& \le {\nu }_A\left(m\right),\hfill \\ \hfill {\mu }_A\left([m_1,m_2]\right)& \ge \min \{{\mu }_A\left(m_1\right),{\mu }_A\left(m_2\right)\},\hfill & \hfill {\nu }_A\left([m_1,m_2]\right)& \le \max \{{\nu }_A\left(m_1\right),{\nu }_A\left(m_2\right)\}.\hfill \end{array}$$

Definition 8

(Bipolar Fuzzy Lie Subalgebra [15]). A bipolar fuzzy Lie subalgebra of M is a bipolar fuzzy set $B=(B^{+},B^{-})$ satisfying for all $m_1,m_2\in M$ and $k\in K$:

$$\begin{array}{cccc}\hfill B^{+}(m_1+m_2)& \ge \min \{B^{+}\left(m_1\right),B^{+}\left(m_2\right)\},\hfill & \hfill B^{-}(m_1+m_2)& \le \max \{B^{-}\left(m_1\right),B^{-}\left(m_2\right)\},\hfill \\ \hfill B^{+}\left(km\right)& \ge B^{+}\left(m\right),\hfill & \hfill B^{-}\left(km\right)& \le B^{-}\left(m\right),\hfill \\ \hfill B^{+}\left([m_1,m_2]\right)& \ge \min \{B^{+}\left(m_1\right),B^{+}\left(m_2\right)\},\hfill & \hfill B^{-}\left([m_1,m_2]\right)& \le \max \{B^{-}\left(m_1\right),B^{-}\left(m_2\right)\}.\hfill \end{array}$$

Definition 9

(Complex Intuitionistic Fuzzy Lie Subalgebra [14]). A complex intuitionistic fuzzy Lie subalgebra of M is a CIFS $E=({\varphi }_E,{\psi }_E)$ on M with ${\varphi }_E\left(m\right)={\rho }_E\left(m\right)e^{i{\zeta }_E\left(m\right)}$, ${\psi }_E\left(m\right)={\widehat{\rho }}_E\left(m\right)e^{i{\widehat{\zeta }}_E\left(m\right)}$, ${\rho }_E+{\widehat{\rho }}_E\le 1$, ${\zeta }_E+{\widehat{\zeta }}_E\le 2\pi$, satisfying for all $m_1,m_2\in M$ and $k\in K$:

$$\begin{array}{cccc}\hfill {\rho }_E(m_1+m_2)& \ge \min \{{\rho }_E\left(m_1\right),{\rho }_E\left(m_2\right)\},\hfill & \hfill {\widehat{\rho }}_E(m_1+m_2)& \le \max \{{\widehat{\rho }}_E\left(m_1\right),{\widehat{\rho }}_E\left(m_2\right)\},\hfill \\ \hfill {\rho }_E\left(km\right)& \ge {\rho }_E\left(m\right),\hfill & \hfill {\widehat{\rho }}_E\left(km\right)& \le {\widehat{\rho }}_E\left(m\right),\hfill \\ \hfill {\rho }_E\left([m_1,m_2]\right)& \ge \min \{{\rho }_E\left(m_1\right),{\rho }_E\left(m_2\right)\},\hfill & \hfill {\widehat{\rho }}_E\left([m_1,m_2]\right)& \le \max \{{\widehat{\rho }}_E\left(m_1\right),{\widehat{\rho }}_E\left(m_2\right)\},\hfill \end{array}$$

and likewise for the phase components ${\zeta }_E,{\widehat{\zeta }}_E$.

3. Bipolar Complex Intuitionistic Fuzzy Lie Algebras

Definition 10

(Bipolar Complex Intuitionistic Fuzzy Lie Subalgebra). Let M be a Lie algebra over K. A BCIFS $\mathcal{E}=({\mu }_{\mathcal{E}}^{+},{\nu }_{\mathcal{E}}^{+},{\mu }_{\mathcal{E}}^{-},{\nu }_{\mathcal{E}}^{-})\in \mathcal{BCIF}\left(M\right)$ is called a bipolar complex intuitionistic fuzzy Lie subalgebra (BCIFLSA) of M if, for all $m_1,m_2\in M$ and $k\in K$:

(S1) 

(Addition)

$$\begin{array}{cc}\hfill r_{\mathcal{E}}^{+}(m_1+m_2)& \ge \min \{r_{\mathcal{E}}^{+}\left(m_1\right),r_{\mathcal{E}}^{+}\left(m_2\right)\},\hfill \\ \hfill w_{r_{\mathcal{E}}^{+}}(m_1+m_2)& \ge \min \{w_{r_{\mathcal{E}}^{+}}\left(m_1\right),w_{r_{\mathcal{E}}^{+}}\left(m_2\right)\},\hfill \\ \hfill k_{\mathcal{E}}^{+}(m_1+m_2)& \le \max \{k_{\mathcal{E}}^{+}\left(m_1\right),k_{\mathcal{E}}^{+}\left(m_2\right)\},\hfill \\ \hfill w_{k_{\mathcal{E}}^{+}}(m_1+m_2)& \le \max \{w_{k_{\mathcal{E}}^{+}}\left(m_1\right),w_{k_{\mathcal{E}}^{+}}\left(m_2\right)\},\hfill \\ \hfill r_{\mathcal{E}}^{-}(m_1+m_2)& \le \max \{r_{\mathcal{E}}^{-}\left(m_1\right),r_{\mathcal{E}}^{-}\left(m_2\right)\},\hfill \\ \hfill w_{r_{\mathcal{E}}^{-}}(m_1+m_2)& \le \max \{w_{r_{\mathcal{E}}^{-}}\left(m_1\right),w_{r_{\mathcal{E}}^{-}}\left(m_2\right)\},\hfill \\ \hfill k_{\mathcal{E}}^{-}(m_1+m_2)& \ge \min \{k_{\mathcal{E}}^{-}\left(m_1\right),k_{\mathcal{E}}^{-}\left(m_2\right)\},\hfill \\ \hfill w_{k_{\mathcal{E}}^{-}}(m_1+m_2)& \ge \min \{w_{k_{\mathcal{E}}^{-}}\left(m_1\right),w_{k_{\mathcal{E}}^{-}}\left(m_2\right)\}.\hfill \end{array}$$

