Dimension and embedding theorems for geometric lattices |
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Authors: | William M Kantor |
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Institution: | University of Oregon, Eugene, Oregon 97403 USA |
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Abstract: | Let G be an n-dimensional geometric lattice. Suppose that 1 ? e, f ? n ? 1, e + f ? n, but e and f are not both n ? 1. Then, in general, there are E, F? G with dim E = e, dim F = f, E ? F = 1, and dim E ∧ F = e + f ? n ? 1; any exception can be embedded in an n-dimensional modular geometric lattice M in such a way that joins and dimensions agree in G and M, as do intersections of modular pairs, while each point and line of M is the intersection (in M) of the elements of G containing it. |
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