Previous |  Up |  Next


vector space; lattice of subspaces; finite field; orthomodular lattice; modular lattice; Boolean lattice; complementation
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.
[1] Beran, L.: Orthomodular Lattices. Algebraic Approach. Mathematics and Its Applications 18 (East European Series). D. Reidel, Dordrecht (1985),\99999DOI99999 10.1007/978-94-009-5215-7 . MR 0784029 | Zbl 0558.06008
[2 ] Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publications 25. AMS, Providence ( 1979). DOI 10.1090/coll/025 | MR 0598630 | Zbl 0505.06001
[3] Chajda, I., Länger, H.: The lattice of subspaces of a vector space over a finite field. Soft Comput. 23 (2019), 3261-3267. DOI 10.1007/s00500-019-03866-y | Zbl 07092395
[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 . MR 0246600 | Zbl 0181.56601
[5] Giuntini, R., Ledda, A., Paoli, F.: A new view of effects in a Hilbert space. Stud. Log. 104 (2016), 1145-1177. DOI 10.1007/s11225-016-9670-3 | MR 3567676 | Zbl 1417.06008
[6] Grätzer, G.: General Lattice Theory. Birkhäuser, Basel (2003). DOI 10.1007/978-3-0348-7633-9 | MR 2451139 | Zbl 1152.06300
Partner of
EuDML logo