The hyperoctant property in orthomodular AC-lattices

Morash, Ronald P.

doi:10.1090/S0002-9939-1976-0417006-5

The hyperoctant property in orthomodular AC-lattices
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Proc. Amer. Math. Soc. 57 (1976), 206-212 Request permission

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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