Observables on perfect MV-algebras

Nola, Antonio Di; Dvurečenskij, Anatolij; Lenzi, Giacomo

doi:10.1016/j.fss.2018.11.018

Cited by 12 publications

(4 citation statements)

References 15 publications

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“…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: 81%

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

Dvurečenskij¹,

Lachman²

2020

Preprint

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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: Characteristic Pointsmentioning

confidence: 81%

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

Dvurečenskij¹,

Lachman²

2020

Preprint

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“…In this paper the categorical equivalence is described, between a classical universal algebra variety, subvariety of the class of P M V -algebras, the P M V falgebras and the category of semi-low f u -rings. This intermediate variety is a proper subvariety of the P M V -algebras defined by Di Nola y Dvurečenskij [5]. On the other hand, the variety of commutative unitary P M V -algebras studied by Montagna [11], to be called in this paper P M V 1 -algebras, is a proper subvariety of the P M V f .…”

Section: Introductionmentioning

confidence: 88%

“…Proof. From theorem 4.2 of [5], it follows that for any P M V -algebra A there exists an l u -ring R such that Γ(R, u) ∼ = A and because of proposition 5.3, A is an M V W -rig. The inclusion is strict because of remark 3.2.…”

Section: The Equivalencementioning

confidence: 97%

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Categorical Equivalence between $PMV_f$- product algebras and semi-low $f_u$-rings

Cruz,

Poveda

2018

Preprint

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An explicit categorical equivalence is defined between a proper subvariety of the class of P M V -algebras, as defined by Di Nola and Dvurečenskij, to be called P M V f -algebras, and the category of semi-low fu-rings. This categorical representation is done using the prime spectrum of the M Valgebras, through the equivalence between M V -algebras and lu-groups established by Mundici, from the perspective of the Dubuc-Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low fu-rings associated to Boolean algebras are characterized. Besides we show that class of P M V f -algebras is coextensive.

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Observables on lexicographic effect algebras

Dvurečenskij

Lachman

2019

Algebra Univers.

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Observables on perfect MV-algebras

Cited by 12 publications

References 15 publications

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

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

Categorical Equivalence between $PMV_f$- product algebras and semi-low $f_u$-rings

Observables on lexicographic effect algebras

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