Observables on perfect MV-algebras

An observable on an MV-algebra is any sigma-homomorphism from the Borel sigma-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 sigma-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 (-infinity, 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 Mas well as, we can define a sum of observables, so that O(M) becomes a lattice-ordered semigroup. (C) 2018 Elsevier B.V. All rights reserved.