The Logic of Lattice Effect Algebras Based on Induced Groupoids

J 2019

The Logic of Lattice Effect Algebras Based on Induced Groupoids

CHAJDA, Ivan, Helmut LAENGER and Jan PASEKA

Basic information

Original name

The Logic of Lattice Effect Algebras Based on Induced Groupoids

Authors

CHAJDA, Ivan (203 Czech Republic), Helmut LAENGER (40 Austria) and Jan PASEKA (203 Czech Republic, guarantor, belonging to the institution)

Edition

JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING, PHILADELPHIA, OLD CITY PUBLISHING INC, 2019, 1542-3980

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 0.703

RIV identification code

RIV/00216224:14310/19:00112273

Organization unit

Faculty of Science

UT WoS

000486419800002

Keywords in English

D-poset; effect algebra; lattice effect algebra; antitone involution; effect groupoid; groupoid-based logic

Abstract

V originále

Effect algebras were introduced by Foulis and Bennett as the so-called quantum structures which describe quantum effects and are determined by the behaviour of bounded self-adjoint operators on the phase space of the corresponding physical system which is a Hilbert space. From the algebraic point of view, the problem is that effect algebras are only partial ones and hence there can be drawbacks when we apply them for a construction of algebraic semantics of the corresponding logic of quantum mechanics. If the effect algebra in question is lattice-ordered this disadvantage can be overcome by using a representation of an equivalent algebra with everywhere defined operations. In our paper, this algebra is a groupoid equipped with one more unary operation which is an antitone involution. It enables us to introduce suitable axioms and inherence rules for the algebraic semantics of the corresponding logic and to prove that this logic is sound and complete.

Links

MUNI/G/1211/2017, interní kód MU

Name: Grupové techniky a kvantová informace (Acronym: GRUPIK)

Investor: Masaryk University, INTERDISCIPLINARY - Interdisciplinary research projects

Citovat

CHAJDA, Ivan, Helmut LAENGER and Jan PASEKA. The Logic of Lattice Effect Algebras Based on Induced Groupoids. JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING. PHILADELPHIA: OLD CITY PUBLISHING INC, 2019, vol. 33, No 3, p. 161-175. ISSN 1542-3980.

@article{1601838,
   author = {Chajda, Ivan and Laenger, Helmut and Paseka, Jan},
   article_location = {PHILADELPHIA},
   article_number = {3},
   keywords = {D-poset; effect algebra; lattice effect algebra; antitone involution; effect groupoid; groupoid-based logic},
   language = {eng},
   issn = {1542-3980},
   journal = {JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING},
   title = {The Logic of Lattice Effect Algebras Based on Induced Groupoids},
   url = {https://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-33-number-3-2019/mvlsc-33-3-p-161-175/},
   volume = {33},
   year = {2019}
}

TY  - JOUR
ID  - 1601838
AU  - Chajda, Ivan - Laenger, Helmut - Paseka, Jan
PY  - 2019
TI  - The Logic of Lattice Effect Algebras Based on Induced Groupoids
JF  - JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING
VL  - 33
IS  - 3
SP  - 161-175
EP  - 161-175
PB  - OLD CITY PUBLISHING INC
SN  - 15423980
KW  - D-poset
KW  - effect algebra
KW  - lattice effect algebra
KW  - antitone involution
KW  - effect groupoid
KW  - groupoid-based logic
UR  - https://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-33-number-3-2019/mvlsc-33-3-p-161-175/
L2  - https://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-33-number-3-2019/mvlsc-33-3-p-161-175/
N2  - Effect algebras were introduced by Foulis and Bennett as the so-called quantum structures which describe quantum effects and are determined by the behaviour of bounded self-adjoint operators on the phase space of the corresponding physical system which is a Hilbert space. From the algebraic point of view, the problem is that effect algebras are only partial ones and hence there can be drawbacks when we apply them for a construction of algebraic semantics of the corresponding logic of quantum mechanics. If the effect algebra in question is lattice-ordered this disadvantage can be overcome by using a representation of an equivalent algebra with everywhere defined operations. In our paper, this algebra is a groupoid equipped with one more unary operation which is an antitone involution. It enables us to introduce suitable axioms and inherence rules for the algebraic semantics of the corresponding logic and to prove that this logic is sound and complete.
ER  -

CHAJDA, Ivan, Helmut LAENGER and Jan PASEKA. The Logic of Lattice Effect Algebras Based on Induced Groupoids. \textit{JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING}. PHILADELPHIA: OLD CITY PUBLISHING INC, 2019, vol.~33, No~3, p.~161-175. ISSN~1542-3980.

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