Continuous geometry
Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.A continuous geometry is a lattice L with the following properties This section summarizes some of the results of von Neumann (1998, Part I).for some positive integer n. Two elements of L have the same image under D if and only if they are perspective, so it gives an injection from the equivalence classes to a subset of the unit interval.This can be restated as saying that the subspaces in the projective geometry correspond to the principal right ideals of a matrix algebra over a division ring.More precisely if the lattice has order n then the von Neumann regular ring can be taken to be an n by n matrix ring Mn(R) over another von Neumann regular ring R. Here a complemented modular lattice has order n if it has a homogeneous basis of n elements, where a basis is n elements a1, ..., an such that ai ∧ aj = 0 if i ≠ j, and a1 ∨ ... ∨ an = 1, and a basis is called homogeneous if any two elements are perspective.