Chains, antichains, and complements in infinite partition lattices
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Chains, antichains, and complements in infinite partition lattices. / Avery, James Emil; Moyen, Jean-Yves; Ruzicka, Pavel; Simonsen, Jakob Grue.
I: Algebra Universalis, Bind 79, Nr. 37, 37, 2018.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › fagfællebedømt
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TY - JOUR
T1 - Chains, antichains, and complements in infinite partition lattices
AU - Avery, James Emil
AU - Moyen, Jean-Yves
AU - Ruzicka, Pavel
AU - Simonsen, Jakob Grue
PY - 2018
Y1 - 2018
N2 - We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$, and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is between the cofinality $\mathrm{cf}(\kappa)$ and $\kappa$, and $\kappa$ always occurs as the cardinality of a maximal well-ordered chain; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every ordinal $\delta$ with $|\delta| $ 2$.
AB - We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$, and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is between the cofinality $\mathrm{cf}(\kappa)$ and $\kappa$, and $\kappa$ always occurs as the cardinality of a maximal well-ordered chain; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every ordinal $\delta$ with $|\delta| $ 2$.
KW - math.RA
KW - 06B05 (Primary), 06C15 (Secondary)
U2 - 10.1007/s00012-018-0514-z
DO - 10.1007/s00012-018-0514-z
M3 - Journal article
VL - 79
JO - Algebra Universalis
JF - Algebra Universalis
SN - 0002-5240
IS - 37
M1 - 37
ER -
ID: 148648055