Series: Penn State Logic Seminar
Date: Tuesday, April 24, 2001
Time: 2:30 - 3:20 PM
Place: 316 Willard Building
Speaker: Natasha Dobrinen, Mathematics, University of Minnesota
Title:
Complete embeddings of the Cohen algebra into three classic examples
of complete, non-measurable, atomless, c.c.c. Boolean algebras.
Abstract:
Von Neumann conjectured that every complete, c.c.c. Boolean algebra
which satisfies the weak (omega,omega)-distributive law carries a
strictly positive, sigma-additive measure. Although consistent
counterexamples have been obtained, whether von Neumann's conjecture
is consistent with ZFC remains an open problem. In view of this, it
is of interest to investigate distributive laws in complete,
c.c.c. Boolean algebras. In this talk, I will construct complete
embeddings of the Cohen algebra into several classic examples of
complete, non-measurable, atomless, c.c.c. Boolean algebras, namely,
the Gaifman, Argyros, and Galvin-Hajnal algebras. Since the Cohen
algebra does not satisfy any form of distributivity, the complete
embedding of the Cohen algebra into each of these three Boolean
algebras implies that they are completely non-distributive. This
leads to the question: Within ZFC, is there a complete,
non-measurable, atomless, c.c.c. Boolean algebra into which the Cohen
algebra does not embed as a complete subalgebra?