Observables on lexicographic effect algebras

We study lexicographic effect algebras which are intervals in lexicographic products H (x) over right arrow G, where (H, u) is a unital po-group and G is a monotone sigma-complete po-group with interpolation. We prove that there is a one-to-one correspondence between observables, which are a special kind of sigma-homomorphisms and analogues of measurable functions, and spectral resolutions which are systems {x(t) : t epsilon R} of elements of a lexicographic effect algebra that are monotone, "left continuous", and going to 0 if t -> -infinity and to 1 if t -> +infinity. We show that this correspondence in lexicographic effect algebras holds only for spectral resolutions with the finiteness property. Otherwise, they do not determine any observable. Whence, the information involved in a spectral resolution with the finiteness property completely describes information about an observable.