|
Compact orthoalgebras
Author(s):
Alexander
Wilce
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2911-2920.
MSC (2000):
Primary 06F15, 06F30;
Secondary 03G12, 81P10
Posted:
May 2, 2005
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
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  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  is determined by that of its space of atoms.
References:
-
- 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)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
06F15, 06F30,
03G12, 81P10
Retrieve articles in all Journals with MSC
(2000):
06F15, 06F30,
03G12, 81P10
Additional Information:
Alexander
Wilce
Affiliation:
Department of Mathematics, Susquehanna University, Selinsgrove, Pennsylvania 17870
Email:
wilce@susqu.edu
DOI:
10.1090/S0002-9939-05-07884-6
PII:
S 0002-9939(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
Posted:
May 2, 2005
Communicated by:
Lance W. Small
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org