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

Ivan Chajda; Helmut Länger

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 2, page 141-153
  • ISSN: 0862-7959

Abstract

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We investigate the lattice of subspaces of an -dimensional vector space over a finite field with a prime power 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 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 is orthomodular. For and we show that contains a -element (non-Boolean) orthomodular lattice as a subposet. Finally, for being a prime and we characterize orthomodularity of by a simple condition.

How to cite

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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 -

References

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  1. Beran, L., Orthomodular Lattices. Algebraic Approach, Mathematics and Its Applications 18 (East European Series). D. Reidel, Dordrecht (1985),9999DOI99999 10.1007/978-94-009-5215-7 . (1985) Zbl0558.06008MR0784029
  2. Birkhoff, G., 10.1090/coll/025, American Mathematical Society Colloquium Publications 25. AMS, Providence ( 1979). (1979) Zbl0505.06001MR0598630DOI10.1090/coll/025
  3. Chajda, I., Länger, H., 10.1007/s00500-019-03866-y, Soft Comput. 23 (2019), 3261-3267. (2019) Zbl07092395DOI10.1007/s00500-019-03866-y
  4. Eckmann, J.-P., Zabey, P. C., Impossibility of quantum mechanics in a Hilbert space over a finite field, Helv. Phys. Acta 42 (1969), 420-424 9999MR99999 0246600 . (1969) Zbl0181.56601MR0246600
  5. Giuntini, R., Ledda, A., Paoli, F., 10.1007/s11225-016-9670-3, Stud. Log. 104 (2016), 1145-1177. (2016) Zbl1417.06008MR3567676DOI10.1007/s11225-016-9670-3
  6. Grätzer, G., 10.1007/978-3-0348-7633-9, Birkhäuser, Basel (2003). (2003) Zbl1152.06300MR2451139DOI10.1007/978-3-0348-7633-9

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