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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_1}$+$\diamond_{\omega_2}(cof(\omega_1))$?\Diamond_{\omega_2}(cof(\omega_1))$?

Thanks!

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# Suslin algebras

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!