Geodesic
Comparison game on Borel ideals
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 2, pp. 191-204.

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We propose and study a “classification” of Borel ideals based on a natural infinite game involving a pair of ideals. The game induces a pre-order $\sqsubseteq$ and the corresponding equivalence relation. The pre-order is well founded and “almost linear”. We concentrate on $F_{\sigma}$ and $F_{\sigma\delta}$ ideals. In particular, we show that all $F_{\sigma}$-ideals are $\sqsubseteq$-equivalent and form the least equivalence class. There is also a least class of non-$F_{\sigma}$ Borel ideals, and there are at least two distinct classes of $F_{\sigma\delta}$ non-$F_{\sigma}$ ideals.
Classification : 03E05, 03E15
Keywords: ideals on countable sets; comparison game; Tukey order; games on integers 
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     title = {Comparison game on {Borel} ideals},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {191--204},
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     number = {2},
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Hrušák, Michael; Meza-Alcántara, David. Comparison game on Borel ideals. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 2, pp. 191-204. http://geodesic.mathdoc.fr/item/CMUC_2011__52_2_a2/