Observables on lexicographic effect algebras

Dvurečenskij, Anatolij; Lachman, Dominik

doi:10.1007/s00012-019-0628-y

Cited by 4 publications

(3 citation statements)

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“…This one-to-one correspondence between observables and spectral resolutions was extended for Rad-Dedekind σ-complete perfect MV-algebras in [DDL], however it is important to note that the notion of a spectral resolution was necessary to strengthen. This one-to-one relation was established also for k-perfect MV-algebras in [DvLa] and for general lexicographic MV-algebras and lexicographic effect algebras in [DvLa1,DvLa2].…”

Section: Introductionmentioning

confidence: 67%

“…Moreover, the elements of the latter equation exist in M , they belong to M i , and they will play an important role by definition of an n-dimensional spectral resolution for lexicographic MV-algebras and lexicographic effect algebras. Moreover, as it was shown in [DDL,DvLa1], if n = 1, then there are spectral resolutions which do not correspond to any observable with for which (4.3) does not exist in M .…”

Section: Characteristic Pointsmentioning

confidence: 84%

“…In [DvLa1], it was shown that even for a one-dimensional mapping F : R → M which satisfies (3.3)-(3.7) it can happen that it has infinitely many characteristic points. For a general two-dimensional spectral resolution, that is a mapping F : R 2 → M satisfying only (3.3)-(3.7), a situation like the one sketched in the Figure 2 could occur.…”

Section: Characteristic Pointsmentioning

confidence: 99%

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

Dvurečenskij¹,

Lachman²

2020

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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.

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Section: Introductionmentioning

confidence: 67%

Section: Characteristic Pointsmentioning

confidence: 84%

Section: Characteristic Pointsmentioning

confidence: 99%

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