Limitations on representing $\mathcal {P}(X)$ as a union of proper subalgebras

Abstract:

For every integer $\mu \geqslant 3$, there exists a function $f_\mu :\mathbb N^+\rightarrow \mathbb N^+$ such that the following holds: (1) $f_\mu (k)=2k-\mu$ for $k$ large enough; (2) if $\mathfrak A$ is a finite nonempty collection of subalgebras of $\mathcal P(X)$ such that $\bigcap \mathfrak B$ is not $f_\mu \left (\#(\mathfrak B)\right )$-saturated, for all nonempty $\mathfrak B\subseteq \mathfrak A$, then $\bigcup \mathfrak A\neq \mathcal P(X)$.