(S2) 

(Scalar multiplication)

$$\begin{array}{cc}\hfill r_{\mathcal{E}}^{+}\left(km\right)& \ge r_{\mathcal{E}}^{+}\left(m\right),w_{r_{\mathcal{E}}^{+}}\left(km\right)\ge w_{r_{\mathcal{E}}^{+}}\left(m\right),\hfill \\ \hfill k_{\mathcal{E}}^{+}\left(km\right)& \le k_{\mathcal{E}}^{+}\left(m\right),w_{k_{\mathcal{E}}^{+}}\left(km\right)\le w_{k_{\mathcal{E}}^{+}}\left(m\right),\hfill \\ \hfill r_{\mathcal{E}}^{-}\left(km\right)& \le r_{\mathcal{E}}^{-}\left(m\right),w_{r_{\mathcal{E}}^{-}}\left(km\right)\le w_{r_{\mathcal{E}}^{-}}\left(m\right),\hfill \\ \hfill k_{\mathcal{E}}^{-}\left(km\right)& \ge k_{\mathcal{E}}^{-}\left(m\right),w_{k_{\mathcal{E}}^{-}}\left(km\right)\ge w_{k_{\mathcal{E}}^{-}}\left(m\right).\hfill \end{array}$$

(S3) 

(Lie bracket)

$$\begin{array}{cc}\hfill r_{\mathcal{E}}^{+}\left([m_1,m_2]\right)& \ge \min \{r_{\mathcal{E}}^{+}\left(m_1\right),r_{\mathcal{E}}^{+}\left(m_2\right)\},\hfill \\ \hfill w_{r_{\mathcal{E}}^{+}}\left([m_1,m_2]\right)& \ge \min \{w_{r_{\mathcal{E}}^{+}}\left(m_1\right),w_{r_{\mathcal{E}}^{+}}\left(m_2\right)\},\hfill \\ \hfill k_{\mathcal{E}}^{+}\left([m_1,m_2]\right)& \le \max \{k_{\mathcal{E}}^{+}\left(m_1\right),k_{\mathcal{E}}^{+}\left(m_2\right)\},\hfill \\ \hfill w_{k_{\mathcal{E}}^{+}}\left([m_1,m_2]\right)& \le \max \{w_{k_{\mathcal{E}}^{+}}\left(m_1\right),w_{k_{\mathcal{E}}^{+}}\left(m_2\right)\},\hfill \\ \hfill r_{\mathcal{E}}^{-}\left([m_1,m_2]\right)& \le \max \{r_{\mathcal{E}}^{-}\left(m_1\right),r_{\mathcal{E}}^{-}\left(m_2\right)\},\hfill \\ \hfill w_{r_{\mathcal{E}}^{-}}\left([m_1,m_2]\right)& \le \max \{w_{r_{\mathcal{E}}^{-}}\left(m_1\right),w_{r_{\mathcal{E}}^{-}}\left(m_2\right)\},\hfill \\ \hfill k_{\mathcal{E}}^{-}\left([m_1,m_2]\right)& \ge \min \{k_{\mathcal{E}}^{-}\left(m_1\right),k_{\mathcal{E}}^{-}\left(m_2\right)\},\hfill \\ \hfill w_{k_{\mathcal{E}}^{-}}\left([m_1,m_2]\right)& \ge \min \{w_{k_{\mathcal{E}}^{-}}\left(m_1\right),w_{k_{\mathcal{E}}^{-}}\left(m_2\right)\}.\hfill \end{array}$$

Remark 2.

The sign conventions are consistent with bipolarity: positive membership components $r^{+},w_{r^{+}}$ increase under Lie operations, while positive non-membership components $k^{+},w_{k^{+}}$ decrease. For the negative components, since values lie in $[-1,0]$, increasing magnitude corresponds to more negative values, so $r^{-},w_{r^{-}}$ decrease and $k^{-},w_{k^{-}}$ increase.

Remark 3.

Although Definition 10 lists eight structurally similar inequalities, the eight components do not reduce to a single scalar-fuzzy computation applied four times: $r_{\mathcal{E}}^{+}$ and $k_{\mathcal{E}}^{-}$ obey one monotonicity pattern under Lie operations (≥ in (S1) and (S3), so that $\mathcal{E}\left(0_M\right)$ attains the pattern’s extremal, most acceptable value—see Proposition 2), while $k_{\mathcal{E}}^{+}$ and $r_{\mathcal{E}}^{-}$ obey the opposite pattern (≤). Correctly tracking which of the four patterns applies to each of the eight components is essential: Example 2 exhibits a three-level BCIFLI in which $r^{-}$ and $k^{-}$ are required to move in opposite directions relative to the same class hierarchy.

Definition 11

(Bipolar Complex Intuitionistic Fuzzy Lie Ideal). A BCIFS $\mathcal{E}\in \mathcal{BCIF}\left(M\right)$ is called a bipolar complex intuitionistic fuzzy Lie ideal (BCIFLI) of M if conditions (S1) and (S2) hold, and condition (S3) is replaced by:

(I3) 

(Ideal Lie bracket)

$$\begin{array}{cc}\hfill r_{\mathcal{E}}^{+}\left([m_1,m_2]\right)& \ge \max \{r_{\mathcal{E}}^{+}\left(m_1\right),r_{\mathcal{E}}^{+}\left(m_2\right)\},\hfill \\ \hfill w_{r_{\mathcal{E}}^{+}}\left([m_1,m_2]\right)& \ge \max \{w_{r_{\mathcal{E}}^{+}}\left(m_1\right),w_{r_{\mathcal{E}}^{+}}\left(m_2\right)\},\hfill \\ \hfill k_{\mathcal{E}}^{+}\left([m_1,m_2]\right)& \le \min \{k_{\mathcal{E}}^{+}\left(m_1\right),k_{\mathcal{E}}^{+}\left(m_2\right)\},\hfill \\ \hfill w_{k_{\mathcal{E}}^{+}}\left([m_1,m_2]\right)& \le \min \{w_{k_{\mathcal{E}}^{+}}\left(m_1\right),w_{k_{\mathcal{E}}^{+}}\left(m_2\right)\},\hfill \\ \hfill r_{\mathcal{E}}^{-}\left([m_1,m_2]\right)& \le \min \{r_{\mathcal{E}}^{-}\left(m_1\right),r_{\mathcal{E}}^{-}\left(m_2\right)\},\hfill \\ \hfill w_{r_{\mathcal{E}}^{-}}\left([m_1,m_2]\right)& \le \min \{w_{r_{\mathcal{E}}^{-}}\left(m_1\right),w_{r_{\mathcal{E}}^{-}}\left(m_2\right)\},\hfill \\ \hfill k_{\mathcal{E}}^{-}\left([m_1,m_2]\right)& \ge \max \{k_{\mathcal{E}}^{-}\left(m_1\right),k_{\mathcal{E}}^{-}\left(m_2\right)\},\hfill \\ \hfill w_{k_{\mathcal{E}}^{-}}\left([m_1,m_2]\right)& \ge \max \{w_{k_{\mathcal{E}}^{-}}\left(m_1\right),w_{k_{\mathcal{E}}^{-}}\left(m_2\right)\}.\hfill \end{array}$$

