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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, what are the sufficient conditions for a given Boolean algebra $\mathcal{A}\subseteq \wp(I)$ with $\mbox{fin}(I)\subseteq \mathcal{A}$ to have $\exp \exp |I|$ many maximal ideals containing $\mbox{fin}(I)$.

Thank you.

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    $\begingroup$ Pospíšil's result is published in Annals of Mathematics 38 (1937), no. 4, 845­-846 (jstor.org/stable/1968840). $\endgroup$
    – Emil Jeřábek
    Apr 5, 2012 at 13:31
  • $\begingroup$ Thanks, I have found the same result in On bicompact spaces. Publ. Fac. Sci. Univ. Masaryk 270 (1939), 3-16. $\endgroup$
    – Tomasz Kania
    Apr 5, 2012 at 14:34
  • $\begingroup$ May I ask why the downvote? The question was asked 9 years ago... $\endgroup$
    – Tomasz Kania
    Nov 18, 2021 at 10:54

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The result on the number of maximal ideals can be found as Corollary 7.4. in the book Comfort & Negrepontis "The Theory of Ultrafilters", 1974. A proof can also be found at PlanetMath here.

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I am sure this is in Koppelberg's Handbook of Boolean Algebras, though I have not checked (I will check later). A proof of Pospisil's result is in this survey-paper of mine that will appear in RIMS Kokyuroku.

I am not sure I completely understand the second part of your question, though. Are you asking for conditions when you have this many maximal ideals? For example, it could happen that $\mathcal A$ is only of size $|I|$ and you have at most $2^{|I|}$ maximal ideals.

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Apart from the obscure Czech journal, Pospíšil published his result in the Annals of Mathematics:

Bedřich Pospíšil: Remark on bicompact spaces, Annals of Mathematics 38 (1937), no. 4, pp. 845–846, doi 10.2307/1968840.

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