Observables on perfect MV-algebras

Abstract

An observable on an MV-algebra is any σ-homomorphism from the Borel σ-algebra B(R) into the MV-algebra which maps a sequence of disjoint Borel sets onto summable elements of the MV-algebra. We establish that there is a one-to-one correspondence between observables on Rad-Dedekind σ-complete perfect MV-algebras with principal radicals and their spectral resolutions. It means that we show that our partial information on an observable known only on all intervals of the form (−∞,t) is sufficient to determine the whole information about the observable. In addition, this correspondence allows us to define the Olson order which is a partial order on the set O(M) of all observables on an MV-algebra M as well as, we can define a sum of observables, so that O(M) becomes a lattice-ordered semigroup.