广义独立性
Generalized independence
摘要：
我们探讨不同的推广的经典概念的独立家庭的$ω$以下的研究发起的Fisher和蒙托亚。我们证明了在$\diamondsuit^*(κ)$下，我们可以得到$κ$上的强独立族，其大小为$2^κ$，并给出了$\mathsf{GCH}$在强独立族方面的等价性。我们合并了两个自然的方式，通过过滤器或理想的推广独立的家庭，我们专注于$\mathcal{C}$ -独立的家庭，其中$\mathcal{C}$是俱乐部过滤器。我们还发现了$\mathcal{J}$独立族的存在与理想$\mathcal{J}$的饱和度之间的关系。
英文摘要：
We explore different generalizations of the classical concept of independent families on $ω$ following the study initiated by Fisher and Montoya. We show that under $\diamondsuit^*(κ)$ we can get strongly independent families on $κ$ of size $2^κ$ and present an equivalence of $\mathsf{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $\mathcal{C}$-independent families, where $\mathcal{C}$ is the club filter. Also we show a relationship between the existence of $\mathcal{J}$-independent families and the saturation of the ideal $\mathcal{J}$.
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