Proposition 1.

Every BCIFLI of M is a BCIFLSA of M.

Proof. 

Since $\max \{a,b\}\ge \min \{a,b\}$ for real numbers $a,b$, condition (I3) implies (S3) for every component. □

Proposition 2

(Values at the Zero Element). Let $\mathcal{E}$ be a BCIFLSA of M. For all $m\in M$:

$$r_{\mathcal{E}}^{+}\left(0_M\right)\ge r_{\mathcal{E}}^{+}\left(m\right),w_{r_{\mathcal{E}}^{+}}\left(0_M\right)\ge w_{r_{\mathcal{E}}^{+}}\left(m\right),k_{\mathcal{E}}^{+}\left(0_M\right)\le k_{\mathcal{E}}^{+}\left(m\right),w_{k_{\mathcal{E}}^{+}}\left(0_M\right)\le w_{k_{\mathcal{E}}^{+}}\left(m\right),$$

$$r_{\mathcal{E}}^{-}\left(0_M\right)\le r_{\mathcal{E}}^{-}\left(m\right),w_{r_{\mathcal{E}}^{-}}\left(0_M\right)\le w_{r_{\mathcal{E}}^{-}}\left(m\right),k_{\mathcal{E}}^{-}\left(0_M\right)\ge k_{\mathcal{E}}^{-}\left(m\right),w_{k_{\mathcal{E}}^{-}}\left(0_M\right)\ge w_{k_{\mathcal{E}}^{-}}\left(m\right).$$

Proof. 

Apply (S2) with $k=0$. □

Proposition 3

(Symmetry). Let $\mathcal{E}$ be a BCIFLSA of M. For all $m\in M$, each of the eight component functions takes the same value at m and at $-m$.

Proof. 

Apply (S2) with $k=-1$ and then replace m by $-m$. □

Proposition 4

(Meet of a Family of BCIFLSAs). Let ${\left\{{\mathcal{E}}_i\right\}}_{i\in \Lambda }$ ($\Lambda \ne \varnothing$) be a family of BCIFLSAs of M. Define ${\bigwedge }_i{\mathcal{E}}_i\in \mathcal{BCIF}\left(M\right)$ componentwise by

$$\begin{array}{cccccccc}\hfill r_{\wedge }^{+}\left(m\right)& =\underset{i}{\inf }{r}_{{\mathcal{E}}_i}^{+}\left(m\right),\hfill & \hfill w_{r_{\wedge }^{+}}\left(m\right)& =\underset{i}{\inf }{w}_{r_{{\mathcal{E}}_i}^{+}}\left(m\right),\hfill & \hfill k_{\wedge }^{+}\left(m\right)& =\underset{i}{\sup }{k}_{{\mathcal{E}}_i}^{+}\left(m\right),\hfill & \hfill w_{k_{\wedge }^{+}}\left(m\right)& =\underset{i}{\sup }{w}_{k_{{\mathcal{E}}_i}^{+}}\left(m\right),\hfill \\ \hfill r_{\wedge }^{-}\left(m\right)& =\underset{i}{\sup }{r}_{{\mathcal{E}}_i}^{-}\left(m\right),\hfill & \hfill w_{r_{\wedge }^{-}}\left(m\right)& =\underset{i}{\sup }{w}_{r_{{\mathcal{E}}_i}^{-}}\left(m\right),\hfill & \hfill k_{\wedge }^{-}\left(m\right)& =\underset{i}{\inf }{k}_{{\mathcal{E}}_i}^{-}\left(m\right),\hfill & \hfill w_{k_{\wedge }^{-}}\left(m\right)& =\underset{i}{\inf }{w}_{k_{{\mathcal{E}}_i}^{-}}\left(m\right).\hfill \end{array}$$

Then, ${\bigwedge }_i{\mathcal{E}}_i\in \mathcal{BCIF}\left(M\right)$ (i.e., it satisfies constraints (1)–(2)) and is a BCIFLSA of M. If, in addition, every ${\mathcal{E}}_i$ is a BCIFLI of M, then ${\bigwedge }_i{\mathcal{E}}_i$ is a BCIFLI of M.

Proof. 

Constraints. Fix $m\in M$ and $i_0\in \Lambda$. Since ${\mathcal{E}}_{i_0}$ satisfies (1), $r_{{\mathcal{E}}_{i_0}}^{+}\left(m\right)\le 1-k_{{\mathcal{E}}_{i_0}}^{+}\left(m\right)$; taking ${\inf }_{i_0}$ on the left and using ${\inf }_{i_0}(1-k_{{\mathcal{E}}_{i_0}}^{+}\left(m\right))=1-{\sup }_{i_0}{k}_{{\mathcal{E}}_{i_0}}^{+}\left(m\right)$ on the right gives $r_{\wedge }^{+}\left(m\right)\le 1-k_{\wedge }^{+}\left(m\right)$, i.e., $r_{\wedge }^{+}\left(m\right)+k_{\wedge }^{+}\left(m\right)\le 1$; non-negativity is inherited pointwise. The negative-pole bound follows the same argument.

