3
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ Pospíšil's result is published in Annals of Mathematics 38 (1937), no. 4, 845­-846 (jstor.org/stable/1968840). $\endgroup$ – Emil Jeřábek Apr 5 '12 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 '12 at 14:34
5
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.