Order

, Volume 34, Issue 1, pp 91–111

On the Strong Freese-Nation Property

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Abstract

We show that there is a boolean algebra that has the Freese-Nation property (FN) but not the strong Freese-Nation property (SFN), thus answering a question of Heindorf and Shapiro. Along the way, we produce some new characterizations of the FN and SFN in terms of sequences of elementary submodels.

Keywords

Freese-Nation Property FN Strong Freese-Nation property SFN Elementary submodel Boolean algebra Davies tree Davies sequence Long ω1-approximation sequence 

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References

  1. 1.
    Davies, R. O.: Covering the plane with denumerably many curves. J. London Math. Soc. 38, 433–438 (1963)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Freese, R., Nation, J. B.: Projective lattices. Pac. J. Math. 75, 93–106 (1978)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Fuchino, S., Koppelberg, S., Shelah, S.: Partial orderings with the weak Freese-Nation property. Ann. Pure Appl. Logic 80(1), 35–54 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Heindorf, L., Shapiro, L. B.: Nearly Projective Boolean Algebras, with an appendix by S. Fuchino, Lecture Notes in Mathematics Springer-Verlag, Berlin 1994 (1596)Google Scholar
  5. 5.
    Jackson, S., Mauldin, R. D.: On a lattice problem of H. Steinhaus. J. Amer. Math. Soc 15(4), 817–856 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Koppelberg, S.: General Theory of Boolean Algebras, vol. 1 of: Monk, J. D. with R. Bonnet (Eds.), Handbook of Algebras, North-Holland, Amsterdam etc (1989)Google Scholar
  7. 7.
    Milovich, D.: Noetherian types of homogeneous compacta and dyadic compacta. Topology Appl. 156, 443–464 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Milovich, D.: The (λ,κ)-Freese-Nation property for Boolean algebras and compacta. Order 29, 361–379 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sapiro, L. B.: The space of closed subsets of \(D^{\aleph }_{2}\) is not a dyadic bicompactum. Soviet Math. Dokl. 17(3), 937–941 (1976)Google Scholar
  10. 10.
    Scepin, E. V.: On κ-metrizable spaces. Math. USSR-Izv 14(2), 406–440 (1980)CrossRefGoogle Scholar
  11. 11.
    Shchepin, E. V.: Functors and uncountable powers of compacta. Russian Math. Surveys 36(3), 1–71 (1981)CrossRefMATHGoogle Scholar
  12. 12.
    Sirota, S.: Spectral representation of spaces of closed subsets of bicompacta. Soviet Math. Doklady 9, 997–1000 (1968)MathSciNetMATHGoogle Scholar
  13. 13.
    Soukup, D.: Davies-trees in infinite combinatorics. Logic Colloquium 2014 (2014). arXiv:1407.3604

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsTexas A&M International UniversityLaredoUSA

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