Comparison game on Borel ideals

Michael Hrušák; David Meza-Alcántara

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 2, page 191-204
  • ISSN: 0010-2628

Abstract

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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 ⊑ and the corresponding equivalence relation. The pre-order is well founded and “almost linear”. We concentrate on F σ and F σ δ ideals. In particular, we show that all F σ -ideals are ⊑ -equivalent and form the least equivalence class. There is also a least class of non- F σ Borel ideals, and there are at least two distinct classes of F σ δ non- F σ ideals.

How to cite

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Hrušák, Michael, and Meza-Alcántara, David. "Comparison game on Borel ideals." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 191-204. <http://eudml.org/doc/246623>.

@article{Hrušák2011,
abstract = {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.},
author = {Hrušák, Michael, Meza-Alcántara, David},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ideals on countable sets; comparison game; Tukey order; games on integers; ideals on countable sets; comparison game; Tukey order; games on integers},
language = {eng},
number = {2},
pages = {191-204},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Comparison game on Borel ideals},
url = {http://eudml.org/doc/246623},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Hrušák, Michael
AU - Meza-Alcántara, David
TI - Comparison game on Borel ideals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 191
EP - 204
AB - 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.
LA - eng
KW - ideals on countable sets; comparison game; Tukey order; games on integers; ideals on countable sets; comparison game; Tukey order; games on integers
UR - http://eudml.org/doc/246623
ER -

References

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  2. Farah I., Analytic quotients: Theory of liftings for quotients over analytic ideals on integers, Mem. Amer. Math. Soc. 148 (2000), no. 702. MR1711328
  3. Kechris A.S., Classical Descriptive Set Theory, Springer, New York, 1995. Zbl0819.04002MR1321597
  4. Laflamme C., Leary C.C., 10.4064/fm173-2-4, Fund. Math. 173 (2002), 159–173. Zbl0998.03038MR1924812DOI10.4064/fm173-2-4
  5. Mazur K., F σ -ideals and ω 1 ω 1 * -gaps in the Boolean algebras P ( ω ) / I , Fund. Math. 138 (1991), no. 2, 103–111. MR1124539
  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., 10.1016/S0168-0072(98)00051-7, Annals of Pure and Applied Logic 99 (1999), 51–72. Zbl0932.03060MR1708146DOI10.1016/S0168-0072(98)00051-7

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