Une Condition Nécessaire et Suffisante de Plongeabilité pour les Treillis Semi-modulaires

We prove a necessary and sufficient condition for a “non-degenerate” semimodular lattice of rank ⩾5 to be isometrically embeddable in a generalized projective space. The condition essentially requires that all intervals [p, 1], above a point p, are isometrically embeddable in such a way that their embeddings are compatible along lines and planes. The result holds, in particular, for geometric lattices, and thus provides a condition for a matroid to be linear