A Complete Boolean Algebra That Has No Proper Atomless Complete Sublagebra

There exists a complete atomless Boolean algebra that has no proper atomless complete subalgebra. An atomless complete Boolean algebra B is simple [5] if it has no atomless complete subalgebra A such that A 6= B. The question whether such an algebra B exists was first raised in [8] where it was proved that B has no proper atomless complete subalgebra if and only if B is rigid andminimal. For more on this problem, see [4], [5] and [1, p. 664]. Properties of complete Boolean algebras correspond to properties of generic models obtained by forcing with these algebras. (See [6], pp. 266–270; we also follow [6] for notation and terminology of forcing and generic models.) When in [7] McAloon constructed a generic model with all sets ordinally definable he noted that the corresponding complete Boolean algebra is rigid, i.e. admitting no nontrivial automorphisms. In [9] Sacks gave a forcing construction of a real number of minimal degree of constructibility. A complete Boolean algebra B that adjoins a minimal set (over the ground model) is minimal in the following sense: If A is a complete atomless subalgebra of B then there exists a partition W of 1 such that for every w ∈ W , Aw = Bw, where Aw = {a · w : a ∈ A}. (1) In [3], Jensen constructed, by forcing over L, a definable real number of minimal degree. Jensen’s construction thus proves that in L there exists rigid minimal The first author was supported in part by an NSF grant DMS-9401275. The second author was partially supported by the U.S.–Israel Binational Science Foundation. Publication No. 566