STATES WITH VALUES IN THE LUKASIEWICZ GROUPOID

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F16%3A00305083" target="_blank" >RIV/68407700:21230/16:00305083 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1515/ms-2015-0139" target="_blank" >http://dx.doi.org/10.1515/ms-2015-0139</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1515/ms-2015-0139" target="_blank" >10.1515/ms-2015-0139</a>

Jazyk výsledku

angličtina

Název v původním jazyce

STATES WITH VALUES IN THE LUKASIEWICZ GROUPOID

Popis výsledku v původním jazyce

In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let * stand for the Lukasiewicz t-norm on [0, 1](2). Let us consider the operation lozenge on [0, 1] by setting x lozenge y = (x(perpendicular to)*y(perpendicular to))(perpendicular to) *(x*y)(perpendicular to), where x(perpendicular to) = 1-x. Let us call the triple L = ([0, 1], lozenge, 1) the Lukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z(2)-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as "quantum logics" in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state. (C) 2016 Mathematical Institute Slovak Academy of Sciences

Název v anglickém jazyce

STATES WITH VALUES IN THE LUKASIEWICZ GROUPOID

Popis výsledku anglicky

In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let * stand for the Lukasiewicz t-norm on [0, 1](2). Let us consider the operation lozenge on [0, 1] by setting x lozenge y = (x(perpendicular to)*y(perpendicular to))(perpendicular to) *(x*y)(perpendicular to), where x(perpendicular to) = 1-x. Let us call the triple L = ([0, 1], lozenge, 1) the Lukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z(2)-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as "quantum logics" in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state. (C) 2016 Mathematical Institute Slovak Academy of Sciences

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

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

Název periodika

Mathematica Slovaca

ISSN

0139-9918

e-ISSN

—

Svazek periodika

66

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

SK - Slovenská republika

Počet stran výsledku

8

Strana od-do

335-342

Kód UT WoS článku

000387220200002

EID výsledku v databázi Scopus

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