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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.

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References

  1. Davies, R. O.: Covering the plane with denumerably many curves. J. London Math. Soc. 38, 433–438 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  2. Freese, R., Nation, J. B.: Projective lattices. Pac. J. Math. 75, 93–106 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fuchino, S., Koppelberg, S., Shelah, S.: Partial orderings with the weak Freese-Nation property. Ann. Pure Appl. Logic 80(1), 35–54 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  5. Jackson, S., Mauldin, R. D.: On a lattice problem of H. Steinhaus. J. Amer. Math. Soc 15(4), 817–856 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  7. Milovich, D.: Noetherian types of homogeneous compacta and dyadic compacta. Topology Appl. 156, 443–464 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Milovich, D.: The (λ,κ)-Freese-Nation property for Boolean algebras and compacta. Order 29, 361–379 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Scepin, E. V.: On κ-metrizable spaces. Math. USSR-Izv 14(2), 406–440 (1980)

    Article  Google Scholar 

  11. Shchepin, E. V.: Functors and uncountable powers of compacta. Russian Math. Surveys 36(3), 1–71 (1981)

    Article  MATH  Google Scholar 

  12. Sirota, S.: Spectral representation of spaces of closed subsets of bicompacta. Soviet Math. Doklady 9, 997–1000 (1968)

    MathSciNet  MATH  Google Scholar 

  13. Soukup, D.: Davies-trees in infinite combinatorics. Logic Colloquium 2014 (2014). arXiv:1407.3604

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  1. Department of Mathematics and Physics, Texas A&M International University, 5201 University Blvd., Laredo, TX 78041, USA

    David Milovich

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  1. David Milovich
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Correspondence to David Milovich.

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Milovich, D. On the Strong Freese-Nation Property. Order 34, 91–111 (2017). https://doi.org/10.1007/s11083-016-9389-9

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  • DOI: https://doi.org/10.1007/s11083-016-9389-9

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