Maximal conditions for locally finite Lie algebras and double chain conditions (Q1327727)

Let \(F\) and \(LF\) denote the classes of Lie algebras which are finite- dimensional and locally finite respectively and \(L(\text{wser})F\) be the class of Lie algebras in which every finite subset is contained in some finite-dimensional weakly serial subalgebra. Let the symbols \(\text{Max- }\triangleleft\), \(\text{Max-}\triangleleft^ 2\), Max-si, Max-ser and Max-sF denote the classes of Lie algebras satisfying the maximal condition for ideals, 2-step subideals, subideals, serial subalgebras, and finite-dimensional subideals respectively. Define \(\text{Min- }\triangleleft\), \(\text{Min-}\triangleleft^ 2\), Min-si similarly for minimal conditions. The author proves that \(L(\text{wser}) F \cap \text{Max-}\triangleleft = F\) and \(\text{Max-} \triangleleft \cap \text{Min- si} \leq \text{Max-si}\) over any field, and \(LF \cap \text{Max- }\triangleleft^ 2 = LF \cap \text{Max-ser}\) and \(\text{Max- }\triangleleft^ 2 \leq \text{Max-sF}\) over fields of characteristic zero. A method of constructing Lie algebras satisfying neither the minimal nor the maximal condition for 2-step subideals is given and it is proved that there exists a Lie algebra \(A\) over any field such that \(A \in LF \cap \text{Max-} \triangleleft \cap \text{Min-}\triangleleft\) and \(A \notin \text{Max-} \triangleleft^ 2 \cup \text{Min-}\triangleleft^ 2\).