The Hyperoctant Property in Orthomodular AC-Lattices
Ronald P. Morash · Proceedings of the American Mathematical Society · 1976
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 $q otin {a_\alpha }$ 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.