Continuous geometry

Summary

A continuous geometry is a complete, complemented, modular lattice in which meets and joins are continuous along chains, and which is irreducible. John von Neumann introduced continuous geometries as analogues of projective geometries. In them the dimension can take every value in $[0, 1]$, not only integer values. The lattice of projections of a type II$_1$ factor is the motivating example. Every continuous geometry carries a dimension function, and von Neumann related these lattices to von Neumann regular rings.