Quantum logics representable as kernels of measures
Mirko Navara · Czechoslovak Mathematical Journal · 1996
Tlic classical Kohnogorov's model of probability assumes that every pair of ev(Tits is siinult.aneouslyobservable.This principle is violated in several applications including quantum mechanics, artificial intelligence, psychology, sociology etc.In these areas iioneoinpatible events are encountered.These are events which can be observed separately, but not simultaneously, so they are not contained in a Boolean subalgebra i = classical subsystem) of the event structure describing the system in question.Various attempts have been made to generalize the probability theory to a more general structure admitting noneonipatibility.Among them, classes of subsets (more generally, concrete logics) wore studied for many years (see e.g.[15, 18]).Although some results were successfully generalized (see e.g.[4. 10, 19]).the theory proceeded slowly and with serious difficulties.Hero we introduce a more special-but still reasonably general-structure, a kernel logic.As it is described in terms of Boolean algebras using measure-theoretic notions, we believe that there is a greater chance to generalize classical results for Boolean algebras to kernel logics.Kernel logics seem to be interesting also from the algebraic point of view as a new const ruction technique for concrete logics.Its usefulness was proved by solutions of several quite nontrivial problems.Besides this, it scorns desirable to describe kernels of measures on Boolean algebras.