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

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Bibliographic Details
Title: Orthogonality and complementation in the lattice of subspaces of a finite vector space
Authors: Ivan Chajda, Helmut Länger
Source: Mathematica Bohemica, Vol 147, Iss 2, Pp 141-153 (2022)
Publisher Information: Institute of Mathematics, Czech Academy of Sciences, 2021.
Publication Year: 2021
Subject Terms: lattice of subspaces, modular lattice, boolean lattice, complementation, orthomodular lattice, vector space, QA1-939, finite field, Mathematics
Description: 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.
Document Type: Article
DOI: 10.21136/mb.2021.0042-20
Access URL: https://doaj.org/article/dad889665ddf407380dee56ae171efc0
Rights: CC BY NC ND
Accession Number: edsair.doi.dedup.....14ad0a2f1b42dddf0770e52d776df552
Database: OpenAIRE
Description
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.
DOI:10.21136/mb.2021.0042-20