A \(\sigma \)-ideal \(\mathcal {I}\) on a Polish group \((X,+)\) has the Smital Property if for every dense set D and a Borel \(\mathcal {I}\)-positive set B the algebraic sum \(D+B\) is a complement of a set from \(\mathcal {I}\). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are \(\mathfrak {c}\) many maximal invariant \(\sigma \)-ideals with Borel bases on the Cantor space \(2^\omega \).

中文翻译：

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具有 Smital 属性的理想

如果对于每个稠密集D和Borel \ ( \ mathcal { I }\) -正集B代数和\(D+B\)是来自\(\mathcal {I}\)的集合的补集。我们考虑了此属性的几种变体，并研究了它们与可数链条件、最大值以及它们通过 Fubini 产品的保存情况的联系。特别地，我们表明在 Cantor 空间\(2^\omega \)上存在\(\mathfrak {c}\)许多具有 Borel 基的最大不变量\(\sigma \)理想。