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Compact orthoalgebras


Author: Alexander Wilce
Journal: Proc. Amer. Math. Soc. 133 (2005), 2911-2920
MSC (2000): Primary 06F15, 06F30; Secondary 03G12, 81P10
DOI: https://doi.org/10.1090/S0002-9939-05-07884-6
Published electronically: May 2, 2005
MathSciNet review: 2159769
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Abstract: We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated $0$ is of finite height. We then focus on stably ordered TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated $0$ is determined by that of its space of atoms.


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  • 1. Bennett, M. K., and Foulis, D. J., Interval effect algebras and unsharp quantum logics, Advances in Applied Math. 19 (1997), 200-219. MR 1459498 (98m:06024)
  • 2. Bruns, G., and Harding, J., The algebraic theory of orthomodular lattices, in B. Coecke, D. J. Moore and A. Wilce (eds.), Current Research in Operational Quantum Logic, Kluwer: Dordrecht (2000). MR 1907155
  • 3. Choe, T. H., and Greechie, R. J., Profinite Orthomodular Lattices, Proc. Amer. Math. Soc. 118 (1993), 1053-1060. MR 1143016 (93j:06008)
  • 4. Choe, T. H., Greechie, R. J., and Chae, Y., Representations of locally compact orthomodular lattices, Topology and its Applications 56 (1994) 165-173. MR 1266141 (95c:06017)
  • 5. Foulis, D. J., Greechie, R. J., and Ruttimann, G. T., Filters and Supports on Orthoalgebras Int. J. Theor. Phys. 31 (1992) 789-807. MR 1162623 (93c:06014)
  • 6. Foulis, D. J., and Randall, C. H., What are quantum logics, and what ought they to be?, in Beltrametti, E., and van Fraassen, B. C. (eds.), Current Issues in Quantum Logic, Plenum: New York, 1981.) MR 0723148 (84k:03142)
  • 7. Greechie, R. J., Foulis, D. J., and Pulmannová, S., The center of an effect algebra, Order 12 (1995), 91-106. MR 1336539 (96c:81026)
  • 8. Johnstone, P. T., Stone Spaces, Cambridge: Cambridge University Press, 1982.MR 0698074 (85f:54002)
  • 9. Kalmbach, G., Orthomodular Lattices, Academic Press, 1983.MR 0716496 (85f:06012)
  • 10. Nachbin, L., Topology and Order, van Nostrand: Princeton 1965.MR 0219042 (36:2125)
  • 11. Priestley, H. A., Ordered Topological Spaces and the Representation of Distributive Lattices, Proc. London Math. Soc. 24 (1972), 507-530. MR 0300949 (46:109)
  • 12. Pulmannová, S., and Riecanova, Z., Block-finite Orthomodular Lattices, J. Pure and Applied Algebra 89 (1993), 295-304. MR 1242723 (94j:06016)
  • 13. Riecanova, Z., Order-Topological Lattice Effect Algebras, preprint, 2003.
  • 14. Wilce, A., Test Spaces and Orthoalgebras, in Coecke et al (eds.) Current Research in Operational Quantum Logic, Kluwer: Dordrecht (2000). MR 1907157 (2003c:81018)

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Additional Information

Alexander Wilce
Affiliation: Department of Mathematics, Susquehanna University, Selinsgrove, Pennsylvania 17870
Email: wilce@susqu.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07884-6
Keywords: Orthoalgebra, effect algebra, orthomodular lattice, topological lattice, quantum logic
Received by editor(s): August 22, 2003
Received by editor(s) in revised form: June 10, 2004
Published electronically: May 2, 2005
Communicated by: Lance W. Small
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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