A NOTE ON LATTICES OF EUCLIDEAN SUBSPACES

Any nonvoid lattice of subspaces from R(n) is known to be a complete lattice, and hence it has a largest and smallest element. Here we show that for a specific class of subspaces also the converse is true. If this class has a largest and a smallest element, then it is a complete lattice. Within the context of algebraic Riccati equations, it follows that the usual classes of real symmetric and positive semidefinite solutions are lattices if and only if these classes contain extremal elements, and if this is the case, then these lattices are modular, yet not necessarily distributive, as is demonstrated by a counterexample.