On Blocks in the Products and Ultraproducts of Orthomodular Lattices

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

On Blocks in the Products and Ultraproducts of Orthomodular Lattices

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

On Blocks in the Products and Ultraproducts of Orthomodular Lattices

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

International Journal of Theoretical Physics

ISSN

0020-7748

e-ISSN

1572-9575

Svazek periodika

62

Číslo periodika v rámci svazku

11

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

8

Strana od-do

—

Kód UT WoS článku

001095536900002

EID výsledku v databázi Scopus

2-s2.0-85175615410