Exact topological analogs to orthoposets
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- by Peter G. Ovchinnikov PDF
- Proc. Amer. Math. Soc. 125 (1997), 2839-2841 Request permission
Abstract:
An arbitrary orthoposet $E$ is shown to be isomorphic to $(\mathcal {E}, \subset ,^c)$, $\mathcal {E}$ being a subbasis of a Hausdorff topological space $\mathcal {S}$ satisfying 1) $\mathcal {S}\in \mathcal {E}$, 2) $\alpha \in \mathcal {E}\Rightarrow \alpha ^c \in \mathcal {E}$, and 3) every covering of $\mathcal {S}$ by elements of $\mathcal {E}$ possesses an at most 2-element subcovering. The couple $(\mathcal {S},\mathcal {E})$ turns out to be unique.References
- Jiří Binder and Pavel Pták, A representation of orthomodular lattices, Acta Univ. Carolin. Math. Phys. 31 (1990), no. 1, 21–26 (English, with Russian and Czech summaries). MR 1098124
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- Stanley P. Gudder, Stochastic methods in quantum mechanics, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York-Oxford, 1979. MR 543489
- Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496
- G. Kalmbach, Measures and Hilbert lattices, World Scientific Publishing Co., Singapore, 1986. MR 867884, DOI 10.1142/0206
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- Pavel Pták and Sylvia Pulmannová, Kvantové logiky, VEDA, Vydavatel′stvo Slovenskej Akadémie Vied, Bratislava, 1989 (Slovak, with English and Russian summaries). MR 1176313
- Roman Sikorski, Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 25, Academic Press, Inc., New York; Springer-Verlag, Berlin-New York, 1964. MR 0177920
Additional Information
- Peter G. Ovchinnikov
- Affiliation: Department of Mathematics, Kazan State University, 420008, Kazan, Russia
- Email: Petr.Ovchinnikov@ksu.ru
- Received by editor(s): April 9, 1996
- Communicated by: Franklin D. Tall
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2839-2841
- MSC (1991): Primary 06C15, 54H10; Secondary 81P10
- DOI: https://doi.org/10.1090/S0002-9939-97-04023-9
- MathSciNet review: 1415360