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

Full-text PDF

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.

**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)**

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:
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.