# On thin-complete ideals of subsets of groups

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Let \( \mathcal{F} \subset {\mathcal{P}_G} \) be a left-invariant lower family of subsets of a group 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*. A subset*A*⊂*G*is called \( \mathcal{F} \)-*thin*if \( xA \cap yA \in \mathcal{F} \) for any distinct elements*x*,*y*∈*G*. 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 })_{ n∈ω }of nonzero elements of*G*, there is*n*∈*ω*such that$$ \bigcap\limits_{{i_0}, \ldots, {i_n} \in \left\{ {0,\;1} \right\}} {g_0^{{i_0}} \ldots g_n^{{i_n}}A \in \mathcal{F}} . $$

*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
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## References

- 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.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.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.A. Kechris,
*Classical Descriptive Set Theory*, Springer (1995).Google Scholar - 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.D. A. Martin, “Measurable cardinals and analytic games,”
*Fund. Math.*,**66**, 287–291 (1970).MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, Inc. 2011