Condition (S1) ($r^{+}$ addition). For every i, $r_{{\mathcal{E}}_i}^{+}(m_1+m_2)\ge \min \{r_{{\mathcal{E}}_i}^{+}\left(m_1\right),r_{{\mathcal{E}}_i}^{+}\left(m_2\right)\}\ge \min \{r_{\wedge }^{+}\left(m_1\right),r_{\wedge }^{+}\left(m_2\right)\}$. Taking ${\inf }_i$ of the left-hand side preserves the right-hand bound.

Condition (S2) (scalar multiplication). $r_{\wedge }^{+}\left(km\right)={\inf }_i{r}_{{\mathcal{E}}_i}^{+}\left(km\right)\ge {\inf }_i{r}_{{\mathcal{E}}_i}^{+}\left(m\right)=r_{\wedge }^{+}\left(m\right)$, and analogously for the remaining seven components.

Condition (S3)/(I3) (Lie bracket). Identical to the additivity argument with $[m_1,m_2]$ in place of $m_1+m_2$; if every ${\mathcal{E}}_i$ satisfies the strengthened ideal inequality (I3), the same inf-of-min ≥ min-of-inf argument carries the strengthened bound through unchanged. □

Corollary 1.

$\left(\mathrm{BCIFLSA}\right(M),\le )$ and $\left(\mathrm{BCIFLI}\right(M),\le )$ are meet-semilattices under the order of Definition 5, with the top element BCIFS 1 given by ${\mu }^{+}\equiv 1$, ${\nu }^{+}\equiv 0$, ${\mu }^{-}\equiv -1$, ${\nu }^{-}\equiv 0$, which trivially satisfies Definitions 10 and 11.

4. Examples

We present two genuine three-level examples that fully illustrate the BCIFLSA and BCIFLI conditions, exposing the gap between them with explicit bracket computations.

Example 1

(Three-Level BCIFLSA on $\mathfrak{sl}(2,\mathbb{R})$ That Is Not a BCIFLI). Let $M=\mathfrak{sl}(2,\mathbb{R})$ with standard basis $\{e,f,h\}$ and bracket relations:

$$[h,e]=2e,[h,f]=-2f,[e,f]=h.$$

Partition M into three classes:

$$C_0=\left\{0_M\right\},C_1=\mathrm{span}\left\{h\right\}\setminus \left\{0_M\right\},C_2=M\setminus (\left\{0_M\right\}\cup \mathrm{span}\left\{h\right\}).$$

Define $\mathcal{E}=({\mu }_{\mathcal{E}}^{+},{\nu }_{\mathcal{E}}^{+},{\mu }_{\mathcal{E}}^{-},{\nu }_{\mathcal{E}}^{-})$ as shown in Table 1.

Table 1. Component values for Example 1.

Class	${\mathit{\mu }}^{+}\left(\mathit{m}\right)$	${\mathit{\nu }}^{+}\left(\mathit{m}\right)$	${\mathit{\mu }}^{-}\left(\mathit{m}\right)$	${\mathit{\nu }}^{-}\left(\mathit{m}\right)$
$C_0$	$0.90+0.80i$	$0.05+0.05i$	$-0.50-0.40i$	$-0.05-0.04i$
$C_1$	$0.60+0.50i$	$0.20+0.20i$	$-0.30-0.25i$	$-0.15-0.12i$
$C_2$	$0.30+0.20i$	$0.40+0.40i$	$-0.10-0.08i$	$-0.30-0.24i$

Step 1: Constraint verification. Constraints (1)–(2) are verified for each class in Table 2.

Table 2. BCIFS constraint verification for Example 1.

Class	${\mathit{r}}^{+}+{\mathit{k}}^{+}$	${\mathit{w}}_{{\mathit{r}}^{+}}+{\mathit{w}}_{{\mathit{k}}^{+}}$	${\mathit{r}}^{-}+{\mathit{k}}^{-}$	${\mathit{w}}_{{\mathit{r}}^{-}}+{\mathit{w}}_{{\mathit{k}}^{-}}$
$C_0$	$0.95\le 1$	$0.85\le 1$	$-0.55\in [-1,0]$	$-0.44\in [-1,0]$
$C_1$	$0.80\le 1$	$0.70\le 1$	$-0.45\in [-1,0]$	$-0.37\in [-1,0]$
$C_2$	$0.70\le 1$	$0.60\le 1$	$-0.40\in [-1,0]$	$-0.33\in [-1,0]$

Step 2: Verification of conditions (S1)–(S3). The bracket condition (S3) is checked in Table 3.

Table 3. Bracket verification for the BCIFLSA condition in Example 1.

${\mathit{m}}_1\in$	${\mathit{m}}_2\in$	Bracket ∈	${\mathit{r}}^{+}$ Check (≥min)	${\mathit{r}}^{-}$ Check (≤max)
$C_0$	any	$C_0$	$0.90\ge 0.90$	$-0.50\le -0.50$
$C_1$	$C_1$	$C_0$	$0.90\ge 0.60$	$-0.50\le -0.30$
$C_1$	$C_2$	$C_2$	$0.30\ge 0.30$	$-0.10\le -0.10$
$C_2$	$C_2$	$C_1$	$0.60\ge 0.30$	$-0.30\le -0.10$

All conditions (S1)–(S3) are satisfied, so $\mathcal{E}$ is a BCIFLSA of $\mathfrak{sl}(2,\mathbb{R})$.

Step 3: Failure of the ideal condition (I3). Take $m_1=h\in C_1$ and $m_2=e\in C_2$. Then, $[h,e]=2e\in C_2$. The ideal condition requires $r_{\mathcal{E}}^{+}\left([h,e]\right)\ge \max \{0.60,0.30\}=0.60$, but $r_{\mathcal{E}}^{+}\left(2e\right)=0.30\ngeqq 0.60$. Hence, $\mathcal{E}$ is not a BCIFLI of $\mathfrak{sl}(2,\mathbb{R})$.

Example 2

(Three-Level BCIFLI on $\mathfrak{t}(2,\mathbb{R})$). Consider the solvable Lie algebra $M=\mathfrak{t}(2,\mathbb{R})$ of upper triangular $2\times 2$ real matrices with bracket $[A,B]=AB-BA$, with basis $e_{11},e_{22},e_{12}$, and nonzero brackets $[e_{11},e_{12}]=e_{12}$, $[e_{22},e_{12}]=-e_{12}$.

