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Chains, antichains, and complements in infinite partition lattices. (English) Zbl 1522.06006

Summary: 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 \(> \lambda \); (III) a regular cardinal \(\lambda\) is strongly inaccessible if and only if every maximal chain in \(\Pi (\lambda)\) has size at least \(\lambda\); if \(\lambda\) is a singular cardinal and \(\mu^{< \kappa } < \lambda \leq \mu^\kappa\) for some cardinals \(\kappa\) and (possibly finite) \(\mu \), then there is a maximal chain of size \(< \lambda\) in \(\Pi (\lambda)\); (IV) every non-trivial maximal antichain in \(\Pi (\lambda)\) 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 \(\kappa \leq \lambda\); (V) all cardinals of the form \(\lambda^\kappa \) with \(0 \leq \kappa \leq \lambda\) occur as the cardinalities of sets of complements to some partition \(\mathcal {P} \in \Pi (\lambda)\), 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.

MSC:

06B05 Structure theory of lattices
03E05 Other combinatorial set theory
06C15 Complemented lattices, orthocomplemented lattices and posets
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