Orthomodular lattices with fully nontrivial commutators
Matoušek, Milan
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992), p. 25-32 / Harvested from Czech Digital Mathematics Library
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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].

Publié le : 1992-01-01
Classification:  06C15,  08C15 
@article{118466,
     author = {Milan Matou\v sek},
     title = {Orthomodular lattices with fully nontrivial commutators},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {33},
     year = {1992},
     pages = {25-32},
     zbl = {0758.06007},
     mrnumber = {1173742},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118466}
}

Matoušek, Milan. Orthomodular lattices with fully nontrivial commutators. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) pp. 25-32. http://gdmltest.u-ga.fr/item/118466/

Beran L. Orthomodular Lattices, Algebraic Approach, D. Reidel, Dordrecht, 1985. | MR 0784029 | Zbl 0558.06008

Bruns G.; Greechie R. Some finiteness conditions for orthomodular lattices, Canadian J. Math. 3 (1982), 535-549. (1982) | MR 0663303 | Zbl 0494.06008

Chevalier G. Commutators and Decomposition of Orthomodular Lattices, Order 6 (1989), 181-194. (1989) | MR 1031654

Godowski R.; Pták P. Classes of orthomodular lattices defined by the state conditions, preprint.

Gudder S. Stochastic Methods in Quantum Mechanics, Elsevier North Holland, Inc., 1979. | MR 0543489 | Zbl 0439.46047

Grätzer G. Universal Algebra, 2nd edition, Springer-Verlag, New York, 1979. | MR 0538623

Kalmbach G. Orthomodular Lattices, Academic Press, London, 1983. | MR 0716496 | Zbl 0554.06009

Mayet R. Varieties of orthomodular lattices related to states, Algebra Universalis, Vol. 20, No 3 (1987), 368-396. (1987) | MR 0811695

Pták P.; Pulmannová S. Orthomodular structures as quantum logics, Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. | MR 1176314

Pulmannová S. Commutators in orthomodular lattices, Demonstratio Math. 18 (1985), 187-208. (1985) | MR 0816029