N-dimensional observables on k-perfect MV-algebras and k-perfect effect algebras. II. One-to-one correspondence

Abstract

This article is a continuation of our 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 Dvurečenskij and Lachman (https://doi.org/10.1016/j.fss.2021.05.005). There we presented the main properties of n-dimensional spectral resolutions and observables, and we studied in depth characteristic points which are crucial for our study. Here we present the 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 lexicographic form of quantum structures such as 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 results obtained are applied to the 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.