Embedding theorems for boolean algebras and consistency results on ordinal definable sets

Abstract

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems.Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C.Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.