STATES WITH VALUES IN THE LUKASIEWICZ GROUPOID

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

STATES WITH VALUES IN THE LUKASIEWICZ GROUPOID

Original language description

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

Czech name

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Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

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Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2016

Confidentiality

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

Name of the periodical

Mathematica Slovaca

ISSN

0139-9918

e-ISSN

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Volume of the periodical

66

Issue of the periodical within the volume

2

Country of publishing house

SK - SLOVAKIA

Number of pages

8

Pages from-to

335-342

UT code for WoS article

000387220200002

EID of the result in the Scopus database

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