On thin-complete ideals of subsets of groups |
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Authors: | T Banakh N Lyaskovska |
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Institution: | 1.Lviv,Ukraine |
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Abstract: | Let F ì PG \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of PG {\mathcal{P}_G} that contains F \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
?i0, ?, in ? { 0, 1 } g0i0 ?gninA ? F . \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}} . |
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