| Title:
|
Orthomodular lattices with fully nontrivial commutators (English) |
| Author:
|
Matoušek, Milan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
1 |
| Year:
|
1992 |
| Pages:
|
25-32 |
| . |
| Category:
|
math |
| . |
| Summary:
|
An orthomodular lattice $L$ is said to have fully nontrivial commutator if the commutator of any pair $x,y \in L$ is different from zero. In this note we consider the class of all orthomodular lattices with fully nontrivial commutators. We show that this class forms a quasivariety, we describe it in terms of quasiidentities and situate important types of orthomodular lattices (free lattices, Hilbertian lattices, etc.) within this class. We also show that the quasivariety in question is not a variety answering thus the question implicitly posed in [4]. (English) |
| Keyword:
|
orthomodular lattice |
| Keyword:
|
commutator |
| Keyword:
|
quasivariety |
| MSC:
|
06C15 |
| MSC:
|
08C15 |
| idZBL:
|
Zbl 0758.06007 |
| idMR:
|
MR1173742 |
| . |
| Date available:
|
2009-01-08T17:53:05Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118466 |
| . |
| Reference:
|
[1] Beran L.: Orthomodular Lattices, Algebraic Approach.D. Reidel, Dordrecht, 1985. Zbl 0558.06008, MR 0784029 |
| Reference:
|
[2] Bruns G., Greechie R.: Some finiteness conditions for orthomodular lattices.Canadian J. Math. 3 (1982), 535-549. Zbl 0494.06008, MR 0663303 |
| Reference:
|
[3] Chevalier G.: Commutators and Decomposition of Orthomodular Lattices.Order 6 (1989), 181-194. MR 1031654 |
| Reference:
|
[4] Godowski R., Pták P.: Classes of orthomodular lattices defined by the state conditions.preprint. |
| Reference:
|
[5] Gudder S.: Stochastic Methods in Quantum Mechanics.Elsevier North Holland, Inc., 1979. Zbl 0439.46047, MR 0543489 |
| Reference:
|
[6] Grätzer G.: Universal Algebra.2nd edition, Springer-Verlag, New York, 1979. MR 0538623 |
| Reference:
|
[7] Kalmbach G.: Orthomodular Lattices.Academic Press, London, 1983. Zbl 0554.06009, MR 0716496 |
| Reference:
|
[8] Mayet R.: Varieties of orthomodular lattices related to states.Algebra Universalis, Vol. 20, No 3 (1987), 368-396. MR 0811695 |
| Reference:
|
[9] Pták P., Pulmannová S.: Orthomodular structures as quantum logics.Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. MR 1176314 |
| Reference:
|
[10] Pulmannová S.: Commutators in orthomodular lattices.Demonstratio Math. 18 (1985), 187-208. MR 0816029 |
| . |
