An old theorem of Pospisil asserts that for any infinite set $I$ the power-set algebra $\wp(I)$ has $\exp \exp |I|$ many maximal ideals containing the ideal of finite sets. This result is published in a rather obscure Czech journal but it seems it should be well-known and described in many textbooks/monographs. I would appreciate any references for that.
Also, I am interested in more general results, that is, when what are the sufficient conditions for a given Boolean algebra $\mathcal{A}\subseteq \wp(I)$ with $\mbox{fin}(I)\subseteq \mathcal{A}$, there are mathcal{A}$ to have $\exp \exp |I|$ many maximal ideals of $\mathcal{A}$ containing $\mbox{fin}(I)$.
Thank you.
