P-Like Properties of Meager Ideals and Cardinal Invariants

Marton, Adam

doi:10.2478/tmmp-2023-0025

Abstract

We study two particular modifications of the P-property of ideals and related cardinal invariants cof^𝒥 (ℐ)and cov⁺(ℐ). We give some results on the existence of P (𝒥)-ideals or non-P (𝒥)-ideals regarding specific classes of ideals, particularly meager ideals on ω. We also provide values of the cardinal invariant cof^𝒥 (ℐ) describing the smallest families ensuring P(𝒥) for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities Fin <_K 〈𝒜 〉 <_K Fin × Fin for any MAD family 𝒜 using the weak P-ideal notion.

References

HAGA, K. B.—SCHRITTESSER, D.—TÖRNQUIST, A.: MAD families, determinacy, and forcing, J. Math. Log. 22 (2021), no. 1, 2150026.
Search in Google Scholar Back to article
BARTOSZYŃSKI, T.—JUDAH, H.: Set Theory: On the Structure of the Real Line. CRC Press (Taylor & Francis Group) London, 1995
Search in Google Scholar Back to article
BLASS, A.: Combinatorial cardinal characteristics of the continuum, In: (M. Foreman, A. Kanamori, eds.), Handbook of Set Theory. Springer-Verlag, Berlin 2010, pp. 395–489.
Search in Google Scholar Back to article
BUKOVSKÁ, Z.: Thin sets in trigonometrical series and quasinormal convergence,Math. Slovaca 40 (1990), no. 1, 53–62.
Search in Google Scholar Back to article
Császár,Á.—Laczkovich, M. Discrete and equal convergence, Stud. Sci. Math. Hungar. 10 (1975), 463–472.
Search in Google Scholar Back to article
FARKAS, B.: Combinatorics of Borel Ideals Ph.D. Thesis, Budapest University of Technology and Economics Department of Algebra, Institute of Mathematics Supervisor: Prof. Lajos Soukup, Alfréd Rényi Institute of Mathematics Hungarian Academy of Sci. 2011.
Search in Google Scholar Back to article
FARKAS, B.—SOUKUP, L.: More on cardinal invariants of analytis P-ideals, Commentationes Mathematicae Universitatis Carolinae 50 (2009), no. 2, 281–295.
Search in Google Scholar Back to article
FARKAS, B.—ZDOMSKYY, L.: Ways of destruction, J. Symb. Log. 87 (2022), no. 3, 938–966.
Search in Google Scholar Back to article
FILIPÓW, R.—STANISZEWSKI, M.: On ideal equal convergence, Cent. Eur. J. Math. 12 (2014), no. 6, 896–910.
Search in Google Scholar Back to article
R. FILIPÓW, R.—STANISZEWSKI, M.: Pointwise versus equal (quasi-normal) convergence via ideals, J. Math. Anal. Appl. 422 (2015), no. 2, 995–1006.
Search in Google Scholar Back to article
R. FILIPÓW, R.—SZUCA, P.: Three kinds of convergence and the associated ℐ-Baire classes, J. Math. Anal. Appl. 391 (2012), no. 1, 1–9.
Search in Google Scholar Back to article
GUZMÁN-GONZÁLEZ, O.—HRUŠÁK, M.— MARTÍNEZ-RANERO, C. A.—RAMOS-GARÍA, U. A.: Generic existence of MAD families, J. Symb. Log. 82 (2017), no. 1, 303–316.
Search in Google Scholar Back to article
HALBEISEN, L.J.: Combinatorial Set Theory With a Gentle Introduction to Forcing. Springer-Verlag, Berlin, 2012.
Search in Google Scholar Back to article
HERNÁNDEZ-HERNÁNDEZ, F.—HRUŠÁK, M.: Cardinal invariants of analytic P-ideals, Canad. J. Math. 59 (2007), no. 3, 575–595.
Search in Google Scholar Back to article
HRUŠÁK, M.: Combinatorics of filters and ideals, Contemp. Math. 533 (2011), 29–69.
Search in Google Scholar Back to article
HRUŠÁK, M.: Katětov order on Borel ideals, Arch. Math. Logic 56 (2017), no. 3, 831–847.
Search in Google Scholar Back to article
HRUŠÁK, M.—GARĆIA-FERREIRA, S.: Ordering MAD families a la Katětov,J. Symb. Log. 68 (2003), no. 4, 1337–1353.
Search in Google Scholar Back to article
JALALI-NAINI, S.-A.: The Monotone Subsets of Cantor Space, Filters and Descriptive Set Theory: Ph.D. Thesis, University of Oxford, 1976.
Search in Google Scholar Back to article
LACZKOVICH, M.—RECŁAW, I.: Ideal limits of sequences of continuous functions, Fundam. Math. 203 (2009), no. 1, 39–46.
Search in Google Scholar Back to article
LAFLAMME, C.: Filter games and combinatorial properties of winning strategies, Contemp. Math. 192 (1995), 51–67.
Search in Google Scholar Back to article
MAČAJ, M.—SLEZIAK, M.: ℐ𝒦-convergence, Real Anal. Exchange 36 (2010/11), no. 1, 177–193.
Search in Google Scholar Back to article
MARTON, A.—ŠUPINA, J.: On P -like ideals induced by disjoint families,J.Math. Anal. Appl. 528 (2023), no. 2, Paper no. 127551.
Search in Google Scholar Back to article
MEZA ALCÁNTARA, D.: Ideals and filters on countable sets: Ph. D. Thesis, Universidad Nacional Autonoma de Mexico, Mexico, 2009.
Search in Google Scholar Back to article
REPICKÝ, M.: Cardinal invariants and the collapse of the continuum by Sacks forcing, J. Symb. Log. 73 (2008), no. 2, 711–727.
Search in Google Scholar Back to article
STANISZEWSKI, M.: On ideal equal convergence II,J.Math. Anal.Appl. 451 (2017), no. 2, 1179–1197.
Search in Google Scholar Back to article
SZEMERÉDI, E.: On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), no. 1, 199–245.
Search in Google Scholar Back to article
ŠUPINA, J.: Ideal QN-spaces, J. Math. Anal. Appl. 435 (2016), no. 1, 477–491.
Search in Google Scholar Back to article
TALAGRAND, M.: Compacts de fonctions mesurables et filtres non mesurables,Acta Arith. 67 (1980), no. 1, 13–43.
Search in Google Scholar Back to article

P-Like Properties of Meager Ideals and Cardinal Invariants

Abstract

Paradigm

My account