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On thin-complete ideals of subsets of groups

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Ukrainian Mathematical Journal Aims and scope

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.

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References

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 6, pp. 741–754, June, 2011.

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Banakh, T., Lyaskovska, N. On thin-complete ideals of subsets of groups. Ukr Math J 63, 865–879 (2011). https://doi.org/10.1007/s11253-011-0549-1

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  • DOI: https://doi.org/10.1007/s11253-011-0549-1

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