We say that $\mathcal {I}$ is an ideal independent family if no element of ${\mathcal {I}}$ is a subset mod finite of a union of finitely many other elements of ${\mathcal {I}}.$ We will show that the minimum size of a maximal ideal independent family is consistently bigger than both $\mathfrak {d}$ and $\mathfrak {u},$ this answers a question of Donald Monk.

中文翻译：

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理想的独立家庭和超滤器数量

我们说$\数学{我}$如果没有任何因素，是一个理想的独立家庭${\数学{I}}$是有限许多其他元素的并集的模有限子集${\mathcal {I}}.$我们将证明最大理想独立家庭的最小规模始终大于两者$\mathfrak {d}$和$\mathfrak {u},$这回答了唐纳德·蒙克的一个问题。