Lifting, n-Dimensional Spectral Resolutions, and n-Dimensional Observables
Abstract
We show that under some natural conditions, we are able to lift an n-dimensional spectral resolution from one monotone σ-complete unital po-group into another one, when the first one is a σ-homomorphic image of the second one. We note that an n-dimensional spectral resolution is a mapping from Rn into a quantum structure which is monotone, left-continuous with non-negative increments and which is going to 0 if one variable goes to -∞ and it goes to 1 if all variables go to +∞. Applying this result to some important classes of effect algebras including also MV-algebras, we show that there is a one-to-one correspondence between n-dimensional spectral resolutions and n-dimensional observables on these effect algebras which are a kind of σ-homomorphisms from the Borel σ-algebra of Rn into the quantum structure. An important used tool are two forms of the Loomis--Sikorski theorem which use two kinds of tribes of fuzzy sets. In addition, we show that we can define three different kinds of n-dimensional joint observables of n one-dimensional observables.