Abstract
If (ℰ, ℒ,P, Ω) is an event-state-operation structure, then the events form an orthomodular ortholattice (ℰ, ≦, ′) and the operations, mappings from the set of states ℒ into ℒ, form a Baer *-semigroup (SΩ, ℴ, *, ′). Additional axioms are adopted which yield the existence of a homomorphism ϑ from (SΩ, ℴ, *, ′) into the Baer *-semigroup (S(ℰ), ℴ, *, ′) of residuated mappings of (ℰ, ≦, ′) such thatx ∈ SΩ maps states while ϑ x ∈S (ℰ) maps supports of states. If (ℰ, ≦, ′) is atomic and there exists a correspondence between atoms and pure states, then the existence of ϑ provides the result: (ℰ, ≦, ′) is semimodular if and only if every operationx ∈ SΩ is a pure operation (maps pure states into pure states).
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Supported in part by the United States Atomic Energy Commission and in part by the Fonds National Suisse.
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Pool, J.C.T. Semimodularity and the logic of quantum mechanics. Commun.Math. Phys. 9, 212–228 (1968). https://doi.org/10.1007/BF01645687
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DOI: https://doi.org/10.1007/BF01645687