Abstract
If (ℰ, Y, 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 Y into Y, form a Baer *-semigroup(S Ω, ∘, *, ′). Additional axioms are adopted which yield the existence of a homomorphism θ from (S Ω, ∘, *, ′) into the Baer *-semigroup (S(ℰ), ∘, *, ′) of residuated mappings of (ℰ, ≦, ′) such that x∈S Ω maps states while θx∈ 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 operation x∈S Ω is a pure operation (maps pure states into pure states).
Supported in part by the United States Atomic Energy Commission and in part by the Fonds National Suisse.
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© 1975 D. Reidel Publishing Company, Dordrecht, Holland
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Pool, J.C.T. (1975). Semimodularity and The Logic of Quantum Mechanics. In: Hooker, C.A. (eds) The Logico-Algebraic Approach to Quantum Mechanics. The University of Western Ontario Series in Philosophy of Science, vol 5a. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1795-4_22
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DOI: https://doi.org/10.1007/978-94-010-1795-4_22
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