Orthomodular lattices with fully nontrivial commutators

Milan Matoušek

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 1, page 25-32
  • ISSN: 0010-2628

Abstract

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An orthomodular lattice L is said to have fully nontrivial commutator if the commutator of any pair x , y ∈ 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].

How to cite

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Matoušek, Milan. "Orthomodular lattices with fully nontrivial commutators." Commentationes Mathematicae Universitatis Carolinae 33.1 (1992): 25-32. <http://eudml.org/doc/247373>.

@article{Matoušek1992,
abstract = {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].},
author = {Matoušek, Milan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {orthomodular lattice; commutator; quasivariety; free lattices; orthomodular lattice; fully nontrivial commutator; Hilbert lattices},
language = {eng},
number = {1},
pages = {25-32},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Orthomodular lattices with fully nontrivial commutators},
url = {http://eudml.org/doc/247373},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Matoušek, Milan
TI - Orthomodular lattices with fully nontrivial commutators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 1
SP - 25
EP - 32
AB - 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].
LA - eng
KW - orthomodular lattice; commutator; quasivariety; free lattices; orthomodular lattice; fully nontrivial commutator; Hilbert lattices
UR - http://eudml.org/doc/247373
ER -

References

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  1. Beran L., Orthomodular Lattices, Algebraic Approach, D. Reidel, Dordrecht, 1985. Zbl0558.06008MR0784029
  2. Bruns G., Greechie R., Some finiteness conditions for orthomodular lattices, Canadian J. Math. 3 (1982), 535-549. (1982) Zbl0494.06008MR0663303
  3. Chevalier G., Commutators and Decomposition of Orthomodular Lattices, Order 6 (1989), 181-194. (1989) MR1031654
  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. Zbl0439.46047MR0543489
  6. Grätzer G., Universal Algebra, 2nd edition, Springer-Verlag, New York, 1979. MR0538623
  7. Kalmbach G., Orthomodular Lattices, Academic Press, London, 1983. Zbl0554.06009MR0716496
  8. Mayet R., Varieties of orthomodular lattices related to states, Algebra Universalis, Vol. 20, No 3 (1987), 368-396. (1987) MR0811695
  9. Pták P., Pulmannová S., Orthomodular structures as quantum logics, Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. MR1176314
  10. Pulmannová S., Commutators in orthomodular lattices, Demonstratio Math. 18 (1985), 187-208. (1985) MR0816029

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