Partition M into three classes: $D_0=\left\{0_M\right\}$, $D_1=\mathrm{span}\left\{e_{12}\right\}\setminus \left\{0_M\right\}$, $D_2=M\setminus \mathrm{span}\left\{e_{12}\right\}$.

Define $\mathcal{F}=({\mu }_{\mathcal{F}}^{+},{\nu }_{\mathcal{F}}^{+},{\mu }_{\mathcal{F}}^{-},{\nu }_{\mathcal{F}}^{-})$ as as given in Table 4.

Table 4. Component values for the BCIFLI in Example 2.

Class	${\mathit{\mu }}^{+}\left(\mathit{m}\right)$	${\mathit{\nu }}^{+}\left(\mathit{m}\right)$	${\mathit{\mu }}^{-}\left(\mathit{m}\right)$	${\mathit{\nu }}^{-}\left(\mathit{m}\right)$
$D_0$	$0.90+0.80i$	$0.05+0.05i$	$-0.50-0.40i$	$-0.05-0.04i$
$D_1$	$0.70+0.60i$	$0.15+0.15i$	$-0.30-0.24i$	$-0.15-0.12i$
$D_2$	$0.40+0.30i$	$0.35+0.35i$	$-0.10-0.08i$	$-0.30-0.24i$

Step 1: Constraint verification. The BCIFS constraints are verified in Table 5.

Table 5. BCIFS constraint verification for Example 2.

Class	${\mathit{r}}^{+}+{\mathit{k}}^{+}$	${\mathit{w}}_{{\mathit{r}}^{+}}+{\mathit{w}}_{{\mathit{k}}^{+}}$	${\mathit{r}}^{-}+{\mathit{k}}^{-}$	${\mathit{w}}_{{\mathit{r}}^{-}}+{\mathit{w}}_{{\mathit{k}}^{-}}$
$D_0$	$0.95\le 1$	$0.85\le 1$	$-0.55\in [-1,0]$	$-0.44\in [-1,0]$
$D_1$	$0.85\le 1$	$0.75\le 1$	$-0.45\in [-1,0]$	$-0.36\in [-1,0]$
$D_2$	$0.75\le 1$	$0.65\le 1$	$-0.40\in [-1,0]$	$-0.32\in [-1,0]$

Step 2: Verification of ideal condition (I3). The $r^{+}$ and $r^{-}$ components are checked in Table 6.

Table 6. Ideal bracket verification ($r^{+}$ and $r^{-}$ components) for Example 2.

${\mathit{m}}_1\in$	${\mathit{m}}_2\in$	Bracket ∈	${\mathit{r}}^{+}$ Check (≥max)	${\mathit{r}}^{-}$ Check (≤min)
$D_1$	$D_1$	$D_0$	$0.90\ge 0.70$	$-0.50\le -0.30$
$D_2$	$D_1$	$D_0$ or $D_1$	≥0.70	≤−0.30
$D_2$	$D_2$	$D_0$ or $D_1$	≥0.40	≤−0.10

All ideal conditions are satisfied for every pair $(m_1,m_2)\in M\times M$, with no restriction. Therefore, $\mathcal{F}$ is a BCIFLI of $\mathfrak{t}(2,\mathbb{R})$.

Since $\mathcal{F}$ has nonzero bipolar components (${\mu }_{\mathcal{F}}^{-}\not\equiv 0$, ${\nu }_{\mathcal{F}}^{-}\not\equiv 0$), it is not a CIFLI [14], witnessing the strict inclusion BCIFLI $⊋$ CIFLI.

5. Homomorphisms

Definition 12.

Let $f:M_1\to M_2$ be a Lie algebra homomorphism and let $\mathcal{P}\in \mathcal{BCIF}\left(M_2\right)$. The inverse image of $\mathcal{P}$ under f is the BCIFS $f^{-1}\left(\mathcal{P}\right)\in \mathcal{BCIF}\left(M_1\right)$ defined componentwise by $r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(m\right)=r_{\mathcal{P}}^{+}\left(f\left(m\right)\right)$, and analogously for all other components.

Theorem 1

(Inverse Image of a BCIFLSA). Let $f:M_1\to M_2$ be a Lie algebra homomorphism. If $\mathcal{P}$ is a BCIFLSA of $M_2$, then $f^{-1}\left(\mathcal{P}\right)$ is a BCIFLSA of $M_1$.

Proof. 

We verify (S1)–(S3) for $f^{-1}\left(\mathcal{P}\right)$, demonstrating the $r^{+}$ component; all others follow identically or dually.

Condition (S1). Since f is linear: $r_{f^{-1}\left(\mathcal{P}\right)}^{+}(m_1+m_2)=r_{\mathcal{P}}^{+}(f\left(m_1\right)+f\left(m_2\right))\ge \min \{r_{\mathcal{P}}^{+}\left(f\left(m_1\right)\right),r_{\mathcal{P}}^{+}\left(f\left(m_2\right)\right)\}=\min \{r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(m_1\right),r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(m_2\right)\}$.

Condition (S2). $r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(km\right)=r_{\mathcal{P}}^{+}\left(kf\left(m\right)\right)\ge r_{\mathcal{P}}^{+}\left(f\left(m\right)\right)=r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(m\right)$.

Condition (S3). Since f is a Lie homomorphism: $r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left([m_1,m_2]\right)=r_{\mathcal{P}}^{+}\left([f\left(m_1\right),f\left(m_2\right)]\right)\ge \min \{r_{\mathcal{P}}^{+}\left(f\left(m_1\right)\right),r_{\mathcal{P}}^{+}\left(f\left(m_2\right)\right)\}$. □

Corollary 2

(Inverse Image of a BCIFLI). If $\mathcal{P}$ is a BCIFLI of $M_2$, then $f^{-1}\left(\mathcal{P}\right)$ is a BCIFLI of $M_1$.

Proof. 

