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Chains, antichains, and complements in infinite partition lattices

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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.

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References

  1. Baumgartner, J.E.: Almost-disjoint sets, the dense set problem and the partition calculus. Ann. Math. Log. 10, 401–439 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Birkhoff, G.: Lattice Theory, Colloquium Publications, vol. 25. American Mathematical Society, Providence (1940)

    Google Scholar 

  3. Blinovsky, V.M., Harper, L.H.: Size of the largest antichain in a partition poset. Probl. Inf. Transm. 38(4), 347–353 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78. Springer, Berlin (1981)

    MATH  Google Scholar 

  5. Canfield, E.R.: The size of the largest antichain in the partition lattice. J. Comb. Theory Ser. A 83(2), 188–201 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chernikov, A., Kaplan, I., Shelah, S.: On non-forking spectra. J. Eur. Math. Soc. 18, 2821–2848 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Czédli, G.: Four-generated large equivalence lattices. Acta. Sci. Math. (Szeged.) 62, 47–69 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Czédli, G.: Lattice generation of small equivalences of a countable set. Order 13(1), 11–16 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Czédli, G.: (1+1+2)-generated equivalence lattices. J. Algebra 221(2), 439–462 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Easton, W.B.: Powers of regular cardinals. Ann. Math. Log. 1(2), 139–178 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  13. Grieser, D.: Counting complements in the partition lattice, and hypertrees. J. Comb. Theory Ser. A 57(1), 144–150 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Harzheim, E.: Ordered Sets, Advances in Mathematics, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Hausdorff, F.: Grundzüge der Mengenlehre. Leipzig (1914)

  16. Holz, M., Steffens, K., Weitz, E.: Introduction to Cardinal Arithmetic. Birkhäuser, Basel (1999)

    Book  MATH  Google Scholar 

  17. König, J.: Über die Grundlage der Mengenlehre und das Kontinuumproblem. Math. Ann. 61, 156–160 (1905)

    Article  MathSciNet  MATH  Google Scholar 

  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. Mitchell, W.: Aronszajn trees and the independence of the transfer property. Ann. Math. Log. 5, 21–46 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nation, J.: Notes on Lattice Theory. http://www.math.hawaii.edu/~jb/books.html

  21. Ore, Ø.: Theory of equivalence relations. Duke Math. J. 9(3), 573–627 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sierpiński, W.F.: Sur un problème concernant les sous-ensembles croissants du continu. Fundam. Math. 3, 109–112 (1922)

    Article  MATH  Google Scholar 

  23. Stern, M.: Semimodular Lattices. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

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Correspondence to James Emil Avery.

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Presented by G. Czédli.

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Avery, J.E., Moyen, JY., Růžička, P. et al. Chains, antichains, and complements in infinite partition lattices. Algebra Univers. 79, 37 (2018). https://doi.org/10.1007/s00012-018-0514-z

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