Joins of weakly ascendant subalgebras of Lie algebras (Q1063678)

The main idea in this paper is to find classes of Lie algebras which ensure that for each member of the class, the set \({\mathcal S}_ L(wasc)\) of all weakly ascendant subalgebras of L is a sublattice (complete sublattice) of the lattice \({\mathcal S}_ L(\leq)\) of all subalgebras of L. A similar investigation is carried out for the set \({\mathcal S}_ L(asc)\) of all ascendant subalgebras of L. Some of the classes involved in the investigation are A [resp. N, F] the class of all abelian [resp. nilpotent, finite-dimensional] Lie algebras, \(ZA\) [resp. \(\acute E(\triangleleft)A]\) the class of all hypercentral [resp. hyperabelian] Lie algebras, \(Min-\triangleleft\) [resp. \(Max-\triangleleft]\) the class of Lie algebras which satisfy the minimal [resp. maximal] condition for ideals, \(m_ 1(wasc)\), \(m_ 2(wasc)\), \(m(wasc)\) [resp. \(m_ 1(asc)\), \(m_ 2(asc)\), \(m(asc)\)] the classes of all Lie algebras in which every weakly ascendant [resp. ascendant] subalgebra has weakly ascendant [resp. ascendant] index less than or equal one, less than or equal two, less than \(\omega\), \(D(wasc)\) [resp. \(D(asc)\)] the class of all Lie algebras in which every subalgebra is a weakly ascendant [resp. ascendant] subalgebra, \(A_ 1\) the class of all Lie algebras L such that either \(L\in A\) or L is metabelian with \(\dim L/L^ 2=1.\) \({\mathcal S}_ L(wasc)\) is a sublattice [resp. complete sublattice] if L belongs to any one of the classes: \(ZA\), \(D(wasc)A\), \(Nm_ 1(wasc)\), \(N(m(wasc)\cap A_ 1)\) [resp. \(m_ 2(asc)\cap \acute E(\triangleleft)A\), \(F\cap (NA_ 1)\), \((F\cap N)(Min-\triangleleft \cap Max-\triangleleft \cap m_ 1(wasc))]\). \({\mathcal S}_ L(asc)\) is a sublattice [resp. complete sublattice] if L belongs to any of the classes: \(ZA\), \(D(asc)A\cap \acute E(\triangleleft)A\), \((F\cap N)m_ 1(asc)\) [resp. \(m_ 2(asc)\), \(F\cap (NA_ 1)\), \((F\cap N)(Min- \triangleleft \cap Max-\triangleleft \cap m_ 1(asc))]\).