Inductive limits of ideals

Abstract

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal $\mathcal{I}$ ($\text{rk}(\mathcal{I})$) as minimal ordinal $\alpha <{\omega }_1$ such that there is $\mathcal{S}\in {\mathbf{\Sigma }}_{1+\alpha }^0$ with $\mathcal{I}\subseteq \mathcal{S}$ and ${\mathcal{I}}^{\star }\cap \mathcal{S}=\varnothing$, where ${\mathcal{I}}^{\star }$ is the filter dual to the ideal $\mathcal{I}$ (actually, the authors use the dual notion of filters instead of ideals). Moreover, they introduced ideals ${\text{Fin}}_{\alpha }$, for all $\alpha <{\omega }_1$, and conjectured that $\text{rk}(\mathcal{I})\ge \alpha$ if and only if $\mathcal{I}$ contains an isomorphic copy of ${\text{Fin}}_{\alpha }$ (${\text{Fin}}_{\alpha }⊑\mathcal{I}$). To define ${\text{Fin}}_{\alpha }$ in the case of limit ordinals $0<\alpha <{\omega }_1$, G. Debs and J. Saint Raymond introduced inductive limits of ideals. We show that the above conjecture is false in the case of $\alpha =\omega$ by constructing an ideal ${\text{Fin}}_{\omega }^{\prime }$ of rank $\omega$ such that ${\text{Fin}}_{\omega }/⊑{\text{Fin}}_{\omega }^{\prime }$. However, we show that ${\text{Fin}}_{\omega }^{\prime }⊑\mathcal{I}$ is equivalent to ${\forall }_{n\in \omega }{\text{Fin}}_n⊑\mathcal{I}$. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.

References in corpus (1)

• Homogeneous ideals on countable sets