Tripolar Complex Fuzzy Lie Subalgebras of Lie Algebras (European Journal of Pure and Applied Mathematics, 2025)
The tripolar complex fuzzy set ($\mathcal{TCFS}$) is an extension of the bipolar complex fuzzy set ($\mathcal{BCFS}$), which itself generalizes traditional fuzzy sets and bipolar fuzzy sets. In this paper, we further develop this framework by introducing the concept of tripolar complex fuzzy Lie brackets and investigating their algebraic properties. Additionally, we demonstrate that the scalar multiplication and addition of tripolar complex fuzzy Lie subalgebras yield another tripolar complex fuzzy Lie subalgebra. Moreover, we establish that the homomorphic image of a nilpotent (or solvable) tripolar complex fuzzy Lie ideal remains a nilpotent (or solvable) tripolar complex fuzzy Lie ideal. Finally, we establish that every nilpotent tripolar complex fuzzy Lie ideal is solvable.
Keywords: Lie algebra; Tripolar complex fuzzy set; Tripolar complex fuzzy Lie subalgebra; Tripolar complex fuzzy Lie ideal; Nilpotent; Solvable