| Title:
|
Comparison game on Borel ideals (English) |
| Author:
|
Hrušák, Michael |
| Author:
|
Meza-Alcántara, David |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
191-204 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
ideals on countable sets |
| Keyword:
|
comparison game |
| Keyword:
|
Tukey order |
| Keyword:
|
games on integers |
| MSC:
|
03E05 |
| MSC:
|
03E15 |
| idZBL:
|
Zbl 1240.03023 |
| idMR:
|
MR2849045 |
| . |
| Date available:
|
2011-05-17T08:34:10Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141498 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Kechris A.S.: Classical Descriptive Set Theory.Springer, New York, 1995. Zbl 0819.04002, MR 1321597 |
| Reference:
|
[4] Laflamme C., Leary C.C.: Filter games on $\omega$ and the dual ideal.Fund. Math. 173 (2002), 159–173. Zbl 0998.03038, MR 1924812, 10.4064/fm173-2-4 |
| Reference:
|
[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 |
| Reference:
|
[6] Meza-Alcántara D.: Ideals and filters on countable sets.Ph.D. Thesis, Universidad Nacional Autónoma de México, Morelia, Michoacán, Mexico, 2009. |
| Reference:
|
[7] Solecki S.: Analytic Ideals and their Applications.Annals of Pure and Applied Logic 99 (1999), 51–72. Zbl 0932.03060, MR 1708146, 10.1016/S0168-0072(98)00051-7 |
| . |
