Abstract
Let (X, #) be an orthogonality space such that the lattice C(X, #) of closed subsets of (X, #) is orthomodular and let (Γ, ⊥) denote the free orthogonality monoid over (X, #). Let C0(Γ, ⊥) be the subset of C(Γ, ⊥), consisting of all closures of bounded orthogonal sets. We show that C0(Γ, ⊥) is a suborthomodular lattice of C(Γ, ⊥) and we provide a necessary and sufficient condition for C0(Γ, ⊥) to carry a full set of dispersion free states.
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The work of the second author on this paper was supported by National Science Foundation Grant GP-9005.
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Randall, C.H., Foulis, D.J. States and the free orthogonality monoid. Math. Systems Theory 6, 268–276 (1972). https://doi.org/10.1007/BF01740718
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DOI: https://doi.org/10.1007/BF01740718