Regular subalgebras of complete Boolean algebras

Abstract

Abstract. It is proved that the following conditions are equivalent:(a) there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore the existence of such algebras is undecidable in ZFC. In "forcing language" condition (a) says that there exists a nontrivial σ-centered forcing not adding Cohen reals A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, sup B X = 1 implies sup A X = 1; see e.g. Heindorf and Shapiro [6]. 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; [2]), 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 infinite free Boolean algebra as a regular subalgebra. Indeed, measure algebras are weakly σ-distributive but free Boolean algebras are not, and a regular subalgebra of a weakly 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 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.

The Modal Logic of -Centered Forcing and Related Forcing Classes

“…which corresponds to the linearity of the forcing class (see [5]), holds over L: by a result of Błaszczyk and Shelah [4], in L there is a -centered forcing notion which does not add a Cohen real. So let P be the L-least such forcing notion, consider the statements:…”

“…is a button for P-forcing. Therefore the statements 2 ℵ 0 ≥ ℵ α | α < form a strong ratchet, and we can use it to construct an n-switch as in Lemma 1.13, to obtain condition (4). So all the conditions of Theorem 4.4 can be met, giving us the result.…”

Inverse systems and I-favorable spaces

“…86-91], gives an another example of a space with no skeletal map onto a dense in itself, separable and metrizable space. In [3] A. Błaszczyk and S. Shelah are considered separable extremally disconnected spaces with no skeletal map onto a dense in itself, separable and metrizable space. They formulated the result in terms of Boolean algebra: There is a nowhere dense ultrafilter on ω if, and only if there is a complete, atomless, σ -centered Boolean algebra which contains no regular, atomless, countable subalgebra.…”

Free Boolean algebras and nowhere dense ultrafilters

“…Theorem 1 (B laszczyk and Shelah [4]). An atomless -centered complete Boolean algebra without any regular free subalgebra exists i there exists a nowhere dense ultraÿlter.…”