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

TL;DR

This paper explores the complex relationship between multidimensional observables and spectral resolutions within lexicographic quantum structures like perfect MV-algebras, focusing on characteristic points and joint observables.

Contribution

It establishes a one-to-one correspondence between n-dimensional observables and spectral resolutions in lexicographic quantum structures, extending the theory to higher dimensions.

Findings

Characterization of characteristic points in spectral resolutions

Existence criteria for n-dimensional joint observables

Extension of one-dimensional results to multidimensional cases

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 $n$-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.