Applications of \sigma-Filtered Boolean Algebras
by
Sabine Koppelberg
Freien Universität Berlin

We apply the notion of a \sigma-filtration to give uniform and transparent proofs of several (mostly known) results on existence of homomorphisms between Boolean algebras: Mokobodzki's theorem and the Dow-Vermeer theorem on homomorphic images of complete Boolean algebras (both assuming the continuum hypothesis), Borel liftings of the measure resp. the category algebra in the Cohen model of 2^\omega = \omega₂, and, again in the Cohen model, a characterization of those algebras which are homomorphic images resp. retracts of the quotient algebra P(\omega)/fin.

A filtration of a Boolean algebra A is an increasing chain (A_\alpha)_{\alpha < \tau}, for some ordinal \tau, of subalgebras of A such that A = \cup _{\alpha < \tau} A_\alpha. It is a \sigma-filtration if, in addition, each A_\alpha is a \sigma-embedded subalgebra of A.

Here, for A a Boolean algebra and C a subalgebra of A, we call C a \sigma-embedded subalgebra of A if for every a in A, the ideal { c in C : c <= a } is countably generated.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-56.