树导出的理想：弗比尼迭代、极限 amalgamations 与 Katetov 障碍
Tree-derived ideals: Fubini iterations, limit amalgamations, and Katetov obstructions
AI总结：
本文研究了通过迹树导出的理想结构，展示了弗比尼迭代、极限 amalgamations 和 Katetov 障碍的性质，并证明了相关理想层次的完全性与正交性。
AI中文摘要：
我们开发了一种机器，用于从一个由 ω^{<ω} 索引的 ω 分割派生可测集上的理想。一个在迹树上定义的导出算子，参数化于一个辅助理想 J，产生一个严格的超限层次 H^J_α，该层次从第一层开始都是适当的理想，并且与所选分割无关。其有限层次正好是弗比尼幂，H_n ≅ Fin⊗(n+1)，而 H_ω = ∪n H_n 合并所有有限幂。我们建立了整个层次的呈现独立性、局部同质性、弗比尼递归性和 Π^1_1-完全性。设 F_ω 是 Kwela 的 canonical inductive limit，F'_ω 是他的 independent-partitions limit。我们证明 F_ω 不 ≤_K H_ω，尽管 F'_ω ⊆ H_ω 并且每个有限的 coherent fragment of 一个潜在的约简都是可实现的 over H_ω。证明引入了 essential depth，一个单调沿 Katetov 减少的 Fin⊗Fin 的不变量，并给出了 sharp non-extension bound N(m)=m+2。因此 H_ω 不包含 F_ω 的同构副本，并且 F_ω 不 ≤_K F'_ω。应用包括 chromatic ideals G_k ∈ H_2 \ H_1，其包含顺序记录除法，其 Katetov 顺序记录算术。我们还证明了 Cantor-Bendixson 和导出等级的正交性。最后，P(ω)/H 是一个 σ-闭的 reduced power B ≅ B^ω/Fin，包含 P(ω)/Fin 正常地，并且在 CH 下与 (P(ω)/Fin)^+ 相同。
英文摘要：
We develop a machinery for deriving ideals on a countable set from a partition of $ω$ indexed by $ω^{<ω}$. A derivative operator on trace trees, parametrized by an auxiliary ideal $\mathcal J$, yields a strict transfinite hierarchy $\mathcal H^{\mathcal J}_α$ of proper ideals, tall from level one onward and independent of the chosen partition. Its finite levels are exactly the Fubini powers, $\mathcal H_n\cong\mathrm{Fin}^{\otimes(n+1)}$, while $\mathcal H_ω=\bigcup_n\mathcal H_n$ amalgamates all finite powers. We establish presentation independence, local homogeneity, Fubini recursion, and $Π^1_1$-completeness of the full hierarchy. Let $\mathcal F_ω$ be Kwela's canonical inductive limit and $\mathcal F'_ω$ his independent-partitions limit. We prove $\mathcal F_ω\not\leq_K\mathcal H_ω$, although $\mathcal F'_ω\sqsubseteq\mathcal H_ω$ and every finite coherent fragment of a putative reduction is realizable over $\mathcal H_ω$. The proof introduces essential depth, an invariant monotone along Katetov reductions of $\mathrm{Fin}\otimes\mathrm{Fin}$, and gives the sharp non-extension bound $N(m)=m+2$. Thus $\mathcal H_ω$ contains no isomorphic copy of $\mathcal F_ω$, and $\mathcal F_ω\not\leq_K\mathcal F'_ω$. Applications include chromatic ideals $\mathcal G_k\in\mathcal H_2\setminus\mathcal H_1$ whose inclusion order records divisibility and whose Katetov order records arithmetic. We also prove the orthogonality of Cantor--Bendixson and derivative ranks. Finally, $\mathcal P(ω)/\mathcal H$ is a $σ$-closed reduced power $\mathbb B\cong\mathbb B^ω/\mathrm{Fin}$, contains $\mathcal P(ω)/\mathrm{Fin}$ regularly, and under CH is forcing-equivalent to $(\mathcal P(ω)/\mathrm{Fin})^+$.