Conditions (S1) and (S2) follow as in Theorem 1. For (I3), since f is a Lie homomorphism:

$$r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left([m_1,m_2]\right)=r_{\mathcal{P}}^{+}\left([f\left(m_1\right),f\left(m_2\right)]\right)\ge \max \left\{r_{\mathcal{P}}^{+}\left(f\left(m_1\right)\right),r_{\mathcal{P}}^{+}\left(f\left(m_2\right)\right)\right\}=\max \{r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(m_1\right),r_{f^{-1}\left(\mathcal{P}\right)}^{+}\left(m_2\right)\}.$$

□

Definition 13.

Let $f:M_1\to M_2$ be a Lie algebra homomorphism and $\mathcal{E}\in \mathcal{BCIF}\left(M_1\right)$. The image $f\left(\mathcal{E}\right)$ on $f\left(M_1\right)$ is defined for $n\in f\left(M_1\right)$ by:

$$\begin{array}{cccc}\hfill r_{f\left(\mathcal{E}\right)}^{+}\left(n\right)& =\underset{m\in f^{-1}\left(\left\{n\right\}\right)}{\sup }{r}_{\mathcal{E}}^{+}\left(m\right),\hfill & \hfill k_{f\left(\mathcal{E}\right)}^{+}\left(n\right)& =\underset{m\in f^{-1}\left(\left\{n\right\}\right)}{\inf }{k}_{\mathcal{E}}^{+}\left(m\right),\hfill \\ \hfill r_{f\left(\mathcal{E}\right)}^{-}\left(n\right)& =\underset{m\in f^{-1}\left(\left\{n\right\}\right)}{\inf }{r}_{\mathcal{E}}^{-}\left(m\right),\hfill & \hfill k_{f\left(\mathcal{E}\right)}^{-}\left(n\right)& =\underset{m\in f^{-1}\left(\left\{n\right\}\right)}{\sup }{k}_{\mathcal{E}}^{-}\left(m\right),\hfill \end{array}$$

and analogously for $w_{r^{+}},w_{k^{+}},w_{r^{-}},w_{k^{-}}$.

Lemma 1

(Constraint Preservation for the Image). Let $\mathcal{E}\in \mathcal{BCIF}\left(M_1\right)$ and $f:M_1\to M_2$, a Lie algebra homomorphism. Then, $f\left(\mathcal{E}\right)\in \mathcal{BCIF}\left(f\left(M_1\right)\right)$.

Proof. 

Let $S=f^{-1}\left(\left\{n\right\}\right)$. For the positive constraint: for every $m\in S$, $r_{\mathcal{E}}^{+}\left(m\right)\le 1-k_{\mathcal{E}}^{+}\left(m\right)$. Taking sup on the left: $r_{f\left(\mathcal{E}\right)}^{+}\left(n\right)\le 1-{\inf }_{m\in S}{k}_{\mathcal{E}}^{+}\left(m\right)=1-k_{f\left(\mathcal{E}\right)}^{+}\left(n\right)$. For the negative constraint: $r_{\mathcal{E}}^{-}\left(m\right)\le -k_{\mathcal{E}}^{-}\left(m\right)$, so $r_{f\left(\mathcal{E}\right)}^{-}\left(n\right)=\inf r_{\mathcal{E}}^{-}\left(m\right)\le -\sup k_{\mathcal{E}}^{-}\left(m\right)=-k_{f\left(\mathcal{E}\right)}^{-}\left(n\right)$, giving $r_{f\left(\mathcal{E}\right)}^{-}\left(n\right)+k_{f\left(\mathcal{E}\right)}^{-}\left(n\right)\le 0$. The lower bound $\ge -1$ follows analogously. □

Remark 4.

Without this lemma, Definition 13 would not by itself guarantee that $f\left(\mathcal{E}\right)\in \mathcal{BCIF}\left(f\left(M_1\right)\right)$: since the image is built by taking a supremum/infimum over a preimage fiber, the eight resulting components could, in principle, fall outside the ranges required by constraints (1)–(2). The proof rules this out via the identity ${\sup }_{m\in S}(1-k\left(m\right))=1-{\inf }_{m\in S}k\left(m\right)$, a genuine (if short) argument rather than a direct substitution into the definitions.

Theorem 2

(Image of a BCIFLSA). If $\mathcal{E}$ is a BCIFLSA of $M_1$, then $f\left(\mathcal{E}\right)$ is a BCIFLSA of $f\left(M_1\right)$.

Proof. 

Feasibility is guaranteed by Lemma 1. For (S1): let $n_1,n_2\in f\left(M_1\right)$ and $m_j\in f^{-1}\left(\left\{n_j\right\}\right)$. Then, $r_{f\left(\mathcal{E}\right)}^{+}(n_1+n_2)\ge r_{\mathcal{E}}^{+}(m_1+m_2)\ge \min \{r_{\mathcal{E}}^{+}\left(m_1\right),r_{\mathcal{E}}^{+}\left(m_2\right)\}$. Taking sup yields the result. Conditions (S2) and (S3) follow by the same argument. □

Corollary 3

(Image of a BCIFLI). If $\mathcal{E}$ is a BCIFLI of $M_1$, then $f\left(\mathcal{E}\right)$ is a BCIFLI of $f\left(M_1\right)$.

Remark 5.

Unlike Theorem 2, whose proof follows directly by taking suprema, the ideal case cannot be verified the same way because max does not distribute over sup in the required direction. The proof below instead argues by contradiction, fixing a threshold t strictly between the two sides of the inequality and using a near-supremum element q with $r_{\mathcal{E}}^{+}\left(q\right)>t$.

Proof. 

