A structural characterization of non-Arguesian lattices

This is the first of a planned series of papers on the structure of non-Arguesian modular lattices. Apart from the (subspace lattices of) non-Arguesian projective planes, the best known examples of such lattices are obtained via the Hall-Dilworth construction by badly gluing together two projective planes of the same order. Our principal result shows that every non-Arguesian modular lattice L retains some of the flavor of these examples: There exist in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a classical non-Arguesian configuration.Research supported by NSERC Operating Grant A8190.Research supported by NSF Grant DMS-8300107.