滤子、理想独立性与理想Mrówka空间
Filters, ideal independence and ideal Mrówka spaces
AI总结：
本文研究了理想独立族，证明了极大理想独立族的等价条件，确立了基数不变量间的独立性与一致性，并引入了相关的自然滤子与Mrówka空间类比。
摘要：
若族$\mathcal{A} \subseteq [ω]^ω$满足：对所有有限子集$\{X_i\}_{i\in n}\subseteq \mathcal A$及任意$A \in \mathcal{A} \setminus \{X_i\}_{i\in n}$，集合$A \setminus \bigcup_{i \in n} X_i$均无限，则称该族为理想独立的。我们证明，理想独立族$\mathcal{A}$是极大的当且仅当对于$ω$上的特定理想$\mathcal J$，$\mathcal A$是$\mathcal J$-完全可分且极大$\mathcal J$-几乎不相交的。我们证明$\mathfrak{u}\leq\mathfrak{s}_{mm}$，其中$\mathfrak{s}_{mm}$是极大理想独立族的最小基数。这特别确立了$\mathfrak{s}_{mm}$与$\mathfrak{i}$的独立性。给定不可数基数的任意集合$C$，我们展示如何通过力迫同时为每个$λ\in C$附加基数为$λ$的极大理想独立族，从而确立$C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$的一致性。假设$\mathsf{CH}$，我们构造一个极大理想独立族，该族在经过任意正规、$^ωω$-有界、保持$p$-点的力迫概念作用后仍保持极大性，并在若干被充分研究的力迫扩充中评估$\mathfrak{s}_{mm}$。我们还研究与理想独立性相关的自然滤子，并引入理想独立族的Mrówka空间类比。
英文摘要：
A family $\mathcal{A} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal A$ and $A \in \mathcal{A} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i \in n} X_i$ is infinite, is said to be ideal independent. We prove that an ideal independent family $\mathcal{A}$ is maximal if and only if $\mathcal A$ is $\mathcal J$-completely separable and maximal $\mathcal J$-almost disjoint for a particular ideal $\mathcal J$ on $ω$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, where $\mathfrak{s}_{mm}$ is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $λ$ for each $λ\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^ωω$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mrówka spaces for ideal independent families.
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