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1991 | 62 | 1 | 7-14
Tytuł artykułu

Partially additive states on orthomodular posets

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also be viewed as a generalization of [6]. Then we prove an extension theorem for B-states giving, as a by-product, a topological proof of a classical Boolean result.
Słowa kluczowe
Rocznik
Tom
62
Numer
1
Strony
7-14
Opis fizyczny
Daty
wydano
1991
otrzymano
1989-01-16
poprawiono
1989-05-15
Twórcy
  • Department of Mathematics, Faculty of Electrical Engineering, Technical University of Prague, 166 27 Praha, Czechoslovakia
Bibliografia
  • [1] J. Binder and P. Pták, A representation of orthomodular lattices, Acta Univ. Carolin.--Math. Phys. 31 (1990), 21-26.
  • [2] E. Čech, Topological Spaces, Publ. House of the Czechoslovak Academy of Sciences, Prague, and Interscience, London 1966.
  • [3] R. J. Greechie, Orthomodular lattices admitting no states, J. Combin. Theory Ser. A 10 (1971), 119-132.
  • [4] S. P. Gudder, Stochastic Methods in Quantum Mechanics, North-Holland, New York 1979.
  • [5] A. Horn and A. Tarski, Measures in Boolean algebras, Trans. Amer. Math. Soc. 64 (1948), 467-497.
  • [6] L. Iturrioz, A representation theory for orthomodular lattices by means of closure spaces, Acta Math. Hungar. 47 (1986), 145-151.
  • [7] G. Kalmbach, Orthomodular Lattices, Academic Press, London 1983.
  • [8] M. J. Mączyński, Probability measures on a Boolean algebra, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 849-852.
  • [9] R. Mayet, Une dualité pour les ensembles ordonnés orthocomplémentés, C. R. Acad. Sci. Paris Sér. I 294 (1982), 63-65.
  • [10] R. Mayet, Varieties of orthomodular lattices related to states, Algebra Universalis 20 (1985), 368-396.
  • [11] P. Pták, Extensions of states on logics, Bull. Polish Acad. Sci. Math. 33 (1985), 493-497.
  • [12] P. Pták, Weak dispersion-free states and the hidden variables hypothesis, J. Math. Phys. 24 (1983), 839-840.
  • [13] N. Zierler and M. Schlessinger, Boolean embeddings of orthomodular sets and quantum logic, Duke Math. J. 32 (1965), 251-262.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv62i1p7bwm
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