Von Neumann coordinatization is not first-order

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: Lattice; complemented; modular; 2-distributive; coordinatizable; ring; von Neumann regular; sheaf; diamond; center; trace; trace-primary; trace-separated

Domaines: Mathématiques générales [math.GM]; Anneaux et algèbres [math.RA]; Logique [math.LO]; Topologie générale [math.GN]

Friedrich Wehrung. Von Neumann coordinatization is not first-order. Journal of Mathematical Logic, 2006, 6 (1), pp.1--24. ⟨10.1142/S0219061306000499⟩. ⟨hal-00002843v3⟩