Complete embeddings of the Cohen algebra into three families of c.c.c., non-measurable Boolean algebras.
type: Item
instance of: scholarly article
title: Complete embeddings of the Cohen algebra into three families of c.c.c., non-measurable Boolean algebras.
zbMATH Open document ID: 1057.03041
author: Natasha Dobrinen
published in: Pacific Journal of Mathematics
publication date: 2004-09-17T00:00:00Z
review text: The question is: which ccc forcings necessarily add Cohen reals? It is known that measure algebras do not. This paper shows that three classes of ccc forcings which are not measure algebras do add Cohen reals: Galvin-Hajnal algebras (in fact, they add cf \(2^{\omega}\) many), Argyros algebras, and the atomless Gaifman algebras (or their atomless parts). In some sense these results are negative: von Neumann conjectured that a Boolean \(\sigma\)-algebra with ccc and the weak \((\omega,\omega)\)-distributive law is a measure algebra. The completions of the Galvin-Hajnal, Argyros, and Gaifman algebras are complete ccc algebras which fail to satisfy certain conditions that measure algebras satisfy, hence were candidates for counterexamples. But the results of this paper show that the weak \((\omega,\omega)\)-distributive law fails in these algebras.
Mathematics Subject Classification ID: 03E40
zbMATH DE Number: 2101151
zbMATH Keywords: ccc Boolean algebra
zbMATH Keywords: Cohen algebra
zbMATH Keywords: ccc forcing
zbMATH Keywords: Cohen reals
reviewed by: Judith Roitman
MaRDI profile type: Publication
full work available at URL: pjm.2004.214.223
OpenAlex ID: W2033613436
DOI: 10.2140/PJM.2004.214.223
description: scientific article; zbMATH DE number 2101151