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The complexity of von Neumann coordinatization; 2-distributive lattices | | Friedrich Wehrung
; | | Date: |
15 Sep 2004 | | Subject: | General Mathematics; Rings and Algebras; Logic; General Topology MSC-class: AMS: 06C20, 06C05, 06B20, 03C10, 03C90, 16E50 | math.GM math.GN math.LO math.RA | | Affiliation: | LMNO | | Abstract: | 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. Jonsson, are first-order. Nevertheless, we prove that coordinatizability of complemented modular lattices is not first-order, even for countable 2-distributive lattices, thus solving a 1960 problem of B. Jonsson. 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 $L_{omega_1,omega}$ but not first-order. In the uncountable case, it is no longer sufficient to characterize coordinatizability. In fact, we prove that there is no $L_{infty,infty}$ 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. Jonsson from 1962. | | Source: | arXiv, math.GM/0409250 | | Services: | Forum | Review | PDF | Favorites | |
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