Suppose for contradiction $r_{f\left(\mathcal{E}\right)}^{+}\left([n_1,n_2]\right)<\max \{r_{f\left(\mathcal{E}\right)}^{+}\left(n_1\right),r_{f\left(\mathcal{E}\right)}^{+}\left(n_2\right)\}$. Choose t between them and $q\in M_1$ with $f\left(q\right)=n_1$, $r_{\mathcal{E}}^{+}\left(q\right)>t$. For any $s\in f^{-1}\left(\left\{n_2\right\}\right)$: $r_{f\left(\mathcal{E}\right)}^{+}\left([n_1,n_2]\right)\ge r_{\mathcal{E}}^{+}\left([q,s]\right)\ge \max \{r_{\mathcal{E}}^{+}\left(q\right),r_{\mathcal{E}}^{+}\left(s\right)\}\ge r_{\mathcal{E}}^{+}\left(q\right)>t$, a contradiction. □

6. Level Subsets and Characterizations

Definition 14

(Level Subset). Let $\mathcal{E}\in \mathcal{BCIF}\left(M\right)$ and ${\alpha }^{+},{\beta }^{+},{\alpha }_k^{+},{\beta }_k^{+}\in [0,1]$, ${\alpha }^{-},{\beta }^{-},{\alpha }_k^{-},{\beta }_k^{-}\in [-1,0]$. The level subset is:

$$\begin{array}{cc}\hfill {\mathcal{E}}_{[\mathit{\alpha },\mathit{\beta }]}=\{m\in M:& r_{\mathcal{E}}^{+}\left(m\right)\ge {\alpha }^{+},w_{r_{\mathcal{E}}^{+}}\left(m\right)\ge {\beta }^{+},k_{\mathcal{E}}^{+}\left(m\right)\le {\alpha }_k^{+},w_{k_{\mathcal{E}}^{+}}\left(m\right)\le {\beta }_k^{+},\hfill \\ \hfill & r_{\mathcal{E}}^{-}\left(m\right)\le {\alpha }^{-},w_{r_{\mathcal{E}}^{-}}\left(m\right)\le {\beta }^{-},k_{\mathcal{E}}^{-}\left(m\right)\ge {\alpha }_k^{-},w_{k_{\mathcal{E}}^{-}}\left(m\right)\ge {\beta }_k^{-}\}.\hfill \end{array}$$

Theorem 3

(Level Subset Characterization). A BCIFS $\mathcal{E}$ on M is a BCIFLSA (resp. BCIFLI) of M if and only if every nonempty level subset ${\mathcal{E}}_{[\mathit{\alpha },\mathit{\beta }]}$ is a Lie subalgebra (resp. Lie ideal) of M.

Proof. 

$(\Rightarrow )$ Let $L={\mathcal{E}}_{[\mathit{\alpha },\mathit{\beta }]}$ be nonempty and $m_1,m_2\in L$, $k\in K$. According to the BCIFLSA conditions, $m_1+m_2$, $km$, and $[m_1,m_2]$ (resp. $[m,n]$ for all $m\in M$) belong to L.

$(\Leftarrow )$ Given $m_1,m_2\in M$, set all thresholds to the componentwise extremum of the values at $m_1,m_2$. Both elements lie in this level subset (a Lie subalgebra/ideal), so $m_1+m_2$ and $[m_1,m_2]$ also belong, yielding the required inequalities. □

Remark 6.

Unlike a classical α-level set of a single membership function, the level subset of Definition 14 is cut simultaneously by eight thresholds spanning both polarities and both real and imaginary parts. The converse direction of the proof above requires that all eight thresholds be set consistently to the pointwise componentwise extremum of $m_1$ and $m_2$ at once, rather than to a single scalar α.

Corollary 4.

$f^{-1}\left({\mathcal{P}}_{[\alpha ,\beta ]}\right)={\left(f^{-1}\left(\mathcal{P}\right)\right)}_{[\mathit{\alpha },\mathit{\beta }]}$.

Definition 15

(Sum of BCIFSs). Let $\mathcal{E},\mathcal{F}\in \mathcal{BCIF}\left(M\right)$. Their sum is defined for all $m\in M$ by:

$$\begin{array}{cc}\hfill r_{(\mathcal{E}+\mathcal{F})}^{+}\left(m\right)& =\underset{m=m_1+m_2}{\sup }\min \{r_{\mathcal{E}}^{+}\left(m_1\right),r_{\mathcal{F}}^{+}\left(m_2\right)\},\hfill \\ \hfill k_{(\mathcal{E}+\mathcal{F})}^{+}\left(m\right)& =\underset{m=m_1+m_2}{\inf }\max \{k_{\mathcal{E}}^{+}\left(m_1\right),k_{\mathcal{F}}^{+}\left(m_2\right)\},\hfill \\ \hfill r_{(\mathcal{E}+\mathcal{F})}^{-}\left(m\right)& =\underset{m=m_1+m_2}{\inf }\max \{r_{\mathcal{E}}^{-}\left(m_1\right),r_{\mathcal{F}}^{-}\left(m_2\right)\},\hfill \\ \hfill k_{(\mathcal{E}+\mathcal{F})}^{-}\left(m\right)& =\underset{m=m_1+m_2}{\sup }\min \{k_{\mathcal{E}}^{-}\left(m_1\right),k_{\mathcal{F}}^{-}\left(m_2\right)\},\hfill \end{array}$$

and analogously for $w_{r^{+}},w_{k^{+}},w_{r^{-}},w_{k^{-}}$.

Theorem 4

(Sum of BCIFLIs). Let $\mathcal{E}$ and $\mathcal{F}$ be BCIFLIs of M. Then $\mathcal{E}+\mathcal{F}$ is a BCIFLI of M.

Remark 7.

Unlike the additivity and scalar-multiplication cases below, the ideal-bracket case is not a routine componentwise verification: it requires expanding $[m_1,m_2]$ by bilinearity into four bracket terms and regrouping them into a new admissible decomposition $(u,v)=\left(\right[a,c]+[a,d],[b,c]+[b,d\left]\right)$ of $[m_1,m_2]$ before Definition 15 can be applied.

Proof. 

We work with the $r^{+}$ component; the remaining seven follow by identical arguments.

Condition (S1). Fix $\epsilon >0$ and choose near-optimal decompositions $m=a+b$ and $n=c+d$. Since $m+n=(a+c)+(b+d)$ is valid, and using (S1) for $\mathcal{E},\mathcal{F}$:

$$r_{(\mathcal{E}+\mathcal{F})}^{+}(m+n)\ge \min \{r_{\mathcal{E}}^{+}\left(a\right),r_{\mathcal{E}}^{+}\left(c\right),r_{\mathcal{F}}^{+}\left(b\right),r_{\mathcal{F}}^{+}\left(d\right)\}\ge \min \{r_{(\mathcal{E}+\mathcal{F})}^{+}\left(m\right),r_{(\mathcal{E}+\mathcal{F})}^{+}\left(n\right)\}-\epsilon .$$

Since $\epsilon >0$ is arbitrary, the result follows.

