Lie algebras satisfying the weak minimal condition on ideals (Q1079645)

The author defines the Lie algebra analogue of the weak minimal condition on subgroups of a group, and investigates Lie algebras satisfying these conditions. Let \({\mathfrak X}\) be a class of Lie algebras and \(\Delta\) be any of the relations \(\leq\), \(\triangleleft^{\sigma}\), si, asc., ser. A Lie algebra L is said to satisfy the weak minimal condition on \(\Delta\) \({\mathfrak X}\)-subalgebras (wmin-\(\Delta\) \({\mathfrak X})\) if for any descending chain \(H_ 1\supseteq H_ 2\supseteq...\supseteq H_ i\supseteq H_{i+1}\supseteq...\) of \(\Delta\) \({\mathfrak X}\)-subalgebras of L there exists \(r\in N\) such that the dimension of the vector space \(H_ i/H_{i+1}\) is finite for any \(i\geq r\). Then wmax-\(\Delta\) \({\mathfrak X}\) is similarly defined. He gives several conditions under which Lie algebras satisfying wmin-\(\Delta\) \({\mathfrak X}\) or wmax-\(\Delta\) \({\mathfrak X}\) are finite-dimensional. He also proves that an ideally soluble hypoabelian Lie algebra satisfying \(w\min\)-\(\triangleleft\) is soluble, a non-abelian ideally finite Lie algebra satisfying the weak minimal condition on non-abelian 2-step subideals satisfies the minimal condition on ideals. Furthermore, he proves that if \({\mathfrak X}\) is any one of the classes abelian, nilpotent or soluble Lie algebras, \(\Delta\) is one of the relations si, asc., \(\triangleleft^{\alpha}\) (\(\alpha\) an infinite ordinal), then \(w\min - \Delta {\mathfrak X}=w\max -\Delta {\mathfrak X}=Min-\Delta {\mathfrak X}.\)