Von Neumann coordinatization is not first-order

Abstract

A lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. This forces L to be complemented modular. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jonsson, are first-order. Nevertheless, we prove that coordinatizability of lattices is not first-order, by finding a non-coordinatizable lattice K with a coordinatizable countable elementary extension L. This solves a 1960 problem of B. Jonsson. We also prove that there is no L\_{infinity, infinity} statement equivalent to coordinatizability. Furthermore, the class of coordinatizable lattices is not closed under countable directed unions; this solves another problem of B. Jonsson from 1962.