A proof of the bundle theorem for certain semimodular locally projective lattices of rank 4
Let L be a projective geometry of rank n and let Q be a nonempty set of points of L. Denote \(L(Q)=\{x\in L:\) \(\exists p\in Q\) with \(p\leq x\}\cup \{0\}\). \textit{R. Wille} [Atti Conv. Geom. Combinat. Appl. Perugia 1970, 421-426 (1971; Zbl 0256.50003)] has shown that every semimodular locally projective lattice of rank \(n>4\), such that the whole space is the join of the set of its points, is projectively embeddedable, i.e. is isomorphic to some L(Q). But the conclusion is false for lattices of rank 4. In the paper, the author obtains an affirmative result for geometric locally projective lattices of rank 4 under some additional assumption which requires, roughly speaking, that for each line \(\ell\) there are not too many lines which are on a common plane with \(\ell\), but on no common point.
scientific article; zbMATH DE number 3918430