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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!

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Jech says that such a thing can be forced using a forcing that generalizes the forcing for adding a Suslin tree with countable approximations. I wonder what he means! –  Monroe Eskew Dec 8 '12 at 1:19

2 Answers 2

up vote 6 down vote accepted

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.

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Preliminary thoughts on (1) and (2)... The answer to (1) ought to be yes but the way to get there is not necessarily the obvious one. For (2), Jensen's construction depends heavily on $\square_{\omega_1}$ in order to "iterate" Souslin trees for length $\omega_2$; I don't think it's known whether such additional devices are necessary. Also, Jensen's construction uses a preparatory step that preserves $\diamondsuit_{\omega_1}$ but kills $\diamondsuit_{\omega_1}^*$; it is an open problem whether that step is necessary. –  François G. Dorais Dec 3 '12 at 6:07
    
Francois, could you elaborate on your thoughts about (1)? –  Monroe Eskew Dec 8 '12 at 1:21

Jech, Thomas J. Some combinatorial problems concerning uncountable cardinals. Ann. Math. Logic 5 (1972/73), 165–198.

Section 5 contains the forcing for arbitrarily big Suslin algebras. See also:

Scharfenberger-Fabian, Gido Chain homogeneous Souslin algebras. (English summary) MLQ Math. Log. Q. 57 (2011), no. 6, 591–610.

According to Gido, his paper, Jech's, and Jensen's result mentioned by Dorais are the only known contsructions of "big" Suslin algebras. Jech's forcing is thus the only thing that makes one bigger than $\aleph_2$.

Note that Jech does not explicitly prove distributivity in his paper, but it is not hard to show given what is there.

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