The Complexity of Von Neumann Coordinatization; 2-distributive Lattices

A complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are first-order. Nevertheless, we prove that coordinatizability of complemented modular lattices is not firstorder, even for countable 2-distributive lattices, thus solving a 1960 problem of B. Jónsson. This is established by expressing the coordinatizability of a countable 2-distributive lattice L as a separation property of the set of all finite homomorphic images of L. This separation property is in Lω1,ω but not first-order. In the uncountable case, it is no longer sufficient to characterize coordinatizability. In fact, we prove that there is no L∞,∞ statement equivalent to coordinatizability. We also prove that the class of coordinatizable lattices is not closed under countable directed unions, thus solving another problem of B. Jónsson from 1962.