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The hyperoctant property in orthomodular AC-lattices


Author: Ronald P. Morash
Journal: Proc. Amer. Math. Soc. 57 (1976), 206-212
MSC: Primary 06A30
DOI: https://doi.org/10.1090/S0002-9939-1976-0417006-5
MathSciNet review: 0417006
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Abstract: The complete atomic orthomodular lattice $ L$ is said to have the hyperoctant property if and only if, for every orthogonal family of atoms $ \{ {a_\alpha }\} $ in $ L$ with cardinality $ \geq 2$, there exists an atom $ q$ such that $ q \leq { \vee _\alpha }{a_\alpha }$ and $ q \notin {a_\alpha }$ for each $ \alpha $. The projection lattice of any separable Hilbert space has the hyperoctant property. In this paper, we show that an abstract complete atomic orthomodular lattice possessing the additional properties, $ M$-symmetry, irreducibility, countably infinite dimension, and the angle bisection property, has the hyperoctant property. Additional remarks are made about the non-$ M$-symmetric case.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0417006-5
Keywords: Orthomodular lattice, Hilbert lattice, coordinatization, hyperoctant property
Article copyright: © Copyright 1976 American Mathematical Society

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