Orthosystems of submodules of a module

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F23%3A00373294" target="_blank" >RIV/68407700:21230/23:00373294 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1080/00927872.2022.2164008" target="_blank" >https://doi.org/10.1080/00927872.2022.2164008</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1080/00927872.2022.2164008" target="_blank" >10.1080/00927872.2022.2164008</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Orthosystems of submodules of a module

Popis výsledku v původním jazyce

Let M be a module over a ring. We first introduce a certain algebraic sub-system Sigma of the lattice of all submodules of M (an orthosystem of submod-ules). We then show that any ortholattice can be represented as a Sigma for a suitable module. Next, we introduce linear (resp. pre-Hilbert) ortholattices as those ortholattices that allow for a "linear" representation sigma (resp. for a meet-preserving "linear" representation sigma). These notions involve a type of splitting property of sigma. As an important example, we show that any Boolean algebra is pre-Hilbertian. We then find that linear orthosystems are orthomodular and that they satisfy the ortho-Arguesian law. In the rest, we consider complete orthosystems. We show that each complete orthosystem can be induced by an orthogonality relation &updatedExpOTTOM; on M. If (M, &updatedExpOTTOM;) is a linear orthospace on M then the collection of all &updatedExpOTTOM;-closed submodules is a complete orthosystem, and vice versa. Finally, we address a natural model theoretic question on the axiomati-zation of orthomodular orthosystems.& mdash;The results obtained may contribute to the algebraic foundation of quantum theory.

Název v anglickém jazyce

Orthosystems of submodules of a module

Popis výsledku anglicky

Let M be a module over a ring. We first introduce a certain algebraic sub-system Sigma of the lattice of all submodules of M (an orthosystem of submod-ules). We then show that any ortholattice can be represented as a Sigma for a suitable module. Next, we introduce linear (resp. pre-Hilbert) ortholattices as those ortholattices that allow for a "linear" representation sigma (resp. for a meet-preserving "linear" representation sigma). These notions involve a type of splitting property of sigma. As an important example, we show that any Boolean algebra is pre-Hilbertian. We then find that linear orthosystems are orthomodular and that they satisfy the ortho-Arguesian law. In the rest, we consider complete orthosystems. We show that each complete orthosystem can be induced by an orthogonality relation &updatedExpOTTOM; on M. If (M, &updatedExpOTTOM;) is a linear orthospace on M then the collection of all &updatedExpOTTOM;-closed submodules is a complete orthosystem, and vice versa. Finally, we address a natural model theoretic question on the axiomati-zation of orthomodular orthosystems.& mdash;The results obtained may contribute to the algebraic foundation of quantum theory.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF20-09869L" target="_blank" >GF20-09869L: Ortomodularita z různých pohledů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

Communications in Algebra

ISSN

0092-7872

e-ISSN

1532-4125

Svazek periodika

51

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

12

Strana od-do

2460-2471

Kód UT WoS článku

000913102800001

EID výsledku v databázi Scopus

2-s2.0-85146701980