Chains, antichains, and complements in infinite partition lattices

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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\$.
Originalsprog Engelsk 37 Algebra Universalis 79 37 21 0002-5240 https://doi.org/10.1007/s00012-018-0514-z Udgivet - 2018

Forskningsområder

• math.RA, 06B05 (Primary), 06C15 (Secondary)

ID: 148648055