Ukrainian Mathematical Journal

, Volume 63, Issue 6, pp 865–879

# On thin-complete ideals of subsets of groups

• T. Banakh
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.

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