$n$-dimensional Observables on $k$-Perfect MV-Algebras and $k$-Perfect Effect Algebras. I. Characteristic Points

Abstract

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 ndimensional 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.