Varieties of Orthocomplemented Lattices Induced by Lukasiewicz-Groupoid-Valued Mappings

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F17%3A00315746" target="_blank" >RIV/68407700:21230/17:00315746 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1007/s10773-017-3411-x" target="_blank" >http://dx.doi.org/10.1007/s10773-017-3411-x</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10773-017-3411-x" target="_blank" >10.1007/s10773-017-3411-x</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Varieties of Orthocomplemented Lattices Induced by Lukasiewicz-Groupoid-Valued Mappings

Popis výsledku v původním jazyce

In the logico-algebraic approach to the foundation of quantum mechanics we sometimes identify the set of events of the quantum experiment with an orthomodular lattice ("quantum logic"). The states are then usually associated with (normalized) finitely additive measures ("states"). The conditions imposed on states then define classes of orthomodular lattices that are sometimes found to be universal-algebraic varieties. In this paper we adopt a conceptually different approach, we relax orthomodular to orthocomplemented and we replace the states with certain subadditive mappings that range in the Aukasiewicz groupoid. We then show that when we require a type of "fulness" of these mappings, we obtain varieties of orthocomplemented lattices. Some of these varieties contain the projection lattice in a Hilbert space so there is a link to quantum logic theories. Besides, on the purely algebraic side, we present a characterization of orthomodular lattices among the orthocomplemented ones. - The intention of our approach is twofold. First, we recover some of the Mayet varieties in a principally different way (indeed, we also obtain many other new varieties). Second, by introducing an interplay of the lattice, measure-theoretic and fuzzy-set notions we intend to add to the concepts of quantum axiomatics.

Název v anglickém jazyce

Varieties of Orthocomplemented Lattices Induced by Lukasiewicz-Groupoid-Valued Mappings

Popis výsledku anglicky

In the logico-algebraic approach to the foundation of quantum mechanics we sometimes identify the set of events of the quantum experiment with an orthomodular lattice ("quantum logic"). The states are then usually associated with (normalized) finitely additive measures ("states"). The conditions imposed on states then define classes of orthomodular lattices that are sometimes found to be universal-algebraic varieties. In this paper we adopt a conceptually different approach, we relax orthomodular to orthocomplemented and we replace the states with certain subadditive mappings that range in the Aukasiewicz groupoid. We then show that when we require a type of "fulness" of these mappings, we obtain varieties of orthocomplemented lattices. Some of these varieties contain the projection lattice in a Hilbert space so there is a link to quantum logic theories. Besides, on the purely algebraic side, we present a characterization of orthomodular lattices among the orthocomplemented ones. - The intention of our approach is twofold. First, we recover some of the Mayet varieties in a principally different way (indeed, we also obtain many other new varieties). Second, by introducing an interplay of the lattice, measure-theoretic and fuzzy-set notions we intend to add to the concepts of quantum axiomatics.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10300 - Physical sciences

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

International Journal of Theoretical Physics

ISSN

0020-7748

e-ISSN

1572-9575

Svazek periodika

56

Číslo periodika v rámci svazku

12

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

13

Strana od-do

4004-4016

Kód UT WoS článku

000414787000028

EID výsledku v databázi Scopus

2-s2.0-85019717696