Combinatorics of filters and ideals. (English) Zbl 1239.03030

Babinkostova, L. (ed.) et al., Set theory and its applications. Annual Boise extravaganza in set theory, Boise, ID, USA, 1995–2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4812-8/pbk). Contemporary Mathematics 533, 29-69 (2011).
The paper under review is mainly a survey which presents key aspects of the combinatorics associated to ideals and filters over a countable set. The aim is to have a whole picture of the structure of ideals and filters. The principal tool in exploring those families is the so-called Katětov order: If \(\mathcal{I}\) and \(\mathcal{J}\) are two ideals over \(\omega\), the natural numbers, then we say that \(\mathcal{I}\leq_K\mathcal{J}\) if there is a function \(f:\omega\rightarrow\omega\) such that \(f^{-1}[I]\in\mathcal{J}\) for each \(I\in\mathcal{I}\). Also used are the Tukey order and the Rudin-Keisler order, which are somehow better known.
Many of the results in the paper are about the cardinal characteristics of ideals (and filters); particular attention is paid to special families of ideals like Borel or analytic \(P\)-ideals. A powerful tool used to analyse those comes from a theorem of S. Solecki saying that an ideal \(\mathcal{I}\) on \(\omega\) is an analytic \(P\)-ideal if and only if there is a lower semicontinuous submeasure \(\varphi:\mathcal{P}(\omega)\rightarrow[0,+\infty]\) such that \(\mathcal{I}=\{A\subseteq\omega:\lim_{n\rightarrow\infty}\varphi(A\setminus n)=0\}\). Another important topic discussed in the paper is on the destruction of ideals by forcings of the type \(\mathbb{P}_{I}=\mathrm{Borel}(X)/I\), where \(X\) is a Polish space and \(I\) is a \(\sigma\)-ideal on \(X\). A nice result announced is for the case when \(I\) is a \(\sigma\)-ideal on \(\omega^\omega\) and \(\mathbb{P}_I\) has continuous reading of names besides being continuously homogeneous, and \(\mathcal{J}\) is an ideal on \(\omega\), then \(\mathbb{P}_I\) destroys \(\mathcal{J}\) (that is, introduces an infinite subset of \(\omega\) which is almost disjoint from every element of \(\mathcal{J}\)) if an only if \(\mathcal{J}\leq_K \mathrm{tr}(I)\), where \(\mathrm{tr}(I)=\{a\in[\omega]^{<\omega}:\{r\in\omega^\omega:(\exists^\infty n\in\omega)(r\upharpoonright n\in a)\}\in I\}\) is the trace ideal of \(I\).
There are several other topics in the paper, like the quest for critical ideals in the Katětov order, characterization of known ideal properties, the so-called comparison game, which, for example, can be used to characterise ideals \(\mathcal{I}=\bigcap_{ n\in\omega}\mathcal{F}_n\), where \(\langle\mathcal{F}_n:n\in\omega\rangle\) is a sequence of hereditarily \(F_\sigma\) sets; the quotient algebras \(\mathcal{P}(\omega)/\mathcal{I}\) are also studied; etc.
Many nice questions are posed, the paper is written in an amenable way, shows important ideas of the theory and, although it presents only few proofs, those that are given show typical ideas. The paper is fundamental for those who want to learn the basics of the theory.
For the entire collection see [Zbl 1205.03004].


03E15 Descriptive set theory
03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results