On the combinatorics of Galois numbers

We define interval decompositions of the lattice of subspaces of a finite-dimensional vector space. We show that such a decomposition exists if and only if there exists a family of linear forms with certain properties. As applications we prove that all finite-dimensional real vector spaces admit an interval decomposition, while GF(2)nGF(2)n has an interval decomposition if and only if n≤4n≤4. On the other hand, we present an interval decomposition of GF(3)5GF(3)5. This partially answers a question of Faigle and Kruse (2004) [1] and [4].