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  • On thin-complete ideals of subsets of groups
    Banakh, Taras, 1968- ; Lyaskovska, Nadia
    Let ▫$\mathcal{F} \subset {\mathcal P}_G$▫ be a left-invariant lower family of subsets of a group ▫$G$▫. A subset ▫$A \subset G$▫ is called ▫$\mathcal{F}$▫-thin if ▫$xA \cap yA \in \mathcal{F}$▫ for ... any distinct elements ▫$x, y \in G$▫: The family of all ▫$\mathcal{F}$▫-thin subsets of ▫$G$▫ is denoted by ▫$\tau(\mathcal{F})$▫. If ▫$\tau(\mathcal{F}) = \mathcal{F}$▫ then ▫$\mathcal{F}$▫ is called thin-complete. The thin-completion ▫$\tau^\ast (\mathcal{F})$▫ 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 ▫$A \subset G$▫ belongs to ▫$\tau^\ast (G)$▫ if and only if, for any sequence ▫$(g_n)_{n \in \omega}$▫ of nonzero elements of ▫$G$▫, there is ▫$n \in \omega$▫ such that ▫$$\bigcup_{i_0, \cdots, \in \{0,1\}} g_0^{i_0} \cdots g_n^{i_n}A \in \mathcal{F}.$$▫ We also prove that, for an additive family ▫$\mathcal{F} \subset {\mathcal P}_G$▫, its thin-completion ▫$\tau^\ast (\mathcal{F})$▫ is additive. If a group ▫$G$▫ is countable and torsion-free, then the completion ▫$\tau^\ast (\mathcal{F}_G)$▫ 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.
    Vir: Ukraïnsʹkij matematičnij žurnal. - ISSN 1027-3190 (Vol. 63, no. 6, 2011, str. 741-754)
    Vrsta gradiva - članek, sestavni del
    Leto - 2011
    Jezik - angleški
    COBISS.SI-ID - 16082009

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