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Inductive limits of ideals
Topology and its Applications ( IF 0.6 ) Pub Date : 2021-07-22 , DOI: 10.1016/j.topol.2021.107798
Adam Kwela 1
Affiliation  

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal I (rk(I)) as minimal ordinal α<ω1 such that there is SΣ1+α0 with IS and IS=, where I is the filter dual to the ideal I. Moreover, they introduced ideals Finα, for all α<ω1, and conjectured that rk(I)α if and only if I contains an isomorphic copy of Finα (FinαI). To define Finα in the case of limit ordinals 0<α<ω1, G. Debs and J. Saint Raymond introduced inductive limits of ideals.

We show that the above conjecture is false in the case of α=ω by constructing an ideal Finω of rank ω such that FinωFinω. However, we show that FinωI is equivalent to nωFinnI. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.



中文翻译:

理想的归纳极限

G. Debs 和 J. Saint Raymond 在 2009 年定义了解析理想的 Borel 分离秩 一世 ((一世)) 作为最小序数 α<ω1 使得有 Σ1+α0一世一世=, 在哪里 一世 是理想的双重过滤器 一世. 此外,他们引入了理想α, 对全部 α<ω1,并推测 (一世)α 当且仅当 一世 包含一个同构的副本 α (α一世)。界定α 在极限序数的情况下 0<α<ω1, G. Debs 和 J. Saint Raymond 介绍了理想的归纳极限。

我们证明上述猜想在以下情况下是错误的 α=ω 通过构建理想 ωω使得ωω. 然而,我们证明ω一世 相当于 nωn一世. 我们讨论(由上述结果表明)对极限序数的原始猜想的可能修改。

更新日期:2021-07-29
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