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Proceedings of the American Mathematical Society

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Some nearly Boolean orthomodular posets

Author: Pavel Pták
Journal: Proc. Amer. Math. Soc. 126 (1998), 2039-2046
MSC (1991): Primary 28A60, 06C15, 81P10
MathSciNet review: 1452822
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Abstract: Let $L$ be an orthomodular partially ordered set (``a quantum logic"). Let us say that $L$ is nearly Boolean if $L$ is set-representable and if every state on $L$ is subadditive. We first discuss conditions under which a nearly Boolean OMP must be Boolean. Then we show that in general a nearly Boolean OMP does not have to be Boolean. Moreover, we prove that an arbitrary Boolean algebra may serve as the centre of a (non-Boolean) nearly Boolean OMP.

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Additional Information

Pavel Pták
Affiliation: Czech Technical University, Faculty of Electrical Engineering, Department of Mathematics, 16627 Prague 6, Czech Republic

Keywords: Orthomodular partially ordered set, Boolean algebra, state (= finitely additive probability measure), subadditivity
Received by editor(s): December 16, 1996
Additional Notes: The author acknowledges the support by the grant GA 201/96/0117 of the Czech Grant Agency.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society