A measure-theoretic characterization of Boolean algebras among orthomodular lattices

Pavel Pták; Sylvia Pulmannová

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 1, page 205-208
  • ISSN: 0010-2628

Abstract

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We investigate subadditive measures on orthomodular lattices. We show as the main result that an orthomodular lattice has to be distributive (=Boolean) if it possesses a unital set of subadditive probability measures. This result may find an application in the foundation of quantum theories, mathematical logic, or elsewhere.

How to cite

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Pták, Pavel, and Pulmannová, Sylvia. "A measure-theoretic characterization of Boolean algebras among orthomodular lattices." Commentationes Mathematicae Universitatis Carolinae 35.1 (1994): 205-208. <http://eudml.org/doc/247580>.

@article{Pták1994,
abstract = {We investigate subadditive measures on orthomodular lattices. We show as the main result that an orthomodular lattice has to be distributive (=Boolean) if it possesses a unital set of subadditive probability measures. This result may find an application in the foundation of quantum theories, mathematical logic, or elsewhere.},
author = {Pták, Pavel, Pulmannová, Sylvia},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {orthomodular lattice; subadditive probability measure; subadditve probability measure; orthomodular lattice; subadditive state; Boolean algebras},
language = {eng},
number = {1},
pages = {205-208},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A measure-theoretic characterization of Boolean algebras among orthomodular lattices},
url = {http://eudml.org/doc/247580},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Pták, Pavel
AU - Pulmannová, Sylvia
TI - A measure-theoretic characterization of Boolean algebras among orthomodular lattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 1
SP - 205
EP - 208
AB - We investigate subadditive measures on orthomodular lattices. We show as the main result that an orthomodular lattice has to be distributive (=Boolean) if it possesses a unital set of subadditive probability measures. This result may find an application in the foundation of quantum theories, mathematical logic, or elsewhere.
LA - eng
KW - orthomodular lattice; subadditive probability measure; subadditve probability measure; orthomodular lattice; subadditive state; Boolean algebras
UR - http://eudml.org/doc/247580
ER -

References

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  1. Beran L., Orthomodular Lattices (Algebraic Approach), Academia, Prague, 1984. Zbl0558.06008MR0785005
  2. Birkhoff G., Lattice Theory, 2nd edition, Amer. Math. Soc. Colloq. Publ., New York, 1948. Zbl0537.06001MR0029876
  3. Foulis D.J., A note on orthomodular lattices, Portugal. Math. 21 (1962), 65-72. (1962) Zbl0106.24302MR0148581
  4. Greechie R.J., Orthomodular lattices admitting no states, Jour. Comb. Theory 10 (1971), 119-132. (1971) Zbl0219.06007MR0274355
  5. Gudder S.P., Stochastic Methods in Quantum Mechanics, North-Holland, Amsterdam, 1979. Zbl0439.46047MR0543489
  6. Kalmbach G., Orthomodular Lattices, Academic Press, London, 1983. Zbl0554.06009MR0716496
  7. Majerník V., Pulmannová S., Bell inequalities on quantum logics, Jour. Math. Phys. 33 (6) (1992), 2173-2178. (1992) MR1164328
  8. Müller V., Jauch-Piron states on concrete quantum logics, Int. Jour. Theor. Phys. 32 (1993), 433-442. (1993) MR1213098
  9. Pták P., Exotic logics, Colloquium Math., Vol. LIV (1987), 1-7. MR0928651
  10. Pták P., Pulmannová S., Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers, Dordrecht-Boston-London, 1991. MR1176314
  11. Riečanová Z., Topology in a quantum logic induced by a measure, Proc. Conf. Topology and Measure V, Greifswald (1988), 126-130. MR1029570
  12. Rüttimann G., Wright J.D.M., Kalmbach outer measures and evaluations, to appear. 
  13. Sarymsakov T., Ajupov S., Khadzhiev D., Chilin V., Ordered Algebras (in Russian), Filial AN, Tashkent, 1983. 

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