Analytic ideals and their quotients

An ideal $\cal I$ of sets of integers is analytic if it is given in an explicit and simple way, for example ${\cal I}\sb{1/n} = \{A:\sum\sb{n\in A} 1/n < \infty\}.$ More generally, if $\varphi$ is a submeasure supported by $\rm I\!N$ satisfying certain continuity properties then$${\cal I} = \{A:\lim\sb{n}\varphi (A\\\lbrack 1,n)) = 0\}$$is a typical analytic P-ideal. These simple objects give rise to quotients ${\cal P}({\rm I\!N})/{\cal I}$ which can be quite complex, or saturated. The basic question considered in this Thesis is: How does a change of the ideal ${\cal I}$ affect the change of its quotient? This question has to be put in a proper setting. We require that a homomorphism $\Phi{:}{\cal P}({\rm I\!N})/{\cal I} \to {\cal P}({\rm I\!N})/{\cal J}$ is given by a continuous map $\Phi\sb*$, a lifting of $\Phi,$ such that the diagram$$\eqalign{{\cal P}(\rm I\!N)\quad &{\buildrel{\Phi\sb*}\over{\longrightarrow}}\quad{\cal P}\ (\rm I\!N)\cr\qquad\pi\sb{\cal I}\quad&\quad\qquad\pi\sb{\cal J}\cr{\cal P}(\rm I\!N)/{\cal I}\quad &{\buildrel{\Phi}\over{\longrightarrow}}\quad{\cal P}(\rm I\!N)/{\cal J}\cr}$$commutes. In Chapter II we study a conjecture of S. Todorcevic about a more canonical form of Baire-measurable liftings when the homomorphism connects quotients over analytic P-ideals. This conjecture would imply that, in this class of ideals, the Baire-embeddability between quotients reduces to the much simpler finite-to-one reduction of the corresponding ideals. One of the main results of this Thesis is a positive answer to this conjecture for a large class of nonpathological ideals (S6). However, we also prove that the conjecture in general has a negative answer. In Chapter III, we consider arbitrary connecting maps but work under an alternative to CH. The right alternative is a two-dimensional Perfect Set Property known as the Open Coloring Axiom (OCA). Most efforts of Chapter III are put in the proof that under OCA every homomorphism $\Phi{:}{\cal P}({\rm I\!N})/\rm Fin \to {\cal P}({\rm I\!N})/{\cal I}$ between quotients over analytic P-ideals is an amalgamation of two homomorphisms, $\Phi\sb1$ and $\Phi\sb2$:$$\eqalign{&{\cal P}(A)/{\cal I}\cr\Phi\sb1\quad\qquad&\qquad\qquad\qquad\rm id\cr {\cal P}(\rm I\!N)/\rm Fin\qquad\qquad&\quad{\buildrel{\Phi}\over{\longrightarrow}}\qquad\qquad {\cal P}(\rm I\!N)/{\cal I},\cr\phi\sb2\quad\qquad&\qquad\qquad\qquad\rm id\cr &{\cal P}(\rm I\!N\\ A)/{\cal I}\cr}$$so that $\Phi\sb1$ has a Baire-measurable lifting, while the kernel of $\Phi\sb2$ is rather large. This implies that under OCA many results from Chapter II apply to arbitrary homomorphisms. The key to our approach to Todorcevic's conjecture is a study of $\varepsilon$-approximate homomorphisms of finite Boolean algebras. This is given in Chapter I. We use this to analyze arbitrary Boolean algebras with submeasures and $\varepsilon$-approximate homomorphisms between them. The main result is: Approximate homomorphisms with respect to nonpathological submeasures are, in some sense, stable: every approximate homomorphism can be approximated by a real homomorphism up to a constant error. In Chapter IV we produce a long coherent sequence of Hausdorff gaps in every analytic quotient. Chapter V is devoted to study of the following general program: Investigate the class of all partially ordered sets which can be isomorphically embedded into ${\cal P}(\rm I\!N)/Fin.$ We describe, applying the method of Forcing, a general setting for analyzing this and related kind of questions. In particular, we use our methods to test the optimality of closure properties of the class of partially ordered sets embedding into ${\cal P}\rm (I\!N)/Fin.$ (Abstract shortened by UMI.)