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Algebra universalis

, 79:37 | Cite as

Chains, antichains, and complements in infinite partition lattices

  • James Emil Avery
  • Jean-Yves Moyen
  • Pavel Růžička
  • Jakob Grue Simonsen
Article
  • 29 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

06B05 06C15 

References

  1. 1.
    Baumgartner, J.E.: Almost-disjoint sets, the dense set problem and the partition calculus. Ann. Math. Log. 10, 401–439 (1976)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, Colloquium Publications, vol. 25. American Mathematical Society, Providence (1940)Google Scholar
  3. 3.
    Blinovsky, V.M., Harper, L.H.: Size of the largest antichain in a partition poset. Probl. Inf. Transm. 38(4), 347–353 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78. Springer, Berlin (1981)MATHGoogle Scholar
  5. 5.
    Canfield, E.R.: The size of the largest antichain in the partition lattice. J. Comb. Theory Ser. A 83(2), 188–201 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chernikov, A., Kaplan, I., Shelah, S.: On non-forking spectra. J. Eur. Math. Soc. 18, 2821–2848 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chernikov, A., Shelah, S.: On the number of Dedekind cuts and two-cardinal models of dependent theories. J. Inst. Math. Jussieu 15, 771–784 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Czédli, G.: Four-generated large equivalence lattices. Acta. Sci. Math. (Szeged.) 62, 47–69 (1996)MathSciNetMATHGoogle Scholar
  9. 9.
    Czédli, G.: Lattice generation of small equivalences of a countable set. Order 13(1), 11–16 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Czédli, G.: (1+1+2)-generated equivalence lattices. J. Algebra 221(2), 439–462 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Easton, W.B.: Powers of regular cardinals. Ann. Math. Log. 1(2), 139–178 (1970)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003)MATHGoogle Scholar
  13. 13.
    Grieser, D.: Counting complements in the partition lattice, and hypertrees. J. Comb. Theory Ser. A 57(1), 144–150 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Harzheim, E.: Ordered Sets, Advances in Mathematics, 2nd edn. Springer, Berlin (2005)MATHGoogle Scholar
  15. 15.
    Hausdorff, F.: Grundzüge der Mengenlehre. Leipzig (1914)Google Scholar
  16. 16.
    Holz, M., Steffens, K., Weitz, E.: Introduction to Cardinal Arithmetic. Birkhäuser, Basel (1999)CrossRefMATHGoogle Scholar
  17. 17.
    König, J.: Über die Grundlage der Mengenlehre und das Kontinuumproblem. Math. Ann. 61, 156–160 (1905)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Malitz, J.: The Hanf number for complete \({{\rm L}}_{\omega _1,\omega }\) sentences. In: Barwise, J. (ed.) The syntax and semantic of infinitary languages. Springer, Berlin (1968)Google Scholar
  19. 19.
    Mitchell, W.: Aronszajn trees and the independence of the transfer property. Ann. Math. Log. 5, 21–46 (1972)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nation, J.: Notes on Lattice Theory. http://www.math.hawaii.edu/~jb/books.html
  21. 21.
    Ore, Ø.: Theory of equivalence relations. Duke Math. J. 9(3), 573–627 (1942)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sierpiński, W.F.: Sur un problème concernant les sous-ensembles croissants du continu. Fundam. Math. 3, 109–112 (1922)CrossRefMATHGoogle Scholar
  23. 23.
    Stern, M.: Semimodular Lattices. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • James Emil Avery
    • 1
  • 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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