1 5 D ec 1 99 7 REGULAR SUBALGEBRAS OF COMPLETE BOOLEAN ALGEBRAS

There is shown that there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC. A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, supB X = 1 implies supA X = 1; see e.g. Heindorf and Shapiro [5]. Clearly, every dense subalgebra is regular. Although, every complete Boolean algebra contains a free Boolean algebra of the same size (see the Balcar-Franek Theorem; [1]), not always such an embedding is regular. For instance, if B is a measure algebra, then it contains a free subalgebra of the same cardinality as B, but B cannot contain any free Boolean algebra as a regular subalgebra. Indeed, measure algebras are weakly σ-distributive and free Boolean algebra are not and a regular subalgebra of a σ-distributive one is again σ-distributive. Thus B does not contain any free Boolean algebra. On the other hand, measure algebras are not σ-centered. So, a natural question arises whether there can exists a σ-centered, complete, atomless Boolean algebra B without regular free subalgebras. Since countable atomless Boolean algebras are free and every free Boolean algebra contains a countable regular free subalgebra, it is enough to ask whether B contains a countable regular subalgebra. In the paper we prove that such an algebra exists iff there exists a nowhere dense ultrafilter. Definition 1 (Baumgartner Principle, see [2]). A filter D on ω is called nowhere dense if for every function f from ω to the Cantor set 2 there exists a set A ∈ D such that f(A) is nowhere dense in 2. In the sequel we will rather interested in nowhere dense ultrafilters. Observe that every P -ultrafilter (i.e. every P -point in ω) is a nowhere dense ultrafilter. The research of the second author was partially supported by the Basic Research Foundation of the Israel Academy of Sciences and Humanities. This publication has Number 640 in S. Shelah’s list. November 17, 1997