Algebra universalis

, 79:37 | Cite as

Chains, antichains, and complements in infinite partition lattices

  • James Emil AveryEmail author
  • Jean-Yves Moyen
  • Pavel Růžička
  • Jakob Grue Simonsen


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 \(\mu ^{< \kappa } < \lambda \le \mu ^\kappa \) 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 \(\kappa \le \lambda \); (V) all cardinals of the form \(\lambda ^\kappa \) with \(0 \le \kappa \le \lambda \) 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.


Partition lattice Equivalence lattice Order theory Chain Antichain Complement Tree Cardinal Ordinal 

Mathematics Subject Classification

06B05 06C15 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • James Emil Avery
    • 1
    Email author
  • Jean-Yves Moyen
    • 2
    • 3
  • Pavel Růžička
    • 4
  • Jakob Grue Simonsen
    • 3
  1. 1.Niels Bohr InstituteUniversity of CopenhagenCopenhagen ØDenmark
  2. 2.Laboratoire d’Informatique de Paris NordUniversité Paris XIIIVilletaneuseFrance
  3. 3.Department of Computer ScienceUniversity of Copenhagen (DIKU)Copenhagen SDenmark
  4. 4.Department of AlgebraCharles University in PraguePragueCzech Republic

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