Advertisement

Ukrainian Mathematical Journal

, Volume 63, Issue 6, pp 865–879 | Cite as

On thin-complete ideals of subsets of groups

  • T. Banakh
  • N. Lyaskovska
Article
Let \( \mathcal{F} \subset {\mathcal{P}_G} \) be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called \( \mathcal{F} \)-thin if \( xA \cap yA \in \mathcal{F} \) for any distinct elements x, yG. The family of all \( \mathcal{F} \)-thin subsets of G is denoted by \( \tau \left( \mathcal{F} \right) \). If \( \tau \left( \mathcal{F} \right) = \mathcal{F} \), then \( \mathcal{F} \) is called thin-complete. The thin-completion \( {\tau^*}\left( \mathcal{F} \right) \) of \( \mathcal{F} \) is the smallest thin-complete subfamily of \( {\mathcal{P}_G} \) that contains \( \mathcal{F} \). Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g n ) nω of nonzero elements of G, there is nω such that
$$ \bigcap\limits_{{i_0}, \ldots, {i_n} \in \left\{ {0,\;1} \right\}} {g_0^{{i_0}} \ldots g_n^{{i_n}}A \in \mathcal{F}} . $$
We also prove that, for an additive family \( \mathcal{F} \subset {\mathcal{P}_G} \), its thin-completion \( {\tau^*}\left( \mathcal{F} \right) \) is additive. If a group G is countable and torsion-free, then the completion \( {\tau^*}\left( {{\mathcal{F}_G}} \right) \) of the ideal \( {\mathcal{F}_G} \) of finite subsets of G is coanalytic and non-Borel in the power-set \( {\mathcal{P}_G} \) endowed with natural compact metrizable topology.

Keywords

Polish Space Inductive Assumption Free Abelian Group Countable Group Measurable Cardinal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ie. Lutsenko and I. V. Protasov, “Relatively thin and sparse subsets of groups,” Ukr. Mat. Zh., 63, No. 2, 216–225 (2011); English translation: Ukr. Math. J., 63, No. 2, 254–263 (2011).CrossRefGoogle Scholar
  2. 2.
    Ie. Lutsenko and I. V. Protasov, “Sparse, thin and other subsets of groups,” Int. J. Algebra Comput., 19, No. 4, 491–510 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    T. Banakh and N. Lyaskovska, “Completeness of invariant ideals in groups,” Ukr. Mat. Zh., 62, No. 8, 1022–1031 (2010); English translation: Ukr. Math. J., 62, No. 8, 1187–1198 (2010).CrossRefGoogle Scholar
  4. 4.
    A. Kechris, Classical Descriptive Set Theory, Springer (1995).Google Scholar
  5. 5.
    A. Bella and V. I. Malykhin, “ On certain subsets of a group,” Questions Answers Gen. Top., 17, No. 2, 183–197 (1999).MathSciNetzbMATHGoogle Scholar
  6. 6.
    D. A. Martin, “Measurable cardinals and analytic games,” Fund. Math., 66, 287–291 (1970).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • T. Banakh
    • 1
  • N. Lyaskovska
    • 1
  1. 1.LvivUkraine

Personalised recommendations