Generalized XOR operation and the categorical equivalence of the Abbott algebras and quantum logics

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F23%3A00367666" target="_blank" >RIV/68407700:21230/23:00367666 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s10773-023-05355-3" target="_blank" >https://doi.org/10.1007/s10773-023-05355-3</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10773-023-05355-3" target="_blank" >10.1007/s10773-023-05355-3</a>

Result language
angličtina
Original language name
Generalized XOR operation and the categorical equivalence of the Abbott algebras and quantum logics
Original language description
Considering the inference rules in generalized logics, J.C. Abbott arrives to the notion of orthoimplication algebra (see Abbott (1970) and Abbott (Stud. Logica. 2:173–177, XXXV)). We show that when one enriches the Abbott orthoimplication algebra with a falsity symbol and a natural XOR-type operation, one obtains an orthomodular difference lattice as an enriched quantum logic (see Matoušek (Algebra Univers. 60:185–215, 2009)). Moreover, we find that these two structures endowed with the natural morphisms are categorically equivalent. We also show how one can introduce the notion of a state in the Abbott XOR algebras strenghtening thus the relevance of these algebras to quantum theories.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GF20-09869L" target="_blank" >GF20-09869L: The many facets of orthomodularity</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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