On a theorem by Benson and Conway
Descriptions of a modular lattice of finite height as subspace lattice of a geometry on the ordered point set of join irreducibles have been given by \textit{U. Faigle} and \textit{C. Herrmann} [Trans. Am. Math. Soc. 266, 319--332 (1981; Zbl 0466.51001)], \textit{D. J. Benson} and \textit{J. H. Conway} [J. Pure Appl. Algebra 37, 111--116 (1985; Zbl 0575.06008)], and \textit{C. Herrmann} and \textit{M. Wild} [J. Algebra 136, 17--36 (1991; Zbl 0725.16014)]. The case of infinite height has been dealt with by \textit{C. Herrmann, D. Pickering} and \textit{M. Roddy} [Algebra Univers. 31, 365--396 (1994; Zbl 0816.06008)]. In particular, the congruence structure of the lattice and subdirect decompositions can be captured in these geometries via connected components. Including all collinearities leads to a handy axiomatization. For the purposes of representation theory and for calculations a minimum number is more convenient: it suffices to consider, for each \(a\) which is a join of two points, a single maximal set of points any two of which have join \(a\), a so-called (dotted) line. The author reviews these results and techniques and points out how they can be used in the computation of submodule lattices and their decomposition into directly indecomposables.
scientific article; zbMATH DE number 5078557