Chains, antichains, and complements in infinite partition lattices

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10383772" target="_blank" >RIV/00216208:11320/18:10383772 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s00012-018-0514-z" target="_blank" >https://doi.org/10.1007/s00012-018-0514-z</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00012-018-0514-z" target="_blank" >10.1007/s00012-018-0514-z</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Chains, antichains, and complements in infinite partition lattices

Popis výsledku v původním jazyce

We consider the partition lattice Pi(lambda) on any set of transfinite cardinality lambda and properties of Pi(lambda) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly lambda; (II) there are maximal chains in Pi(lambda) of cardinality &gt; lambda; (III) a regular cardinal lambda is strongly inaccessible if and only if every maximal chain in II(lambda) has size at least lambda; if lambda is a singular cardinal and mu(&lt;kappa) &lt; lambda &lt;= mu(kappa) for sonic cardinals kappa and (possibly finite) mu, then there is a maximal chain of size &lt; lambda in Pi(lambda); (IV) every non-trivial maximal antichain in II(A) has cardinality between lambda and 2 lambda, and these bounds are realised. Moreover, there are maximal antichains of cardinality max(lambda, 2(kappa)) for any kappa &lt;= lambda; (V) all cardinals of the form lambda(kappa) with 0 &lt;= kappa &lt;= lambda occur as the cardinalities of sets of complements to some partition P is an element of II(lambda), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.

Název v anglickém jazyce

Chains, antichains, and complements in infinite partition lattices

Popis výsledku anglicky

We consider the partition lattice Pi(lambda) on any set of transfinite cardinality lambda and properties of Pi(lambda) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly lambda; (II) there are maximal chains in Pi(lambda) of cardinality &gt; lambda; (III) a regular cardinal lambda is strongly inaccessible if and only if every maximal chain in II(lambda) has size at least lambda; if lambda is a singular cardinal and mu(&lt;kappa) &lt; lambda &lt;= mu(kappa) for sonic cardinals kappa and (possibly finite) mu, then there is a maximal chain of size &lt; lambda in Pi(lambda); (IV) every non-trivial maximal antichain in II(A) has cardinality between lambda and 2 lambda, and these bounds are realised. Moreover, there are maximal antichains of cardinality max(lambda, 2(kappa)) for any kappa &lt;= lambda; (V) all cardinals of the form lambda(kappa) with 0 &lt;= kappa &lt;= lambda occur as the cardinalities of sets of complements to some partition P is an element of II(lambda), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA14-15479S" target="_blank" >GA14-15479S: Teorie reprezentací (strukturní rozklady a jejich meze)</a><br>

Návaznosti

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

Rok uplatnění

2018

Kód důvěrnosti údajů

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

Název periodika

Algebra Universalis

ISSN

0002-5240

e-ISSN

—

Svazek periodika

79

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

21

Strana od-do

—

Kód UT WoS článku

000431737200020

EID výsledku v databázi Scopus

2-s2.0-85045969891