Orthogonality and complementation in the lattice of subspaces of a finite-dimensional vector space over a finite field

Abstract

We investigate the lattice L(V) of subspaces of an m-dimensional vector space V over a finite field GF(q) with q being the n-th power of a prime p. It is well-known that this lattice is modular and that orthogonality is an antitone involution. The lattice L(V) satisfies the Chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when L(V) is orthomodular. For m > 1 and m not divisible by p we show that L(V) contains a certain (non-Boolean) orthomodular lattice as a subposet. Finally, for q being a prime and m = 2 we characterize orthomodularity of L(V) by a simple condition.