id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-6489	Balamurugan, M. ; Ellammal, G. ; Iampan, Aiyared	Tripolar Complex Fuzzy Lie Subalgebras of Lie Algebras	2025	26	.pdf	application/pdf	8070	591	90	Then f ( ⟨⟨ξP̃ג , ξ P ̃ג ⟩⟩ ) (η̃) = sup { ⟨⟨ξP̃ג , ξ P ̃ג ⟩⟩(ϕ̃) | f(ϕ̃) = η̃ } = sup { sup { min { ξP̃ג (ε̃), ξ P ̃ג (ς̃) } | ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃ }} = sup { min { ξP̃ג (ε̃), ξ P ̃ג (ς̃) } | ε̃, ς̃ ∈ L̃, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃ } = sup { min { ξP̃ג (ε̃), ξ P ̃ג (ς̃) } | ε̃, ς̃ ∈ L̃1, [f(ε̃), f(ς̃)] = η̃ } = sup { min { ξP̃ג (ε̃), ξ P ̃ג (ς̃) } | ε̃, ς̃ ∈ L̃1, f(ε̃) = ϖ̃, f(ς̃) = ϑ̃, [ϖ̃, ϑ̃] = η̃ } ≥ sup { min { sup ε̃∈f−1(ϖ̃) ξP̃ג (ε̃), sup ς̃∈f−1(ϑ̃) ξP̃ג (ς̃) } = η̃ } = inf { inf { max { ξÑג (ε̃), ξÑג (ς̃) } | ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃ }} = inf { max { ξÑג (ε̃), ξÑג (ς̃) } | ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃ } = inf { max { ξÑג (ε̃), ξÑג (ς̃) } | ε̃, ς̃ ∈ L̃1, [f(ε̃), f(ς̃)] = η̃ } = inf { max { ξÑג (ε̃), ξÑג (ς̃) } | f(ε̃) = ϖ̃, f(ς̃) = ϑ̃, [ϖ̃, ϑ̃] = η̃ } ≤ inf { max { inf ε̃∈f−1(ϖ̃) ξÑג (ε̃), inf ς̃∈f−1(ϑ̃) ξÑג (ς̃) } | [ϖ̃, ϑ̃] = η̃ } = inf { max { f(ξÑג (ϖ̃)), f(ξÑג (ϑ̃)) } | [ϖ̃, ϑ̃] = η̃ } = ⟨⟨f(ξÑג ), f(ξÑג )⟩⟩(η̃).	cache/ejpam-6489.pdf	txt/ejpam-6489.txt