Chains, antichains, and complements in infinite partition lattices

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Chains, antichains, and complements in infinite partition lattices

Original language description

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.

Czech name

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Czech description

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Type

J_imp - 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

Project

<a href="/en/project/GA14-15479S" target="_blank" >GA14-15479S: Representation Theory (Structural Decompositions and Their Constraints)</a><br>

Continuities

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

Publication year

2018

Confidentiality

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

Name of the periodical

Algebra Universalis

ISSN

0002-5240

e-ISSN

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Volume of the periodical

79

Issue of the periodical within the volume

2

Country of publishing house

CH - SWITZERLAND

Number of pages

21

Pages from-to

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UT code for WoS article

000431737200020

EID of the result in the Scopus database

2-s2.0-85045969891