Skip to main content



We investigate ideals of the form {Aω: ΣnAxn is unconditionally convergent} where (xn)nω is a sequence in a Polish group or in a Banach space. If an ideal on ω can be seen in this form for some sequence in X, then we say that it is representable in X.

After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.

We focus on the family of ideals representable in c0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable in c0, and that a tall Fσ P-ideal is representable in c0 iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in 1 but not in ℝ.

Finally, we summarize some open problems of this topic.

Hide All
[1]Banach, Stefan, Théorie Des Opérations Linéaires, Chelsea, New York, 1955.
[2]Diestel, Joseph, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984.
[3]Drewnowski, Lech and Paúl, Pedro J., The Nikodým property for ideals of sets defined by matrix summability methods. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.), vol. 94 (2000), no. 4, pp. 485503. Perspectives in mathematical analysis (Spanish.)
[4]Farah, Ilijas, Ideals induced by Tsirelson submeasures. Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 243258.
[5]Farah, Ilijas, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702.
[6]Freedman, A. R. and Sember, J. J., Densities and summability. Pacific Journal of Mathematics, vol. 95 (1981), no. 2, pp. 293305.
[7]Farkas, Barnabás and Soukup, Lajos, More on cardinal invariants of analytic P-ideals. Commentationes Mathematicae Universitatis Carolinae, vol. 50 (2009), no. 2, pp. 281295.
[8]Filipów, Rafał and Szuca, Piotr, Rearrangement of conditionally convergent series on a small set. Journal of Mathematical Analysis and Applications, vol. 362 (2010), no. 1, pp. 6471.
[9]Heil, Christopher, A basis theory primer. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, expanded edition, 2011.
[10]Hernández-Hernández, Fernando and Hrušák, Michael, Cardinal invariants of analytic P-ideals. Canadian Journal of Mathematics, vol. 59 (2007), no. 3, pp. 575595.
[11]Hrušák, Michael, Combinatorics of filters and ideals, Set theory and its applications, vol. 533, American Mathematical Society, Providence, RI, 2011, pp. 2969.
[12]Klee, V. L. Jr., Invariant metrics in groups (solution of a problem of Banach). Proceedings of the American Mathematical Society, vol. 3 (1952), pp. 484487.
[13]Kwela, Adam and Sabok, Marcin, Topological representations, doi:10.1016/j.jmaa.2014.09.059, preprint.
[14]Meza-Alcántara, D., Ideals and filters on countable sets, PhD thesis, Universidad Nacional Autónoma México, Mexico City, 2009.
[15]Mátrai, Tamás, More cofinal types, preprint.
[16]Mazur, Krzysztof, F σ-ideals and -gaps in the Boolean algebras P(ω)/I. Fundamenta Mathematicae, vol. 138 (1991), no. 2, pp. 103111.
[17]Solecki, Sławomir, Analytic ideals. Bulletin of Symbolic Logic, vol. 2 (1996), no. 3, pp. 339348.
[18]Solecki, Sławomir and Todorcevic, Stevo, Cofinal types of topological directed orders. Annales de l’institut Fourier (Grenoble), vol. 54 (2004), no. 6, pp. 18771911.
[19]Solecki, Sławomir and Todorcevic, Stevo, Avoiding families and Tukey functions on the nowhere-dense ideal. Journal of the Institute of Mathematics of Jussieu, vol. 10 (2011), no. 2, pp. 405435.
[20]Veličković, Boban, A note on Tsirelson type ideals. Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 259268.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 144 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th August 2018. This data will be updated every 24 hours.