The book mainly consists of four chapters (Chapters 1, 3, 4 and 5), each of which corresponds to a different line of research. (Chapter 2 is very short and it contains only some background material.)

The basic question considered in this monograph is:

The ideal I is always assumed to be analytic (as a matter of fact, all of the considered ideals are Borel).

The main recent development on the subject shows that this was not the most general (or most interesting) question along these lines that one could ask. My result that

Remarks below are intended for a reader familiar with

Both ideals NWD(Q) and NULL(Q) have the Radon-Nikodym property. NWD(Q) is the first known ideal that has the RNP that is not an F-ideal.

This provides some information on Problem 3.14.1: all nonpathological F-sigma ideals, as well as the ideals NWD(Q) and NULL(Q) belong to the class of analytic ideals described in this problem. Unfortunately, Problem 3.14.1 unneccessarily involves a more general form of (wide open) Question 1.14.1. This problem was separated into two parts in

Question 3.14.3 has been solved:

In 2001 Juris Steprans has proved that there is a family of continuum many coanalytic ideals that have pairwise non-isomorphic quotient algebras. Moreover, the regular open algebras of these quotient algebras are nonisomorphic.

Mike Oliver (a student of Greg Hjorth at UCLA) has proved (December 2002) that there are continuum many pairwise nonisomorphic quotients over analytic P-ideals. This solves Question 3.14.3.

All quotients over ideals I-alpha and W-alpha (see

Consider the class of dense ideals of the form Exh(phi), where phi is a supremum of a family of pairwise orthogonal lower semicontinuous measures. This class includes all dense density ideals and all summable ideals. Under CH there are exactly six nonisomorphic quotients over ideals in this class.

These results appear in

Question 3.7.6 turned out to have a simple answer (yes). E.g., the ideal NWD(Q) of all nowhere dense subsets of the rationals is homogeneous (thanks to Micha(e)l Hrusak for turning my attention to this interesting ideal).

This question was motivated by a question of Just and Krawczyk: if I, J are F-sigma-delta ideals that are not F-sigma and such that their quotients are homogeneous, are their quotients isomorphic? The answer turns out to be negative (see

In 2004 I have announced there are no analytic Hausdorff gaps over Z_0 or any other EU-ideal. I am retracting this claim, for a partial result see

Regarding Question 5.13.9: It turns out that the answer is positive, as there is a natural separation property (weaker than Todorcevic's, yet still rather useful) that all analytic P-ideals satisfy. These results appear in

Last but not least - a compact, self-contained, and error-free exposition of the `OCA lifting theorem' was written by David Fremlin and it is available from his web site (among `notes').

(Last revised: November 2007.)

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