Compatibility of observables on effect algebras

Abstract

We compare different notions of simultaneous measurability (compatibility) of observables on lattice $$\sigma $$ź-effect algebras and more generally, on $$\sigma $$ź-effect algebras that can be covered by $$\sigma $$ź-MV-algebras. We prove that every $$\sigma $$ź-MV-algebra is the range of a $$\sigma $$ź-additive observable, and we compare the following notions of compatibility of observables: joint measurability, coexistence, joint measurability of binarizations, coexistence of binarizations, smearings of the same observable. We prove that if there is a faithful state on the effect algebra, then any two standard observables that are smearings of the same (sharp) observable admit a generalized joint observable.