Geodesic
Orthogonality and complementation in the lattice of subspaces of a finite vector space
Mathematica Bohemica, Tome 147 (2022) no. 2, pp. 141-153
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We investigate the lattice ${\bf L}({\bf V})$ of subspaces of an $m$-dimensional vector space ${\bf V}$ 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 ${\bf L}({\bf V})$ 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 ${\bf L}({\bf V})$ is orthomodular. For $m>1$ and $p\nmid m$ we show that ${\bf L}({\bf V})$ 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 ${\bf L}({\bf V})$ by a simple condition.
We investigate the lattice ${\bf L}({\bf V})$ of subspaces of an $m$-dimensional vector space ${\bf V}$ 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 ${\bf L}({\bf V})$ 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 ${\bf L}({\bf V})$ is orthomodular. For $m>1$ and $p\nmid m$ we show that ${\bf L}({\bf V})$ 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 ${\bf L}({\bf V})$ by a simple condition.
DOI : 10.21136/MB.2021.0042-20
Classification : 06C05, 06C15, 12D15, 12E20, 15A03
Keywords: vector space; lattice of subspaces; finite field; orthomodular lattice; modular lattice; Boolean lattice; complementation 
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Chajda, Ivan; Länger, Helmut. Orthogonality and complementation in the lattice of subspaces of a finite vector space. Mathematica Bohemica, Tome 147 (2022) no. 2, pp. 141-153. doi: 10.21136/MB.2021.0042-20

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