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ideals on countable sets; comparison game; Tukey order; games on integers
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.
[1] Bartoszyński T., Judah H.: Set Theory: On the Structure of the Real Line. A.K. Peters, Wellesley, Massachusetts, 1995. MR 1350295
[2] Farah I.: Analytic quotients: Theory of liftings for quotients over analytic ideals on integers. Mem. Amer. Math. Soc. 148 (2000), no. 702. MR 1711328
[3] Kechris A.S.: Classical Descriptive Set Theory. Springer, New York, 1995. MR 1321597 | Zbl 0819.04002
[4] Laflamme C., Leary C.C.: Filter games on $\omega$ and the dual ideal. Fund. Math. 173 (2002), 159–173. DOI 10.4064/fm173-2-4 | MR 1924812 | Zbl 0998.03038
[5] Mazur K.: $F_\sigma $-ideals and $\omega_1\omega_1^*$-gaps in the Boolean algebras $P(\omega)/I$. Fund. Math. 138 (1991), no. 2, 103–111. MR 1124539
[6] Meza-Alcántara D.: Ideals and filters on countable sets. Ph.D. Thesis, Universidad Nacional Autónoma de México, Morelia, Michoacán, Mexico, 2009.
[7] Solecki S.: Analytic Ideals and their Applications. Annals of Pure and Applied Logic 99 (1999), 51–72. DOI 10.1016/S0168-0072(98)00051-7 | MR 1708146 | Zbl 0932.03060
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