Dimension and embedding theorems for geometric lattices

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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 ε{lunate} 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. © 1974.

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Kantor, W. M. (1974). Dimension and embedding theorems for geometric lattices. Journal of Combinatorial Theory, Series A, 17(2), 173–195. https://doi.org/10.1016/0097-3165(74)90005-3

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