Orthogonality and complementation in the lattice of subspaces of a finite vector space

Abstract

We investigate the lattice $ L( V)$ of subspaces of an $m$-dimensional vector space $ V$ over a finite field ${\rm GF}(q)$ with a prime power $q=p^n$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the 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 $p\nmid m$ we show that $ L( V)$ contains a $(2^m+2)$-element (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.

Keywords

• vector space
• lattice of subspaces
• finite field
• orthomodular lattice
• modular lattice
• boolean lattice
• complementation