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

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