Condition (S2). For any decomposition $m=a+b$, $(ka,kb)$ decomposes $km$, and $r_{\mathcal{E}}^{+}\left(ka\right)\ge r_{\mathcal{E}}^{+}\left(a\right)$, $r_{\mathcal{F}}^{+}\left(kb\right)\ge r_{\mathcal{F}}^{+}\left(b\right)$. Taking sup gives the result.

Condition (I3). Suppose for contradiction $r_{(\mathcal{E}+\mathcal{F})}^{+}\left([m_1,m_2]\right)<\max \{r_{(\mathcal{E}+\mathcal{F})}^{+}\left(m_1\right),r_{(\mathcal{E}+\mathcal{F})}^{+}\left(m_2\right)\}$, with threshold t between them. Choose $a,b$ with $m_1=a+b$ and $\min \{r_{\mathcal{E}}^{+}\left(a\right),r_{\mathcal{F}}^{+}\left(b\right)\}>t$. For any decomposition $m_2=c+d$, bilinearity gives $[m_1,m_2]=[a,c]+[a,d]+[b,c]+[b,d]=u+v$ where $u=[a,c]+[a,d]$, $v=[b,c]+[b,d]$. By the ideal condition and (S1) for $\mathcal{E}$: $r_{\mathcal{E}}^{+}\left(u\right)\ge r_{\mathcal{E}}^{+}\left(a\right)$. Similarly $r_{\mathcal{F}}^{+}\left(v\right)\ge r_{\mathcal{F}}^{+}\left(b\right)$. Hence, $r_{(\mathcal{E}+\mathcal{F})}^{+}\left([m_1,m_2]\right)\ge \min \{r_{\mathcal{E}}^{+}\left(u\right),r_{\mathcal{F}}^{+}\left(v\right)\}\ge \min \{r_{\mathcal{E}}^{+}\left(a\right),r_{\mathcal{F}}^{+}\left(b\right)\}>t$, a contradiction. □

Theorem 5

(Homomorphism and Sum). Let $f:M_1\to M_2$ be a surjective Lie algebra homomorphism and let $\mathcal{E},\mathcal{F}$ be BCIFLIs of $M_1$. Then, $f(\mathcal{E}+\mathcal{F})=f\left(\mathcal{E}\right)+f\left(\mathcal{F}\right)$.

Proof. 

For the $r^{+}$ component and $n\in M_2$: since f is linear, every decomposition $m_1+m_2=m$ with $f\left(m\right)=n$ gives $f\left(m_1\right)+f\left(m_2\right)=n$, so $r_{f(\mathcal{E}+\mathcal{F})}^{+}\left(n\right)\le r_{f\left(\mathcal{E}\right)+f\left(\mathcal{F}\right)}^{+}\left(n\right)$. Conversely, surjectivity of f guarantees that for any $q+s=n$, we can find $m_1,m_2$ with $f\left(m_1\right)=q$, $f\left(m_2\right)=s$, giving the reverse inequality. Hence, equality holds, and the argument extends to all eight components. □

7. Hierarchy of Special Cases

The BCIFLA framework strictly generalizes all previously known fuzzy Lie algebra theories. The specializations are summarized in Table 7.

Table 7. Hierarchy of special cases recovered from BCIFLA.

Specialization of BCIFLA	Recovered Structure
${\mu }^{-}\equiv {\nu }^{-}\equiv 0$	Complex Intuitionistic Fuzzy Lie Algebra [14]
${\mu }^{-}\equiv {\nu }^{-}\equiv 0$ + all imaginary parts $\equiv 0$	Intuitionistic Fuzzy Lie Algebra [12]
${\mu }^{-}\equiv {\nu }^{-}\equiv 0$ + im. parts $\equiv 0$ + ${\nu }^{+}\equiv 1-{\mu }^{+}$	Fuzzy Lie Algebra [11]
${\nu }^{+}\equiv {\nu }^{-}\equiv 0$ + all imaginary parts $\equiv 0$	Bipolar Fuzzy Lie Algebra [15]

The resulting hierarchy of strict inclusions is as follows:

$$\mathrm{BCIFLA}⊋\mathrm{CIFLA}⊋\mathrm{IFLA}⊋\mathrm{FLA},\mathrm{BCIFLA}⊋\mathrm{Bipolar}\mathrm{FLA}.$$

All inclusions are proper: Example 1 witnesses BCIFLA ⊋ CIFLA (since ${\mu }_{\mathcal{E}}^{-}\not\equiv 0$). Example 2 witnesses BCIFLA ⊋ Bipolar FLA (since ${\nu }_{\mathcal{F}}^{+}\not\equiv 0$). Witnesses for the remaining inclusions appear in [12,14].

8. Conclusions

We have introduced bipolar complex intuitionistic fuzzy Lie algebras (BCIFLAs), grounded in the Cartesian coordinate BCIFS framework of [10]. The main results are as follows:

(1)

Well-defined BCIFLA concepts with elementary properties, including the key observation that the zero element attains extremal membership values (Proposition 2) and all component functions are symmetric (Proposition 3).

(2)

Closure under arbitrary meets: BCIFLSA(M) and BCIFLI(M) form meet-semilattices (Proposition 4, Corollary 1).

(3)

A constraint-preservation lemma (Lemma 1) for the image construction.

(4)

Full preservation under Lie algebra homomorphisms: preimages (Theorem 1, Corollary 2) and images (Theorem 2, Corollary 3).

(5)

A level-subset characterization (Theorem 3) with compatibility formula (Corollary 4).

(6)

Closure of BCIFLIs under sums (Theorem 4) with surjective homomorphism compatibility (Theorem 5).

(7)

A strict hierarchy encompassing all known fuzzy Lie algebra frameworks (Section 7).

(8)

Non-degenerate three-level examples on $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{t}(2,\mathbb{R})$ demonstrating the structural gap between BCIFLSAs and BCIFLIs.

Future directions include: isomorphism theorems for BCIFLAs following [30]; extensions to n-Lie algebras [31] and Hom-Lie algebras [32]; bipolar complex Pythagorean fuzzy Lie algebras; extension of the MCDM framework of [10] to a Lie-algebraic interaction structure; and applications to quantum information theory.