2020/10/23 by Dvurečenskij, Anatolij, Lachman, Dominik
#06D35 #06F20 #81P10 #Commutative Algebra (math.AC) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2011.05882
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 \citeDvLa4. 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.