Topological Orthoalgebras

Details Topological Orthoalgebras Topological Orthoalgebras

2003-01-08

arxiv.org/abs/math/0301072v2

Mathematics

Rings and Algebras

16 pp, LaTex. Minor changes and corrections in sections 1; more substantial corrections in section 5

Scientific paper

10.1007/BFb0119441

We define topological orthoalgebras (TOAs) and study their properties. While every topological orthomodular lattice is a TOA, the lattice of projections of a Hilbert space is an example of a lattice-ordered TOA that is not a toplogical lattice. On the other hand, we show that every compact Boolean TOA is a topological Boolean algebra. We also show that a compact TOA in which 0 is an isolated point is atomic and of finite height. We identify and study a particularly tractable class of TOAs, which we call {\em stably ordered}: those in which the upper-set generated by an open set is open. This includes all topological OMLs, and also the projection lattices of Hilbert spaces. Finally, we obtain a topological version of the Foulis-Randall representation theory for stably ordered TOAs

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