Complemented modular lattices and projective spaces of infinite dimension

Abstract

Garrett Birkhoff [1](1) has shown that every complemented modular lattice of finite dimension is the direct union of lattices associated with projective geometries of finite dimension. The present paper is an attempt to generalize this characterization to the case of complemented modular lattices and projective spaces in general, with no restriction on the dimensionality. There is one respect in which the infinite-dimensional case is more complicated than the finite-dimensional case. The general projective spaces we consider are atomic; they contain points or atoms. Consequently the lattices associated with them are atomic. But complemented modular lattices need not be atomic, as is shown by the example of continuous geometries. Our procedure is to show that every complemented modular lattice determines a complete atomic complemented modular lattice in which it is imbedded. This extension to an atomic lattice is accomplished by the use of maximal dual ideals. The resulting atomic lattice is then shown to be the direct union of irreducible projective spaces of a particular kind. The final characterization theorem we obtain states that every complemented modular lattice is the subdirect union of projective planes and irreducible projective coordinate spaces. A projective coordinate space is determined by an arbitrary cardinal number (its dimension) and an arbitrary division ring. This result is not as neat as Birkhoff's characterization theorem, since it involves subdirect unions instead of direct unions. On the other hand, the preliminary step of imbedding the original lattice in an atomic lattice (a step which is not needed for finite-dimensional lattices, which are atomic) promises to be of use in connection with the theory of continuous geometries. Continuous geometries have no points, lines, or planes. It seems to have been assumed further that it is not possible to introduce points, lines, and planes into a continuous geometry in a reasonable way. Such an assumption would be incorrect. The introduction is not only possible, but it has the advantage that it allows one to associate a division ring with the continuous geometry by the method of von Staudt. This should lead to a simpler coordinatization of continuous geometries than the coordinatization by regular rings used by von