On Blocks in the Products and Ultraproducts of Orthomodular Lattices
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F23%3A00372779" target="_blank" >RIV/68407700:21230/23:00372779 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s10773-023-05488-5" target="_blank" >https://doi.org/10.1007/s10773-023-05488-5</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10773-023-05488-5" target="_blank" >10.1007/s10773-023-05488-5</a>
Alternative languages
Result language
angličtina
Original language name
On Blocks in the Products and Ultraproducts of Orthomodular Lattices
Original language description
Let OML denote the class of orthomodular lattices (OMLs, quantum logics). Let L be an OML and let B be a maximal Boolean subalgebra of L. Then B is called a block of L. In the algebraic investigation of OMLs a natural question is whether the blocks of a product (resp. ultraproduct) of OMLs are products (resp. ultraproducts) of the blocks of the respective "coordinate" OMLs. We first add to the study of this question as regards the products and the centres of the products (a special mention deserves the result that the centre of the ultraproduct is the ultraproduct of the centres of the respective OMLs). Then we pass to the analogous questions for ultraproducts where we present main results of this note. Though this question on the "regular" behaviour of blocks in ultraproducts remains open in general, we provide a positive partial solution. This contributes to the understanding of varieties important to quantum theories - to the varieties that contain both set-representable OMLs and projection OMLs. We consider an axiomatizable class of the OMLs, OMLn, whose blocks uniformly intersect in finite sets of the maximal cardinality of 2(n). It is worth realizing within the connection to quantum logic theory that, for instance, the OMLs given by Greechie diagrams belong to OML2. The importance of the results is commented on in relation to the state space properties of OMLs.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
International Journal of Theoretical Physics
ISSN
0020-7748
e-ISSN
1572-9575
Volume of the periodical
62
Issue of the periodical within the volume
11
Country of publishing house
US - UNITED STATES
Number of pages
8
Pages from-to
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UT code for WoS article
001095536900002
EID of the result in the Scopus database
2-s2.0-85175615410