On the structure of Borel ideals in-between the ideals $\mathcal{E}\mathcal{D}$ and $\mathrm{Fin}\otimes \mathrm{Fin}$ in the Katětov order

Abstract

For a family $\mathcal{F}\subseteq {\omega }^{\omega }$ we define the ideal $\mathcal{I}(\mathcal{F})$ on $\omega \times \omega$ to be the ideal generated by the family $\{A\subseteq \omega \times \omega :\exists f\in \mathcal{F}{\forall }^{\infty }n(|\{k:(n,k)\in A\}|\le f(n))\}.$ Using ideals of the form $\mathcal{I}(\mathcal{F})$, we show that the structure of Borel ideals in-between two well known Borel ideals $\mathcal{E}\mathcal{D}=\{A\subseteq \omega \times \omega :\exists m{\forall }^{\infty }n(|\{k:(n,k)\in A\}|<m))\}$ and $\mathrm{Fin}\otimes \mathrm{Fin}=\{A\subseteq \omega \times \omega :{\forall }^{\infty }n(|\{k:(n,k)\in A\}|<{\aleph }_0))\}$ in the Katětov order is fairly complicated. Namely, there is a copy of $\mathcal{P}(\omega )/\mathrm{Fin}$ in-between $\mathcal{E}\mathcal{D}$ and $\mathrm{Fin}\otimes \mathrm{Fin}$, and consequently there are increasing and decreasing chains of length $\mathfrak{b}$ and antichains of size $\mathfrak{c}$.

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References in corpus (4)

• Rothberger gaps in fragmented ideals
• Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters
• Ideal weak QN-spaces
• Baire theorem for ideals of sets