Forcing a Boolean Algebra with Predesigned Automorphism Group
نویسندگان
چکیده
For suitable groups G we will show that one can add a Boolean algebra B by forcing in such a way that Aut(B) is almost isomorphic to G . In particular, we will give a positive answer to the following question due to J. Roitman: Is אω a possible number of automorphisms of a rich Boolean algebra? In [Ro], J. Roitman asked, is אω a possible number of automorphisms of a rich Boolean algebra? A Boolean algebra is rich if the number of automorphisms is greater than the size of the algebra and possible means consistent with ZFC (since GCH implies trivially that the answer is no). We will answer this question positively. For this we give a method of adding a Boolean algebra by forcing in such a way that we have a lot of control on the automorphism group of the Boolean algebra. Notice that by [Mo] Theorem 4.3, one can not hope of giving a positive answer to Roitmans question from the assumption that 2 > אω . Partially, our methods are similar to those in [Ro]. We say that a po-set P is κ-Knaster if for all pi ∈ P , i < κ , there is Z ⊆ κ of power κ such that for all i, j ∈ Z , pi and pj are compatible. ∗ Partially supported by the Academy of Finland, grant 40734, and the MittagLeffler Institute. † This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publ. 756
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