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

Chajda, Ivan, and Länger, Helmut. "Orthogonality and complementation in the lattice of subspaces of a finite vector space." Mathematica Bohemica 147.2 (2022): 141-153. <http://eudml.org/doc/298221>.

@article{Chajda2022,
abstract = {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.},
author = {Chajda, Ivan, Länger, Helmut},
journal = {Mathematica Bohemica},
keywords = {vector space; lattice of subspaces; finite field; orthomodular lattice; modular lattice; Boolean lattice; complementation},
language = {eng},
number = {2},
pages = {141-153},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Orthogonality and complementation in the lattice of subspaces of a finite vector space},
url = {http://eudml.org/doc/298221},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Chajda, Ivan
AU - Länger, Helmut
TI - Orthogonality and complementation in the lattice of subspaces of a finite vector space
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 2
SP - 141
EP - 153
AB - 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.
LA - eng
KW - vector space; lattice of subspaces; finite field; orthomodular lattice; modular lattice; Boolean lattice; complementation
UR - http://eudml.org/doc/298221
ER -