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Some Structural Aspects of the Katětov Order on Borel Ideals

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Abstract

We prove that the Katětov order on Borel ideals (1) contains a copy of \(\mathcal {P}(\omega )/\mathbf {Fin}\), consequently it has increasing and decreasing chains of lenght 𝔟; (2) the sequence F i n α (α < ω 1) is a strictly increasing chain; and (3) in the Cohen model, Katětov order does not contain any increasing nor decreasing chain of length 𝔠, answering a question of Hrušák (2011).

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References

  1. Barbarski, P., Filipów, R., MroŻek, N., Szuca, P.: When does the Katětov order imply that one ideal extends the other? Colloq. Math. 130(1), 91–102 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Farah, I.: Analytic quotients: theory of liftings for quotients over analytic ideals on integers. Memoirs of the American Mathematical Society 148(702) (2000)

  3. Flašková, J.: Description of some ultrafilters via I-ultrafilters. RIMS Kôkyûroku 1619, 20–31 (2008)

    Google Scholar 

  4. Hrušák, M.: Combinatorics of ideals and filters on ω. In: Set Theory and its Applications, Contemporary Mathematics, vol. 533, pp. 29–69. American Mathematical Society, Providence, RI (2011)

  5. Hrušák, M.: Katětov order on Borel ideals. To appear in Arch. Math. Logic

  6. Hrušák, M., Ferreira, S.G.: Ordering MAD families a la Katětov. J. Symbolic Logic 68(4), 1337–1353 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hrušák, M., Meza-Alcántara, D.: Katětov order, Fubini property and Hausdorff ultrafilters. Rend. Istit. Mat. Univ. Trieste 44, 503–511 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Hrušák, M., Meza-Alcántara, D., Thümmel, E., Uzcátegui, C.: Ramsey type properties of ideals. In preparation

  9. Katětov, M.: On descriptive classification of functions. General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Third Prague Topological Sympos., 1971), pp. 235–242 (1972)

  10. Kunen, K.: Set Theory An Introduction to Independence Proofs. North Holland, Amsterdam (1980)

    Google Scholar 

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Authors and Affiliations

  1. Centro de Ciencias Matemáticas UNAM, Morelia, Mexico

    Osvaldo Guzmán-González

  2. Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio Alpha, Ciudad Universitaria, 58060, Morelia, Michoacán, México

    David Meza-Alcántara

Authors
  1. Osvaldo Guzmán-González
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  2. David Meza-Alcántara
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Correspondence to David Meza-Alcántara.

Additional information

Second author was supported by grants UMSNH-CIC-9.30 and CONACYT-CB-169078

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Guzmán-González, O., Meza-Alcántara, D. Some Structural Aspects of the Katětov Order on Borel Ideals. Order 33, 189–194 (2016). https://doi.org/10.1007/s11083-015-9358-8

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  • DOI: https://doi.org/10.1007/s11083-015-9358-8

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