$n$-dimensional Observables on $k$-Perfect MV-Algebras and $k$-Perfect Effect Algebras. II. One-to-one Correspondence

TL;DR

This paper extends the theory of $n$-dimensional observables and spectral resolutions on lexicographic quantum structures, establishing a one-to-one correspondence and exploring their properties and applications in perfect MV-algebras and effect algebras.

Contribution

It develops a multidimensional framework for observables and spectral resolutions on $k$-perfect MV-algebras and effect algebras, generalizing previous one-dimensional results.

Findings

Established a one-to-one correspondence between $n$-dimensional observables and spectral resolutions.

Analyzed characteristic points in multidimensional spectral resolutions.

Applied results to joint observables and sums of $n$-dimensional observables.

Abstract

The paper is a continuation of the research on a one-to-one correspondence between $n$-dimensional spectral resolutions and $n$-dimensional observables on lexicographic types of quantum structures which started in \cite{DvLa4}. In Part I, we presented the main properties of $n$-dimensional spectral resolutions and observables, and we deeply studied characteristic points which are crucial for our study. In present Part II, there is a main body of our research. We investigate a one-to-one correspondence between $n$-dimensional observables and $n$-dimensional spectral resolutions with values in a kind of a lexicographic form of quantum structures like perfect MV-algebras or perfect effect algebras. The multidimensional version of this problem is more complicated than a one-dimensional one because if our algebraic structure is $k$-perfect for $k>1$, then even for the two-dimensional case of spectral resolutions we have more characteristic points. The obtained results are applied to existence of an $n$-dimensional meet joint observable of $n$ one-dimensional observables on a perfect MV-algebra and a sum of $n$-dimensional observables.