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Some nearly Boolean orthomodular posets. (English) Zbl 0894.06003

Let \(L\) be an orthomodular poset (OMP). \(L\) is called set-representable if it is isomorphic to an OMP of sets with set-theoretically defined relation and operations. A state on \(L\) is a finitely additive normed measure on \(L\). A state \(s\) on \(L\) is called subadditive if for all \(x,y\in L\) there exists some \(z\in L\) with both \(z\geq x,y\) and \(s(z)\leq s(x)+s(y)\). \(L\) is called nearly Boolean if it is set-representable and every state on \(L\) is subadditive. Sufficient conditions for a nearly Boolean OMP to be Boolean are stated. The main result of the paper is the construction of nearly Boolean OMPs that are not Boolean. Since the constructed examples have trivial centre it follows from another construction that an arbitrary Boolean algebra may serve as the centre of a suitable nearly Boolean OMP that is not Boolean.
Reviewer: H.Länger (Wien)

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
28A60 Measures on Boolean rings, measure algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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