Von Neumann coordinatization is not first-order
Friedrich Wehrung
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.
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