Semimodularity and the logic of quantum mechanics

James T. Pool

pp. 395-414

in: Clifford A. Hooker (ed), The logico-algebraic approach to quantum mechanics I, Berlin, Springer, 1975

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 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 operation x∈S ^Ω is a pure operation (maps pure states into pure states).