Geodesic
Orthomodular lattices with fully nontrivial commutators
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 1, pp. 25-32 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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].
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].
Classification : 06C15, 08C15
Keywords: orthomodular lattice; commutator; quasivariety 
@article{CMUC_1992_33_1_a2,
     author = {Matou\v{s}ek, Milan},
     title = {Orthomodular lattices with fully nontrivial commutators},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {25--32},
     year = {1992},
     volume = {33},
     number = {1},
     mrnumber = {1173742},
     zbl = {0758.06007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1992_33_1_a2/}
}
TY  - JOUR
AU  - Matoušek, Milan
TI  - Orthomodular lattices with fully nontrivial commutators
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 1992
SP  - 25
EP  - 32
VL  - 33
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMUC_1992_33_1_a2/
LA  - en
ID  - CMUC_1992_33_1_a2
ER  - 
%0 Journal Article
%A Matoušek, Milan
%T Orthomodular lattices with fully nontrivial commutators
%J Commentationes Mathematicae Universitatis Carolinae
%D 1992
%P 25-32
%V 33
%N 1
%U http://geodesic.mathdoc.fr/item/CMUC_1992_33_1_a2/
%G en
%F CMUC_1992_33_1_a2
Matoušek, Milan. Orthomodular lattices with fully nontrivial commutators. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 1, pp. 25-32. http://geodesic.mathdoc.fr/item/CMUC_1992_33_1_a2/

[1] Beran L.: Orthomodular Lattices, Algebraic Approach. D. Reidel, Dordrecht, 1985. | MR | Zbl

[2] Bruns G., Greechie R.: Some finiteness conditions for orthomodular lattices. Canadian J. Math. 3 (1982), 535-549. | MR | Zbl

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

[4] Godowski R., Pták P.: Classes of orthomodular lattices defined by the state conditions. preprint.

[5] Gudder S.: Stochastic Methods in Quantum Mechanics. Elsevier North Holland, Inc., 1979. | MR | Zbl

[6] Grätzer G.: Universal Algebra. 2nd edition, Springer-Verlag, New York, 1979. | MR

[7] Kalmbach G.: Orthomodular Lattices. Academic Press, London, 1983. | MR | Zbl

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

[9] Pták P., Pulmannová S.: Orthomodular structures as quantum logics. Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. | MR

[10] Pulmannová S.: Commutators in orthomodular lattices. Demonstratio Math. 18 (1985), 187-208. | MR