Fuzzy Nilpotent Lie Algebras: Bases, Isomorphisms, and a Measure of Nilpotency

This paper investigates the structure of fuzzy Lie subalgebras, with particular emphasis on isomorphisms and nilpotency. Building on two prior conference contributions, one of which established foundational results on fuzzy bases of Lie algebras, we develop here a more complete and unified treatment of these themes. We introduce a notion of isomorphism between fuzzy Lie subalgebras based on the transfer principle via t-cut sets, and we prove that isomorphic fuzzy Lie subalgebras necessarily share the same nilpotency measure. The central contribution of the paper is a fuzzy measure of nilpotency 𝑁⁡(𝜇)∈[0,1], defined for any non-constant fuzzy Lie subalgebra 𝜇 of a Lie algebra 𝔤. This invariant equals 1 precisely when 𝜇 is fuzzy nilpotent, and decreases as the subalgebra departs from nilpotency. We show that nilpotency of the underlying Lie algebra implies 𝑁⁡(𝜇)=1, but that the converse fails in general, as witnessed by an explicit counterexample.