A characterization of subgroup lattices of finite Abelian groups

We show that every primary lattice can be considered a glueing of intervals having geometric dimension at least 3 and with a skeleton of breadth at most 2. We call this geometric decomposition. In the Arguesian case, we analyse the sub-glueings corresponding to cover preserving sublattices of the skeleton which are 2-element chains or a direct product of 2 such. We show that these admit a cover preserving embedding into the submodule lattice of a finitely generated module over a completely primary uniserial ring. It follows, that a primary Arguesian lattice can be cooordinatized by such module if the skeleton of the geometric decomposition is a chain. This fails due to an example of G.S. Monk [29] if the skeleton has breadth 2. Moreover, there are non-isomorphic eaxmples of this type having isomorphic skeletons and isomorphism between the corresponding intervals. Hence, most of the statements in sections 9 and 10 are wrong. The main results of Antonov and Nazyrova [25, 26] are wrong, too, since the subgroup lattice of Cn pk (k ≥ 2, n ≥ 3) cannot be embedded into the subspace lattice of any vector space [28], Credit for Thm.3.1 below should be also given to [31, 27, 30]. The due Corrigenda and Addenda are given in: On the coordinatizationm of primary arguesian lattices of low geometric dimension, Beitr. Geom.u.Algebra 55, Issue 2 (2014), Page 649–668