A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega\_1}$. A proof is given in the current edition of his book "Set Theory." He also mentions that it is consistent to have Suslin algebras of size $2^{\omega\_1}$ but does not sketch an argument. I have been having trouble finding a proof of this fact. I looked in the older edition of Jech's book, and he says a little more about it there, but still no proof. I would be very grateful if someone could point me to a paper that has a construction of a large Suslin algebra, or sketch a proof here in an answer. I am also curious about the following:


1) Is it consistent to have Suslin trees but no Suslin algebras of density larger than $\omega\_1$?


2) Can the existence of large Suslin algebras be proved from several diamonds, say $\Diamond\_{\omega\_1}$ + $\Diamond\_{\omega\_2}(cof(\omega\_1))$?


Thanks!



Suslin algebras
Jensen constructed a Suslin algebra of size $2^{\aleph\_1}$ ($= \aleph\_2$ in that model) to prove the consistency of the Suslin Hypothesis with the Continuum Hypothesis. You can find a detailed account of his intricate construction in *The Souslin Problem* (Lecture Notes in Mathematics 405) by Devlin and Johnsbraten. This is essentilally the only construction I am aware of.


