Frechet Borel Ideals with Borel orthogonal

Abstract

We study Borel ideals $I$ on $\mathbb{N}$ with the Fr\'echet property such its orthogonal $I^\perp$ is also Borel (where $A\in I^\perp$ iff $A\cap B$ is finite for all $B\in I$ and $I$ is Fr\'echet if $I=I^{\perp\perp}$). Let $\mathcal{B}$ be the smallest collection of ideals on ${\mathbb{N}}$ containing the ideal of finite sets and closed under countable direct sums and orthogonal. All ideals in $\mathcal{B}$ are Fr\'echet, Borel and have Borel orthogonal. We show that $\mathcal{B}$ has exactly $\aleph_1$ non isomorphic members. The family $\mathcal{B}$ can be characterized as the collection of all Borel ideals which are isomorphic to an ideal of the form $I_{wf}\up A$, where $I_{wf}$ is the ideal on $\mathbb{N}^{<\omega}$ generated by the wellfounded trees. Also, we show that $A\subseteq \mathbb{Q}$ is scattered iff $WO({\mathbb{Q}})\up A$ is isomorphic to an ideal in $\mathcal{B}$, where $WO(\mathbb{Q})$ is the ideal of well founded subset of $\mathbb{Q}$.