Lifting, $n$-Dimensional Spectral Resolutions, and $n$-Dimensional Observables

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 $\mathbb R^n$ 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 $σ$-homomorphisms from the Borel $σ$-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.

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