Observables on lexicographic effect algebras

Abstract

We study lexicographic effect algebras which are intervals in lexicographic products $H\,\overrightarrow{\times }\,G$, 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 $\{x_t : t \in {\mathbb {R}}\}$ of elements of a lexicographic effect algebra that are monotone, “left continuous”, and going to 0 if $t\rightarrow -\infty$ and to 1 if $t\rightarrow +\infty$. 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.