1/kappa-Homogeneous long solenoids

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17610%2F16%3AA1701BJD" target="_blank" >RIV/61988987:17610/16:A1701BJD - isvavai.cz</a>

Result on the web

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DOI - Digital Object Identifier

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Result language

čeština

Original language name

1/kappa-homogeneous long solenoids

Original language description

We study nonmetric analogues of Vietoris solenoids. Let ? be an ordered continuum, and let p =?p1,p2,??p?=?p1,p2,?? be a sequence of positive integers. We define a natural inverse limit space S(?,p), where the first factor space is the nonmetric ?circle? obtained by identifying the endpoints of ?, and the nth factor space, n>1, consists of p1p2?pn?1 copies of ? laid end to end in a circle. We prove that for every cardinal ??1, there is an ordered continuum ? such that S(?,p) is 1/?-homogeneous; for ?>1, ? is built from copies of the long line. Our example with ?=2 provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with ?=1 provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal ? and for each fixed p, there are 2^?-many 1/?-homogeneous solenoids of the form S(?,p) as varies over ordered continua of weight. Finally, we show that for every ordered continuum ? the shape of S(?,p) depends only on the equivalence class of p for a relation similar to one used to classify the additive subgroups of the rational numbers. Consequently, for each fixed ?, as p varies, there are exactly c-many different shapes, where c=2^? (and there are also exactly that many homeomorphism types) represented by S(?,p).

Czech name

1/kappa-homogeneous long solenoids

Czech description

We study nonmetric analogues of Vietoris solenoids. Let ? be an ordered continuum, and let p =?p1,p2,??p?=?p1,p2,?? be a sequence of positive integers. We define a natural inverse limit space S(?,p), where the first factor space is the nonmetric ?circle? obtained by identifying the endpoints of ?, and the nth factor space, n>1, consists of p1p2?pn?1 copies of ? laid end to end in a circle. We prove that for every cardinal ??1, there is an ordered continuum ? such that S(?,p) is 1/?-homogeneous; for ?>1, ? is built from copies of the long line. Our example with ?=2 provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with ?=1 provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal ? and for each fixed p, there are 2^?-many 1/?-homogeneous solenoids of the form S(?,p) as varies over ordered continua of weight. Finally, we show that for every ordered continuum ? the shape of S(?,p) depends only on the equivalence class of p for a relation similar to one used to classify the additive subgroups of the rational numbers. Consequently, for each fixed ?, as p varies, there are exactly c-many different shapes, where c=2^? (and there are also exactly that many homeomorphism types) represented by S(?,p).

Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

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Project

<a href="/en/project/LQ1602" target="_blank" >LQ1602: IT4Innovations excellence in science</a><br>

Continuities

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

Publication year

2016

Confidentiality

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

Name of the periodical

MONATSH MATH

ISSN

0026-9255

e-ISSN

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Volume of the periodical

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Issue of the periodical within the volume

180

Country of publishing house

AT - AUSTRIA

Number of pages

20

Pages from-to

171-192

UT code for WoS article

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EID of the result in the Scopus database

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