2020/10/23 by Anatolij Dvurečenskij, Dvurečenskij, Anatolij, Dominik Lachman +1
Mathematics · Physics and Astronomy · #06D35 #06F20 #81P10 #Commutative Algebra (math.AC) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #math-ph #math.AC #math.FA #math.MP #math.OA #msc:06D35 #msc:06F20 #msc:81P10
paper · pdf · doi:10.48550/arxiv.2011.01133
arxiv created 2020/10/23 · arxiv updated 2020/11/03
In the paper, 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 we have more characteristic points. The obtained results are also applied to existence of an n-dimensional meet joint observable of n one-dimensional observables on a perfect MV-algebra. The results are divided into two parts. In Part I, we present notions of n-dimensional observables and n-dimensional spectral resolutions with accent on lexicographic type effect algebras and lexicographic MV-algebras. We concentrate on characteristic points of spectral resolutions and the main body is in Part II where one-to-one relations between observables and spectral resolutions are presented.