| Title:
|
Orthogonality and complementation in the lattice of subspaces of a finite vector space (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Länger, Helmut |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
2 |
| Year:
|
2022 |
| Pages:
|
141-153 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
vector space |
| Keyword:
|
lattice of subspaces |
| Keyword:
|
finite field |
| Keyword:
|
orthomodular lattice |
| Keyword:
|
modular lattice |
| Keyword:
|
Boolean lattice |
| Keyword:
|
complementation |
| MSC:
|
06C05 |
| MSC:
|
06C15 |
| MSC:
|
12D15 |
| MSC:
|
12E20 |
| MSC:
|
15A03 |
| idZBL:
|
Zbl 07547246 |
| idMR:
|
MR4407348 |
| DOI:
|
10.21136/MB.2021.0042-20 |
| . |
| Date available:
|
2022-04-14T13:39:20Z |
| Last updated:
|
2022-09-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150322 |
| . |
| Reference:
|
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| Reference:
|
[2 ] Birkhoff, G.: Lattice Theory.American Mathematical Society Colloquium Publications 25. AMS, Providence ( 1979). Zbl 0505.06001, MR 0598630, 10.1090/coll/025 |
| Reference:
|
[3] Chajda, I., Länger, H.: The lattice of subspaces of a vector space over a finite field.Soft Comput. 23 (2019), 3261-3267. Zbl 07092395, 10.1007/s00500-019-03866-y |
| Reference:
|
[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 \99999MR99999 0246600 . Zbl 0181.56601, MR 0246600 |
| Reference:
|
[5] Giuntini, R., Ledda, A., Paoli, F.: A new view of effects in a Hilbert space.Stud. Log. 104 (2016), 1145-1177. Zbl 1417.06008, MR 3567676, 10.1007/s11225-016-9670-3 |
| Reference:
|
[6] Grätzer, G.: General Lattice Theory.Birkhäuser, Basel (2003). Zbl 1152.06300, MR 2451139, 10.1007/978-3-0348-7633-9 |
| . |
