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 $\sigma$-complete unital po-group into another one, when the first one is a $\sigma$-homomorphic image of the second one. We note that an n-dimensional spectral resolution is a mapping from $\mathbb R^n$ into a quantum structure which is monotone, left-continuous with non-negative increments and which is going to 0 if one variable goes to $-\infty$ and it goes to 1 if all variables go to $+\infty$. 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 $\sigma$-homomorphisms from the Borel $\sigma$-algebra of ${\mathbb {R}}^n$ 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.