Semilogics, Quasilogics and Other Quantum Structures

We give an axiomatic formulation of quantum structures like semilogics and quasilogics which generalize the boolean semirings of events and fuzzy logics. The notions of distributions, states, representations observables and semiobservables are introduced and their Hilbert space realizations are found. The closed and open structures in semilogics are introduced and the regular distributions on the semilogics are studied. 1 The logic of the systems 1.1 Logical relations In this section we summarize the logical notions, which were discussed in connection with the foundations of quantum mechanics in [2]. The set l of all propositions a, b, c, . . . about an arbitrary system is a partially ordered set with respect to the relation of logical inclusion: the proposition a ∈ l