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Existence of states on quantum structures. (English) Zbl 1162.81323

Summary: Orthomodular lattices occurred as generalized event structures in the models of probability for quantum mechanics. Here we contribute to the question of existence of states (=probability measures) on orthomodular lattices. We prove that known techniques do not allow to find examples with less than 19 blocks (=maximal Boolean subalgebras). This bound is achieved by the example by Mayet [R. Mayet, Personal communication, 1993]. Although we do not finally exclude the existence of other techniques breaking this bound, existence of smaller examples is highly unexpected.

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06C15 Complemented lattices, orthocomplemented lattices and posets
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