Invariant linear manifolds for CSL-algebras and nest algebras

Every invariant linear manifold for a CSL-algebra, $\operatorname{Alg} \mathcal{L}$, is a closed subspace if, and only if, each non-zero projection in $\mathcal{L}$ is generated by finitely many atoms associated with the projection lattice. When $\mathcal{L}$is a nest, this condition is equivalent to the condition that every non-zero projection in $\mathcal{L}$ has an immediate predecessor ( $\mathcal{L}^{\perp}$ is well ordered). The invariant linear manifolds of a nest algebra are totally ordered by inclusion if, and only if, every non-zero projection in the nest has an immediate predecessor.