论文标题
关于理想之间的borel理想的结构$ \ mathcal {ed} $和$ \ mathrm {fin} \ otimes \ otimes \ mathrm {fin} $
On the structure of Borel ideals in-between the ideals $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$ in the Katětov order
论文作者
论文摘要
对于一个家庭$ \ MATHCAL {F} \ subseteqω^ω$，我们定义了$ω\timesmomΩ$上的理想$ \ Mathcal {i}（\ Mathcal {F}）$ \ Mathcal {f} \，\ forall^\ infty n \，（| \ {k：（n，k）\在A \} | \ leq f（n））中使用$ \ \ nathcal {i}（i}（i}（i}（i}（i Mathcal {f}）的理想，我们在bet nondount of Unsuce n offication n offication n office n offication n office n office n office offorms $ \ MATHCAL {ED} = \ {A \subseteqΩ\timsΩ：\已存在m \，\ forall^\ forall^\ infty n \，（| \ {k：\ {n，k）\ in A \} | <m）| <m）） \ {a \subseteqΩ\timesΩ：\ forall^\ infty n \，（| \ {k：（n，k）\ in a \} | <\ aleph_0）\} $ inKatětov订单中的\} $很复杂。即，有$ \ Mathcal {p}（ω）/\ Mathrm {fin} $ in-bete $ \ mathcal {ed} $和$ \ m atrm {fin} \ otimes \ mathrm {fin} $的副本在$ \ otimem {fin} $，因此$ \ nsime $ \ m \ \ mathaak的尺寸越来越大。 $ \ mathfrak {c} $。
For a family $\mathcal{F}\subseteq ω^ω$ we define the ideal $\mathcal{I}(\mathcal{F})$ on $ω\timesω$ to be the ideal generated by the family $\{A\subseteq ω\timesω:\exists f\in \mathcal{F}\,\forall^\infty n\, (|\{k:(n,k)\in A\}|\leq f(n))\}.$ Using ideals of the form $\mathcal{I}(\mathcal{F})$, we show that the structure of Borel ideals in-between two well known Borel ideals $\mathcal{ED} = \{A\subseteqω\timesω:\exists m \, \forall^\infty n\, (|\{k:(n,k)\in A\}|<m))\}$ and $\mathrm{Fin}\otimes\mathrm{Fin} = \{A\subseteqω\timesω:\forall^\infty n \, (|\{k:(n,k)\in A\}|<\aleph_0))\}$ in the Katětov order is fairly complicated. Namely, there is a copy of $\mathcal{P}(ω)/\mathrm{Fin}$ in-between $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$, and consequently there are increasing and decreasing chains of length $\mathfrak{b}$ and antichains of size $\mathfrak{c}$.
